Generalized Locally Toeplitz sequences: a review and anextension

Carlo Garonia,b and Stefano Serra-Capizzanob,c

aUniversity of Roma ‘Tor Vergata’, Department of Mathematics, Via della Ricerca Scientifica, 00133 Roma, Italy.Email: garoni@mat.uniroma2.it.

bUniversity of Insubria, Department of Science and High Technology, Via Valleggio 11, 22100 Como, Italy.Email: carlo.garoni@uninsubria.it, stefano.serrac@uninsubria.it.

cUppsala University, Department of Information Technology, Division of Scientific Computing,Box 337, SE-751 05 Uppsala, Sweden. Email: stefano.serra@it.uu.se.

May 18, 2015

Abstract

We review the theory of Generalized Locally Toeplitz (GLT) sequences, hereinafter called ‘the GLT theory’,which goes back to the pioneering work by Tilli on Locally Toeplitz (LT) sequences and was developed bythe second author during the last decade: every GLT sequence has a measurable symbol; the singular valuedistrbution of any GLT sequence is identified by the symbol (also the eigenvalue distribution if the sequence ismade by Hermitian matrices); the GLT sequences form an algebra, closed under linear combinations, (pseudo)-inverse if the symbol vanishes in a set of zero measure, product and the symbol obeys to the same algebraicmanipulations.

As already proved in several contexts, this theory is a powerful tool for computing/analyzing the asymp-totic spectral distribution of the discretization matrices arising from the numerical approximation of continuousproblems, such as Integral Equations and, especially, Partial Differential Equations, including variable coeffi-cients, irregular domains, different approximation schemes such as Finite Differences, Finite Elements, Collo-cation/Galerkin Isogeometric Analysis etc. However, in this review we are not concerned with the applicativeinterest of the GLT theory, for which we limit to refer the reader to the numerous applications available in theliterature. On the contrary, we focus on the theoretical foundations. We propose slight (but relevant) modifi-cations of the original definitions, which allow us to enlarge the applicability of the GLT theory. In particular,we remove a certain ‘technical’ hypothesis concerning the Riemann-integrability of the so-called ‘weight func-tions’, which appeared in the statement of many spectral distribution and algebraic results for GLT sequences.With the new definitions, we introduce new technical and useful results and we provide a formal proof of the factthat sequences formed by multilevel diagonal sampling matrices, as well as multilevel Toeplitz sequences, fallin the class of LT sequences; the latter results were mentioned in previous papers, but no direct proof was givenespecially regarding the case of multilevel diagonal sampling matrix-sequences. As a final step, we extend theGLT theory: we first prove an approximation result, which is particularly useful to show that a given sequenceof matrices is a GLT sequence; by using this result, we provide a new and easier proof of the fact that A−1

n nis a GLT sequence with symbol κ−1 whenever Ann is a GLT sequence of invertible matrices with symbol κand κ 6= 0 almost everywhere; finally, using again the approximation result, we prove that f(An)n is a GLTsequence with symbol f(κ), as long as f : R → R is continuous and Ann is a GLT sequence of Hermitianmatrices with symbol κ. This latter theoretical property has important implications, e.g. in proving that thegeometric means of GLT sequences are still GLT, so obtaining for free that the spectral distribution of the meanis just the geometric mean of the symbols.

1

Contents1 Introduction 2

1.1 GLT and approximated PDEs: basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1 The model of a rod with variable section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 The linear elasticity in saddle-point form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3 Some intermediate remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Technical introduction: new vs previous approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Mathematical background 112.1 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Multi-index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Matrix-sequences and multilevel diagonal sampling matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.3 Separable functions and multivariate trigonometric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Convergence in measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Preliminaries on Linear Algebra and Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Tensor products and direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Singular value and eigenvalue distributions of matrix-sequences: the symbol . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Multilevel Toeplitz matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Approximating classes of sequences (a.c.s.) 233.1 The a.c.s. machinery as a tool for computing singular value and eigenvalue distributions . . . . . . . . . . . . . . . . . . 233.2 The a.c.s. algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Some criterions to identify a.c.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 LT and sLT sequences 374.1 The Locally Toeplitz operator LTm

n (a, f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.1 Properties of LTm

n (a, f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 LT and sLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2.1 Definition and basic examples: Toeplitz and diagonal sampling matrix-sequences . . . . . . . . . . . . . . . . . 444.2.2 Properties of LT and sLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 GLT sequences 575.1 Definition and characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2 Approximation results for GLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3 Singular value and eigenvalue distribution of GLT sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.4 The GLT algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.4.1 The algebra generated by Toeplitz sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6 Conclusions: GLT as a Generalized Fourier Analysis 696.1 The algebra generated by diagonal sampling matrix-sequences, by zero distributed sequences, and by Toeplitz sequences:

a tool for computing the symbol of PDE approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2 Future work: reduced GLT, block GLT, and an automatic (symbolic) calculus of the symbol . . . . . . . . . . . . . . . . 70

1 IntroductionWe review, improve, and extend the theory of Generalized Locally Toeplitz (GLT) sequences, hereinafter called‘the GLT theory’, which stems from Tilli’s work on Locally Toeplitz (LT) sequences [54] and from the theory ofclassical Toeplitz operators [8, 31, 53], and was developed by the second author in [45, 46].

As already proved in several contexts, this theory is a powerful tool for computing/analyzing the asymptoticspectral distribution of the discretization matrices arising from the numerical approximation of continuous prob-lems, such as Integral Equations (IEs) and, especially, Partial Differential Equations (PDEs). Let us explain thispoint in more detail. When discretizing a linear PDE by means of a linear numerical method, the actual computa-tion of the numerical solution un reduces to solving a linear system Anun = bn. The size dn of this linear systemincreases when the discretization parameter n tends to infinity, so that we are left with a sequence of discretization

2

matrices Ann with increasing size. What is often verified in practice is that Ann enjoys an asymptotic spectraldistribution in the Weyl sense, which is somehow related to the spectrum of the differential operator associatedwith the considered PDE. More precisely, it often happens that, for a large set of test functions F (usually, for allcontinuous functions F with bounded support), the following limit relations hold (see Subsection 2.4):

limn→∞

1

dn

dn∑j=1

F (λj(An)) =1

µδ(D)

∫D

F (κ(y))dy, (1.1)

limn→∞

1

dn

dn∑j=1

F (σj(An)) =1

µδ(D)

∫D

F (|κ(y)|)dy, (1.2)

where λj(An), j = 1, . . . , dn, are the eigenvalues of An, σj(An), j = 1, . . . , dn, are the singular values of An,µδ is the Lebesgue measure in Rδ, µδ(D) ∈ (0,∞), and κ : D ⊂ Rδ → C is a measurable function. If relations(1.1) hold, then κ is called the spectral symbol of the sequence of matrices Ann and we write Ann ∼λ (κ,D);in such a case, κ provides a ‘compact’ description of the asymptotic spectral distribution of Ann. If relations(1.2) hold, then κ is called the singular value symbol of the sequence of matrices Ann; in this setting, thefunction |κ| provides a ‘compact’ description of the asymptotic singular value distribution of Ann and we writeAnn ∼σ (κ,D).

Often, for the sake of of brevity and when the information is clear from the context, we will write Ann ∼σ κin place of Ann ∼σ (κ,D) and Ann ∼λ κ in place of Ann ∼λ (κ,D), respectively.

The informal, but important, meaning of (1.1) can be given as follows: if κ is continuous, then a suitable order-ing of the eigenvalues λj(An), in correspondence with a equispaced gridding on D, reconstructs approximatelythe surface t → κ(t). On the other hand, the informal meaning of (1.2) can be summarized in perfect analogy:if κ is continuous, then a suitable ordering of the singular values σj(An), in correspondence with a equispacedgridding on D, reconstructs approximately the surface t→ |κ(t)|.

The GLT theory, in combination with the result of [29, Theorem 3.4] concerning the spectral distribution ofperturbed sequences of matrices, is one of the most powerful and successful tools for computing the spectralsymbol κ for wide classes of approximated IEs and PDEs. Indeed, the sequence Ann is often a GLT sequence.We refer the reader to [45, 46] for applications of the GLT theory in the context of Finite Difference discretizationsof PDEs; to [6, 46, 49] for the Finite Element and Finite Difference collocation settings; to [1, 38] for the GLTaaproach to IE matrix-sequences; and to [15, 24, 25] for recent applications to the case of B-spline IsogeometricAnalysis (IgA) approximations of PDEs, both in the collocation and Galerkin frameworks. We remind at this pointthat IgA is a modern paradigm introduced in [12, 33] for analyzing problems governed by PDEs, where the mainidea is to combine the Computed Aided Geometric Design (CAGD), widely used by engineers for modelling, andthe numerical methods for the solution of PDEs: the core of the success of the quoted approach is that the functionsused for describing the numerical solution of the considered PDE are exactly the same as those used for modellingthe geometric objects, so resulting in a saving of more than the 80% of the CPU time usually employed in thetranslation between two different languages (e.g. either between Finite Elements and CAGD or between FiniteDifferences and CAGD).

Now we summarize five main theoretical features of the GLT class of matrix-sequences and then, in Subsection1.1, we show the reason why they have high relevance to the approximation of PDEs, by using some elementaryexamples.

GLT1 Each GLT sequence Ann has a symbol κ = κ(x,θ), x = (x1, . . . , xd) ∈ [0, 1]d physical variables,θ = (θ1, . . . , θd) ∈ (−π, π)d Fourier variables, δ = 2d, and relation (1.2) holds, that is κ is the singularvalue symbol of the sequence of matrices Ann. If, in addition, the sequence is Hermitian then (1.1) holdsand κ is also the spectral symbol of the sequence of matrices Ann.

3

GLT2 The set of GLT sequences form a ∗-algebra that is close under linear combinations, conjugation, products,(pseudo-) inversion, whenever the sequence is invertible. In the case of a sequence having a singular valuedistribution κ, as in the the GLT setting thanks item GLT1, a sequence is invertible if κ vanishes, at most,in a set of zero Lebesgue measure. Hence, the sequence obtained via algebraic operations on a finite set ofinput GLT sequences is still a GLT sequence and its symbol is obtained by the same algebraic manipulationson the corresponding symbols of the input GLT sequences.

GLT3 Every multilevel Toeplitz sequence generated by a L1 function f is a GLT sequence and its symbol isκ(x,θ) ≡ f(θ), under the conditions given in item GLT1.

GLT4 Every sequence formed by multilevel diagonal sampling matrices related to the Riemann integrable func-tion a is a GLT sequence and its symbol is κ(x,θ) ≡ a(x), with the eigenvalue distribution formula holdingalso in the non Hermitian case.

GLT5 Xn which is distributed as the zero function in the singular value sense, i.e. Xn ∼σ (0, D) accordingto (1.2), is a GLT sequence with symbol κ ≡ 0.

1.1 GLT and approximated PDEs: basic examplesWe first consider a scalar 1D boundary value problem and then a simple vector problem, that is, the linear elasticityin saddle-point form, both in 1D and 2D. The idea is to show that items GLT1-GLT5 are a tool for giving anautomatic way of deducing the asymptotic spectral features of approximated PDEs, just using (multilevel) diagonalsampling matrix-sequences, (multilevel) Toeplitz matrix-sequences, and zero distributed sequences as buildingblocks.

1.1.1 The model of a rod with variable section

Let us consider the simple differential operator L(u) = −u(′′) and the corresponding boundary value problemL(u) = g on (0, 1) with Dirichlet boundary conditions. If we use centered equispaced standard Finite Differencesfor approximating the previous equation, then we obtain a linear system whose coefficient matrix has the form

∆n =

2 −1

−1. . . . . .. . . . . . −1−1 2

.

Such a matrix is a Toeplitz matrix according to the notation in (2.20)–(2.21), is Hermitian (in fact real symmetric),and its generating function, directly reconstructed from the entries of the matrix is f(θ) = 2−2 cos(θ); see Section2.5 for the general definitions. Therefore

∆n = Tn(f)

and according to GLT3 the sequence ∆n is a GLT sequence with symbol κ(x, θ) = f(θ) and finally by itemGLT1, the eigenvalues of ∆n behave as a uniform sampling of f(θ). However this is known also algebraically and

indeed the eigenvalues of ∆n are exactly given by f(θ

(n)i

)= 2− 2 cos

(θ

(n)i

), θ(n)

i =iπ

n+ 1.

Now let us consider the variable coefficient version of L(u) = −u(′′) in divergence form that is La(u) =

−(a(x)u

′)′ , a(x) bounded and positive on [0, 1], which models the behavior of a rod with a variable section. By

4

employing the same equispaced Finite Differences, setting aj = a((j − 1/2)h), dj = aj−1 + aj , we find that thecorresponding coefficient matrix is given by

∆(2)n (a) =

d1 −a2

−a2. . . . . .. . . . . . −an

−an dn

.

Even if ∆(2)n (1) = ∆n = Tn(f), for nonconstant a the Toeplitz character seems to be completely lost. In fact, we

find it again ‘in a approximated sense’ and at ‘local scale’ as explained below. First of all we have a nice ‘dyadicrepresentation’ i.e.

∆(2)n (a) =

n+1∑j=1

ajΨj =n+1∑j=1

ajdiag(0, . . . , 0,Θ, 0, . . . , 0),

Ψi =O⊕Θ⊕O, Θ =

(1 −1−1 1

)=

(1−1

)(1 −1

).

From the latter it is easy to deduce the inequalities

[min a]Tn(f) ≤ ∆(2)n (a) ≤ [max a]Tn(f),

which implies that the minimal eigenvalue of ∆(2)n (a) is positive and is larger than [min a]λmin(Tn(f)) and the

maximal eigenvalue is bounded from above by [max a]λmax(Tn(f)) ≤ max a(x)f(θ). These bounds show that thenew function a(x)f(θ) has an important role, as deduced in the next derivations using the GLT theory. In fact,setting Dn(a) the diagonal sampling matrix related to the function a that is

Dn(a) =

a(h)

a(2h). . .

a(nh)

, h =1

n,

and defining En = ∆(2)n (a) − Dn(a)Tn(f), Fn = ∆

(2)n (a) − D

1/2n (a)Tn(f)D

1/2n (a), we can plainly see that the

spectral norms ‖En‖, ‖Fn‖ are infinitesimal as n tends to infinity, under the assumption the a is continuous. There-fore

1. En ∼σ (0, D), Fn ∼σ (0, D) for everyD and hence by item GLT5 both En, Fn are GLT sequenceswith symbol κ ≡ 0;

2. since D1/2n (a) = Dn(

√a), by item GLT4 , D1/2

n (a), Dn(a) are GLT sequences with symbols κ(x, θ) =√a(x), κ(x, θ) = a(x), respectively.

In conclusion, since Fn is Hermitian and

∆n(a) = D1/2n (a)Tn(f)D1/2

n (a) + Fn,

also ∆n(a) is a Hermitian GLT sequence with symbol κ(x, θ) = a(x)f(θ) by item GLT2, which states thatGLT sequences form a ∗-algebra. Thus, as a consequence of item GLT1, we deduce that κ(x, θ) = a(x)f(θ) isboth the spectral and the singular value symbol according to (1.1) and (1.2). In other words, for n = p2 large

5

enough, the eigenvalues of ∆n(a) are given approximately by the values a(xi)(2 − 2 cos(θj)), i, j = 1, . . . , p,xi = i/p, θj = jπ/p.

Along the same lines, we may consider the same problem with non-equaspaced grid points and the same FiniteDifference approximation. In fact, if the internal nodes are given as tj = g(jh), j = 1, . . . , n, h = 1

n+1, and g such

that g([0, 1]) = [0, 1], diffeomorfism, then setting

φ(a, g) =a(g)

(g′)2,

we find that the coefficient matrix ∆n(a, g) has the form

∆n(a, g) = ∆n(φ(a, g)) + En,

∆n(φ(a, g)) = Dn(φ(a, g))Tn(f) + Fn,

with En ∼σ (0, D), Fn ∼σ (0, D) for every D. Therefore, by following the same reasoning as before andusing items GLT1-GLT5, we obtain that ∆n(a, g) is a GLT sequence with symbol κ(x, θ) = φ(a, g)(x)f(θ):thus κ(x, θ) = φ(a, g)(x)f(θ) is both the spectral and the singular value symbol of ∆n(a, g), according to (1.1)and (1.2). We observe that the map g is an example of the Geometric Maps used in the IgA, Finite Elements with’graded’ grids, etc, and that the above machinery has a natural extension in the multidimensional.

In this direction, as an example, we just mention that the same Finte Difference approximation of the operator

La(u) = −d∑

i,j=1

∂

∂xi

(ai,j(x)

∂u

∂xj

), a(x) = (ai,j(x))di,j=1 ,

leads to a matrix ∆n(a) that can be seen, up to squences distributed as the zero function, as∑d

i,j=1Dn(ai,j)∆n,i,j ,where, n = (n1, . . . , nd) is a multi-index and each ∆n,i,j is a proper multilevel Toeplitz matrix generated bymulti-variate trigonometric polynomial pi,j (see Subsection 2.5): again. by items GLT1-GLT5, this is enough toconclude that ∆n(a) is a GLT sequence with symbol κ(x,θ) =

∑di,j=1 .ai,j(x)pi,j(θ).

1.1.2 The linear elasticity in saddle-point form

In this section we introduce the with the help ofWe present two simplified examples, both in connection with the linear elasticity in saddle-point form: in

the first case we consider one dimension and linear Finite Elements, while in the the second we consider twodimensions and standard Finite Differences. We use again the GLT technique for deriving spectral symbol of therelated matrices and of the corresponding Schur complement: we recall that Schur complement, key tool for thenumerical treatment of the underlying linear systems, implies the inversion of a block, but this is no problem sincethe GLT sequences are stable also under (pseudo)-inversion, as long as the symbol of the inverted block vanishesin a set of zero Lebesgue measure.

The analysis of the elasticity problem, using the GLT tools and taking into account stable approximationtechniques, is given in [21].

Consider the coupled system of scalar equations−(ψ(x)u′)′ + v′ = g1(x),

u′ − ρv = g2(x),(1.3)

with Dirichlet boundary conditions. Here ρ > 0 and the function ψ(x) is positive and continuous on the domainΩ = [0, 1].

6

Assume first that ψ(x) = κ0 is constant in Ω. The use of linear Finite Element basis functions on a uniformmesh with a step size h and a proper scaling leads to a linear system of equations with a coefficient matrix thatadmits the following structure

A =

[K BT

B −ρM

].

Here, the blocks K,B,M are square of size n and are given below

K = κ0 ·

2 −1−1 2 −1

−1. . . . . .. . . . . . −1−1 2

, M =h2

6

4 11 4 1

1. . . . . .. . . . . . 1

1 4

, (1.4)

B = h

1−1 1

−1. . .. . . . . .−1 1

. (1.5)

Following the definitions in (2.20)-(2.21) as in Section 2.5, we find

K = κ0 Tn(2− 2 cos(θ)), B = hTn(1− eiθ),

BT = hTn(1− e−iθ), M = h2

3Tn(2 + cos(θ)),

(1.6)

where the negative Schur complement of A is expressed as

Sn = ρM +BTK−1B. (1.7)

Since κ0 > 0, the matrix K is invertible. Considering (1.7), we deduce that Sn is given in terms of Toeplitzstructures

S =ρ

3Tn(2 + cos(θ)) +

1

κ0

Tn(1− e−iθ)T−1n (2− 2 cos(θ))Tn(1− eiθ). (1.8)

According to GLT3, the sequences Tn(2 + cos(θ)), Tn(1 − eiθ), Tn(1 − e−iθ) and Tn(2− 2 cos(θ)) areGLT sequences with symbols 2 + cos(θ), 1− eiθ, 1− e−iθ, 2− 2 cos(θ), respectively. As a consequence, by GLT2,taking into account that 2 − 2 cos(θ) vanishes in a set of zero Lebesgue measure, we obtain that the inverse ofTn(2− 2 cos(θ)) is still in the GLT algebra and therefore also Snn is a GLT sequence generated by the symbol

κS(θ) =ρ

3(2 + cos(θ)) +

1

κ0

(1− e−iθ)1

2− 2 cos(θ)(1− eiθ) =

ρ

3(2 + cos(θ)) +

1

κ0

. (1.9)

Since Sn is Hermitian independently of its size, according to GLT1, we deduce that (1.1) holds for Snn, i.e.,Snn ∼λ (κS , (−π, π), where κS is an even trigonometric polynomial. Figure 1 shows the agreement betweenthe asymptotic forecast and the eigenvalues of Sn for a couple of values of the parameters κ0 and ρ, where boththe evaluations of κS over θ(n)

j = πjn+1

, j = 1, . . . , n. In the plots, the eigenvalues of Sn are sorted in an increasingorder.

Next we consider equation (1.3) with a non-constant diffusivity coefficient ψ(x). The resulting matrix K is nolonger Toeplitz but, as shown in [8] and in analogy with the Finite Difference case reported in Subsection 1.1.1, the related sequence Knn belongs to the GLT class with symbol ψ(x)(2 − 2 cos(θ)). Therefore, following

7

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

3.5

eig(Tn(f)), ρ = 0.2

eig(S), ρ = 0.2eig(T

n(f)), ρ = 1

eig(S), ρ = 1

0 50 100 150 2000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

eig(Tn(f)), ρ = 0.2

eig(S), ρ = 0.2eig(T

n(f)), ρ = 1

eig(S), ρ = 1

Figure 1: 1D Elasticity Problem: Spectrum of Sn vs sampling of its symbol κS , const. coeff. κ0 = 0.4 and κ0 = 1

Figure 2: 1D Elasticity Problem: Spectrum of S vs sampling of its symbol κS , variable coefficient ψ(x) = 1 + x.

verbatim the reasoning for the case of ψ(x) = κ0, we deduce that the sequence of Schur complements forms aGLT sequence with the symbol

κS(x, θ) =ρ

3(2 + cos(θ)) +

1

ψ(x)(1− e−iθ)

1

ψ(x)(2− 2 cos(θ))(1− eiθ) (1.10)

=ρ

3(2 + cos(θ)) +

1

ψ(x).

The latter is illustrated in Figure 2, showing the agreement between the asymptotic forecast and the eigenvaluesof Sn for a pair of choices of ρ and ψ(x), where both the evaluations of κS over a n-sized uniform gridding over[0, 1]× [0, π] of size n and the eigenvalues of S have been sorted in an increasing order.

We formulate next a two-dimensional coupled test system of two PDEs and two unknowns, the first of which,referred to as the displacements is a vector with two components.

Consider an elasticity-like problem in saddle point form, defined in Ω = [0, 1]2,

A =

[K BT

B −ρM

]displacementspressure,

where K and M are symmetric and positive definite.

8

Under the so-called separate displacement ordering (SDO) of the components of the displacements, K itselfattains a two-by-two block structure,

K =

[K11 K12

K21 K22

]displacements in xdisplacements in y.

The SDO ordering of the displacements induces a corresponding block structure in the block B, which we denoteas B =

[B1 B2

]. Thus, we have

A =

K11 K12 BT1

K21 K22 BT2

B1 B2 −ρM

. (1.11)

Further, we assume that, respectively,K11, K22 approximate two anisotropic Laplacians of the form−(

2 ∂2

∂x2+ ∂2

∂y2

),

−(∂2

∂x2+ 2 ∂2

∂y2

),K12, K21 approximate the operators− ∂2

∂y∂x,− ∂2

∂x∂yandBT

1 , BT2 approximate the operators ∂

∂x, ∂∂y

.M is the mass matrix, approximating the identity operator.

We use standard Finite Differences on a square mesh, even though this approximation does not possess thestability properties, required for mixed problems. The blocks of A are described as follows:

K11 = 2Tn(2− 2 cos(θ1))⊗ In + In ⊗ Tn(2− 2 cos(θ2)),

K12 = Tn(1− e−iθ1)⊗ Tn(1− e−iθ2),

K21 = KT12,

K22 = Tn(2− 2 cos(θ1))⊗ In + 2In ⊗ Tn(2− 2 cos(θ2)),

B1 = hTn(1− e−iθ1)⊗ In,B2 = h In ⊗ Tn(1− e−iθ2),

M = h2 In.

Here⊗ denotes the Kronecker product (cf. [7]). We notice that the Kronecker product induces a two-level Toeplitzstructure and in fact every matrix is Toeplitz, where each‘’entry’ along the diagonal is a standard unilevel Toeplitzmatrix, see Section 2.5. Hence, using the two- level notation introduced in [26, 56], we construct the two- variategenerating functions as in Section 2.5, see (2.20)-(2.21), associated with each block. More precisely,

K11 = Tn((6− 4 cos(θ1)− 2 cos(θ2))), (1.12)K12 = Tn((1− e−iθ1)(1− e−iθ2)), (1.13)K21 = Tn((1− eiθ1)(1− eiθ2)), (1.14)K22 = Tn((6− 2 cos(θ1)− 4 cos(θ2))), (1.15)B1 = hTn(1− e−iθ1), (1.16)B2 = hTn(1− e−iθ2), (1.17)M = h2 Tn(1). (1.18)

Consider next the negative Schur complement ofA, S = ρM+BK−1BT . Given the rich block structure of thematrixA, the formal expression of the Schur complement involves inversion of the spd blockK and multiplicationby rectangular blocks. Since we want to use the symbols (1.12)–(1.18) of the related sub-blocks, we utilize thefollowing exact block-factorization of K and of its inverse:

K =

[K11 K12

K21 K22

]=

[I

K21K−111 I

] [K11

SK

] [I K−1

11 K12

I

], (1.19)

9

where SK = K22 −K21K−111 K12. Since the positive definite character of SK is guaranteed by the positive definite

character of K, we find

K−1 =

[I −K−1

11 K12

I

] [K−1

11

S−1K

] [I

−K21K−111 I

]. (1.20)

CLearly, the latter factorization holds for any nonsingular matrix K. Therefore, as it is well known, the explicitformal expression of S is given by

S = ρM +[B1 B2

]K−1

[BT

1

BT2

]= ρM +B1K

−111 B

T1 + (B2 −B1K

−111 K12)S−1

K (B2 −B1K−111 K12)T . (1.21)

As expected, constructing the symbol in 2D is by far more complicated than that in 1D. Nevertheless, the expressionof S in (1.21) can be seen as a‘rational noncommutative’ formula involving the blocks in (1.12)–(1.18), whichare two-level Toeplitz structures with trigonometric polynomial symbols. As a consequence, in view of GLT1,GLT2, GLT3, exactly as done in the one-dimensional example (1.3), we deduce that S is a GLT sequence withS ∼λ κS , since S is Hermitian independently of its size, and κS is a rational function of the symbols given in(1.12)–(1.18). Interestingly enough, setting δ(θ) = 2− 2 cos(θ) the symbol of the unilevel Laplacian and makingtedious algebraic manipulations, we find

κS = ρ+ µq1(δ(θ1), δ(θ2))

q2(δ(θ1), δ(θ2)),

where q1, q2 are two nonnegative homogeneous polynomials of degree two. Therefore, ρ ≤ f(θ1, θ2) ≤ ρ+φ withφ being the maximum of q1/q2.

Following the same procedure as in the one-dimensional setting we can treat the case of variable coefficients.

1.1.3 Some intermediate remarks

There are two further features of the GLT class of matrices that are a consequence of items GLT1-GLT5, as brieflysketched in Subsection 1.1 and that represent a powerful and general tool for identifying the spectral behavior ofmatrix-sequences, coming from the approximation by local methods of PDEs and IEs. We summarize them in thefollowing.

GLT6 The approximation of PDEs with non-constant coefficients, general domains, nonuniform gridding by lo-cal methods (Finite Difference, Finite Elements, IgA etc), under very mild assumptions leads also to GLTsequences (see [54, 45, 46] for the case of Finite Differences, [8, 26] for the Finite Elements setting, and[15, 23] for the case of IgA approximations).

GLT7 We encounter GLT structures when dealing with preconditioning of iterative and semi-iterative solvers(see e.g. [4]), when the dealing with the stabilty analysis of implicit numerical methods for PDEs (seee.g. [34, 35]), when considering multigrid methods applied to approximations of PDEs and IEs by localmethods (see [1, 2, 3, 20, 18] and references therein). Moreover, the symbol includes information aboutthe coefficients and the domain of the PDE, as well as information on the discretization schemes for thederivatives including the used meshes, which have to be described, at least asymptotically, as a map of areference equispaced mesh (see [36, 54, 48, 39] for the one-dimensional setting and [42, 40, 49, 45, 46, 8,13, 14, 26] for the two-dimensional and multi-dimensional settings. Furthermore, also in presence of non-dominating advection terms the distribution result for the eigenvalues can be recovered, thanks to ad hocresults in [29, 27], heavily based on the majorization theory well explained in the remarkable book [7].

10

1.2 Technical introduction: new vs previous approachThe focus of this paper is on the theoretical foundations of the GLT theory and therefore we are not concernedwith its applicative interest, already briefly emphasized in the former subsections.

Here we first propose slight (but relevant) modifications of the original definitions of separable Locally Toeplitz(sLT) sequences and GLT sequences appeared in [45, 46]. With the new definitions, which are based on thenotion of approximating class of sequences [28, 41], we are able to enlarge the applicability of the GLT theory.In particular, we remove a certain ‘technical’ hypothesis concerning the Riemann-integrability of the so-called‘weight functions’, which appeared in the statement of fundamental spectral distribution and algebraic results forGLT sequences, such as [45, Theorems 4.5 and 4.8] and [46, Theorem 2.2]. We also show that the product of twoLT sequences with symbols a⊗f, a⊗ f is a LT sequence with symbol aa⊗ff , under the only assumption that thetwo generating functions f, f are conjugate (i.e., one in Lp and the other in Lq, being p, q conjugate exponents). Inthis way, we remove from [45, Theorem 5.3] both the assumption that f, f are in L∞ and the technical conditionin [45, Eq. (41)]. In addition, we provide a formal proof of the fact that sequences formed by multilevel diagonalsampling matrix-sequences, as well as multilevel Toeplitz sequences, fall in the class of LT sequences; this resultwas often used in previous papers, but no formal proof was given. The latter two results allows us to show thatthe sequence Dn(a)Tn(f)n, obtained as the product of the diagonal sampling matrix Dn(a) associated with thefunction a and the Toeplitz matrix generated by f , is a LT sequence with symbol a⊗f , under the only assumptionsthat a is Riemann-integrable and f is in L1.

As a final step, we also extend the GLT theory. This is the completely new part of the the review. We firstprovide an approximation result in Section 5.2, which is particularly useful to show that a given sequence ofmatrices is a GLT sequence. Then, by using this result, we provide in Section 5.4 a new and easier proof ofthe fact that A†nn is a GLT sequence with symbol κ−1 whenever Ann is a GLT sequence with symbol κ andκ 6= 0 almost everywhere; here, A†n denotes the (Moore–Penrose) pseudoinverse of An. Finally, using again theapproximation result of Section 5.2, we prove in Section 5.4 that f(An)n is a GLT sequence with symbol f(κ),as long as f : R→ R is continuous and Ann is a GLT sequence of Hermitian matrices with symbol κ.

The paper is concluded with a final section where we summarize the results, we put in evidence the applica-tive interest, especially regarding the approaximation of PDEs, and we state open problems, both concrete andtheoretical, to be considered in a future reserches.

2 Mathematical backgroundThe section contains the mathematical background, including notations, terminology, preliminaries in analysis,linear algebra, and matrix theory. Of special importance is the definition of singular value and spectral symbol andthe definition of multilevel Toeplitz matrices.

2.1 Notation and terminology• Rm×n (resp. Cm×n) is the space of real (resp. complex) m× n matrices.

• χE is the characteristic (or indicator) function of the set E, so χE(x) = 1 if x ∈ E and χE(x) = 0 otherwise.

• µd denotes the Lebesgue measure in Rd. Throughout this work, all the terminology of measure theory (such as‘measure’, ‘measurable’, ‘a.e.’, ‘in Lp’, etc.) is always referred to the Lebesgue measure.

• If f : D ⊆ Rd → C is in Lp(D) and if the domain D is clear from the context, we write ‖f‖Lp instead of‖f‖Lp(D) to indicate the Lp-norm of f , which is defined by ‖f‖Lp = (

∫D|f |p)1/p for 1 ≤ p < ∞, and by

‖f‖L∞ = ess supD|f | for p =∞.

11

• We use a notation that is common in probability theory to indicate the sets defined in terms of functions. For ex-ample, if f : D ⊆ Rd → C, then f 6= 1 = x ∈ D : f(x) 6= 1, 0 ≤ f ≤ 1 = x ∈ D : 1 ≤ f(x) ≤ 1,µdf > 0 is the measure of the set x ∈ D : f(x) > 0, χf=0 is the characteristic function of the set wheref vanishes, and so on.

• Om and Im denote, respectively, the m ×m zero matrix and the m ×m identity matrix. Sometimes, when thedimension m can be inferred from the context, O and I are used instead of Om and Im.

• If x is a vector and X is a matrix, then xT and x∗ (resp. XT and X∗) are the transpose and the transposeconjugate of x (resp. X).

• GivenX ∈ Cm×m, Λ(X) is the spectrum ofX and ρ(X) is the spectral radius ofX , i.e., ρ(X) = maxλ∈Λ(X) |λ|.The eigenvalues of X are denoted by λj(X), j = 1, . . . ,m.

• Let X ∈ Cm×m be a matrix with only real eigenvalues (e.g., a Hermitian matrix). Unless otherwise stated, it isunderstood that the eigenvalues of X are labeled in non-increasing order: λmax(X) = λ1(X) ≥ . . . ≥ λm(X) =λmin(X). In addition, we set λj(X) = +∞ if j < 1 and λj(X) = −∞ if j > m.

• If X ∈ Cm×m, we denote by σj(X), j = 1, . . . ,m, the singular values of X labeled, as usual, in non-increasingorder: σmax(X) = σ1(X) ≥ . . . ≥ σm(X) = σmin(X). In addition, we set σj(X) = +∞ if j < 1 andσj(X) = −∞ if j > m.

• Given X ∈ Cm×m and 1 ≤ p ≤ ∞, ‖X‖p denotes the Schatten p-norm of X , which is defined as the p-norm ofthe vector (σ1(X), . . . , σm(X)) formed by the singular values of X; see [7]. The Schatten 1-norm is also calledthe trace-norm. The Schatten∞-norm ‖X‖∞ = σmax(X) coincides with the spectral (Euclidean) norm of X; itwill be preferably denoted by ‖X‖. Note that the Schatten norms are unitarily invariant, i.e., ‖UXV ‖p = ‖X‖pfor all p ∈ [1,∞], all X ∈ Cm×m and all unitary matrices U, V ∈ Cm×m; this follows from the fact that X andUXV have the same singular values.

• <(X) and =(X) are, respectively, the real and the imaginary part of the (square) matrix X:

<(X) =X +X∗

2, =(X) =

X −X∗

2i,

where i is the imaginary unit (i2 = −1).

• If wi : Di → C, i = 1, . . . , d, are arbitrary functions, w1 ⊗ · · · ⊗ wd : D1 × · · · × Dd → C denotes thetensor-product function

(w1 ⊗ · · · ⊗ wd)(ξ1, . . . , ξd) = w1(ξ1) · · ·wd(ξd), ξi ∈ Di, i = 1, . . . , d.

• Cc(C) (resp. Cc(R)) is the space of complex-valued continuous functions defined on C (resp. R) and withbounded support. Moreover, C1

c (R) = Cc(R) ∩ C1(R), where C1(R) is the space of complex-valued functionsF defined on R whose real and imaginary parts <(F ), =(F ) are of class C1 over R in the classical sense.

• If H : R→ R and the limit limx→∞H(x) exists, we denote it by H(∞). Similarly, H(−∞) = limx→−∞H(x).

• ann→∞−→ a means that an → a as n→∞.

• A function a : [0, 1]d → C is said to be Riemann-integrable if <(a),=(a) : [0, 1]d → R are Riemann-integrablein the classical sense. Recall that any Riemann-integrable function is bounded.

• If g : D → C and E ⊆ D, we set ‖g‖∞,E = supx∈E |g(x)|. If E = D and D can be inferred from the context,we often write ‖g‖∞ instead of ‖g‖∞,D. Clearly, ‖g‖∞ <∞ if and only if g is bounded over D.

12

2.1.1 Multi-index notation

Throughout this work, we will systematically use the multi-index notation, expounded by Tyrtyshnikov in [56,Section 6]. A multi-index i is simply a vector in Zd; its components are denoted by i1, . . . , id. A multi-indexi ∈ Zd is also called a d-index.

• 0, 1, 2, . . . are the vectors of all zeros, all ones, all twos, . . . (their size will be clear from the context).

• For any d-indexm, N(m) =∏d

j=1 mj andm→∞ means that minj=1,...,dmj →∞.

• If i, j are d-indices, i ≤ j means that ir ≤ jr for all r = 1, . . . , d.

• If h,k are d-indices such that h ≤ k, the multi-index range h, . . . ,k is the set j ∈ Zd : h ≤ j ≤ k. Weassume for the multi-index range h, . . . ,k the standard lexicographic ordering:[

. . .[

[ (j1, . . . , jd) ]jd=hd,...,kd

]jd−1=hd−1,...,kd−1

. . .

]j1=h1,...,k1

. (2.1)

For instance, in the case d = 2 the ordering is

(h1, h2), (h1, h2 + 1), . . . , (h1, k2), (h1 + 1, h2), (h1 + 1, h2 + 1), . . . , (h1 + 1, k2),

. . . . . . , (k1, h2), (k1, h2 + 1), . . . , (k1, k2).

• When a d-index j varies over a multi-index range h, . . . ,k (this is sometimes written as j = h, . . . ,k), it isalways understood that j varies from h to k following the specific ordering (2.1). For instance, if m ∈ Nd andif we write X = [xij ]

mi,j=1, then X is a N(m)×N(m) matrix whose components are indexed by two d-indices

i, j, both varying over the multi-index range 1, . . . ,m according to (2.1). Similarly, if x = [xi]mi=1 then x is

a vector of size N(m) whose components xi, i = 1, . . . ,m, are ordered in accordance with (2.1): the firstcomponent is x1 = x(1,...,1,1), the second component is x(1,...,1,2), and so on until the last component, which isxm = x(m1,...,md).

• When a multi-index appears as subscript or superscript, we often suppress the parentheses to simplify the nota-tion. For instance, the component of the vector x = [xi]

mi=1 corresponding to the multi-index i is denoted by xi

or by xi1,...,id , and we preferably avoid the heavy notation x(i1,...,id).

• Given h,k ∈ Zd with h ≤ k, the notation∑kj=h indicates the summation over all j in h, . . . ,k.

• Operations involving multi-indices that do not have a meaning when considering multi-indices as normal vec-tors must always be interpreted in the componentwise sense. For instance, np = (n1p1, . . . , ndpd), αi/j =(αi1/j1, . . . , αid/jd) for all α ∈ C (of course, the division is defined when j1, . . . , jd 6= 0), i2 = (i21, . . . , i

2d),

max(i, j) = (max(i1, j1), . . . ,max(id, jd)), imodm = (i1 modm1, . . . , id modmd), and so on.

2.1.2 Matrix-sequences and multilevel diagonal sampling matrices

In all this work, by sequence of matrices (or matrix-sequence) we mean a sequence of the form Ann, where:

• n varies in some infinite subset of N;

• n = (n1, . . . , nd) ∈ Nd is a multi-index which depends on n, i.e., n = n(n);

• An is a square matrix with size(An) = N(n);

13

• n→∞ when n→∞.

Two classes of matrix-sequences, which can be regarded as the building blocks of the theory of GLT sequences,will be of particular interest in the following: sequences of multilevel diagonal sampling matrices and sequences ofmultilevel Toeplitz matrices. Here, we introduce multilevel diagonal sampling matrices, while multilevel Toeplitzmatrices will be considered in Section 2.5. For n ∈ Nd and a : [0, 1]d → C, we define the d-level diagonalsampling matrix Dn(a) as the following diagonal matrix of size N(n):

Dn(a) = diagi=1,...,n

a( in

),

where we recall that i varies from 1 to n according to the lexicographic ordering (2.1). For example, if d = 2 then

Dn(a) = diagi1=1,...,n1

[diag

i2=1,...,n2

a( i1n1

,i2n2

)].

Note that Dn(a) can also be defined through a sort of recursive formula: if d = 1 then

Dn(a) = diagi=1,...,n

a( in

);

if d > 1, then

Dn(a) = Dn1,...,nd(a) = diagi1=1,...,n1

Dn2,...,nd

(a( i1n1

, ·)), (2.2)

where a(i1/n1, ·) : [0, 1]d−1 → C is defined by (x2, . . . , xd) 7→ a(i1/n1, x2, . . . , xd).

2.1.3 Separable functions and multivariate trigonometric polynomials

Let I1, . . . , Id ⊆ R be measurable sets and let f : I1 × · · · × Id → C be measurable. We say that f is separable ifthere exist measurable functions fi : Ii → C, i = 1, . . . , d, such that f = f1 ⊗ · · · ⊗ fd. In this case, the functionsf1, . . . , fd are called factors of f and f1⊗· · ·⊗fd is said to be a factorization of f . Note that the factorization is notunique: it suffices to choose d constants c1, . . . , cd such that c1 · · · cd = 1 in order to obtain another factorizationf = c1f1 ⊗ · · · ⊗ cdfd.

Let f : I1 × · · · × Id → C be separable and take a factorization f = f1 ⊗ · · · ⊗ fd. If f ∈ Lp(I1 × · · · × Id)and f is not a.e. equal to 0, then fi ∈ Lp(Ii) for all i = 1, . . . , d. Indeed, for p <∞ we have∫

I1×···×Id|f |p =

d∏i=1

∫Ii

|fi|p.

Since∫Ii|fi|p 6= 0 for all i (otherwise fi = 0 a.e. for some i and f = 0 a.e., contrary to the assumption), it follows

that f ∈ Lp(I1× · · · × Id) if and only if fi ∈ Lp(Ii) for all i. For the case p =∞, we only prove that f1 ∈ L∞(I1)(the proof for the other factors is similar). Since f is not a.e. equal to 0, in particular f2⊗ · · · ⊗ fd is not a.e. equalto 0, hence µd−1 |f2 ⊗ · · · ⊗ fd| ≥ ε > 0 for some ε > 0. If we assume by contradiction that f1 /∈ L∞(I1), thenµ1|f1| ≥ α > 0 for all α > 0, implying that

µd |f | ≥ α ≥ µd (|f1| ≥ α/ε ∩ |f2 ⊗ · · · ⊗ fd| ≥ ε)= µ1|f1| ≥ α/εµd−1|f2 ⊗ · · · ⊗ fd| ≥ ε > 0,

for all α > 0. This is a contradiction to the assumption that f ∈ L∞(I1× · · · × Id). In conclusion, we have provedthat, for any 1 ≤ p ≤ ∞, the factors f1, . . . , fd appearing in any factorization of a function f ∈ Lp(I1×· · ·×Id) are

14

themselves inLp, provided that f is not a.e. equal to 0. In particular, for any separable function f : I1×· · ·×Id → Cin Lp there always exists a factorization f = f1 ⊗ · · · ⊗ fd such that f1, . . . , fd are in Lp.

A (d-variate) trigonometric polynomial is a finite linear combination of the Fourier frequencies eij·θ, j ∈ Zd.Let f : Rd → C be a separable trigonometric polynomial, i.e., a trigonometric polynomial which is separable inthe sense specified above. Let f = f1 ⊗ · · · ⊗ fd be a factorization of f . If f is not identically 0 then f1, . . . , fdare (univariate) trigonometric polynomials. Indeed, since f2, . . . , fd are not identically 0, there exists (ϑ2, . . . , ϑd)such that f2(ϑ2) · · · fd(ϑd) 6= 0. For obvious properties of trigonometric polynomials, θ1 7→ f(θ1, ϑ2, . . . , ϑd) =f1(θ1)f2(ϑ2) · · · fd(ϑd) is a (univariate) trigonometric polynomial, i.e., f1 is a trigonometric polynomial. Withthe same argument, one can prove that f2, . . . , fd are trigonometric polynomials as well. This shows that, for aseparable d-variate trigonometric polynomial f there always exists a factorization f = f1 ⊗ · · · ⊗ fd in whichf1, . . . , fd are trigonometric polynomials.

2.2 Convergence in measureThe convergence in measure is of particular interest in probability theory, and it plays an important role also inthe study of GLT sequences. In this section, we recall the definition and provide some basic properties of thisconvergence that we shall use in the following.

Definition 2.1 (convergence in measure). Let D ⊆ Rk be any measurable subset of Rk, and let fm, f : D → Cbe measurable functions. We say that fm → f in measure over D – or simply that fm → f in measure, if thedomain D can be inferred from the context – when the following condition is met: for every ε > 0,

limm→∞

µk|fm − f | > ε = 0.

Using the definition, it can be shown that, if fm → f in measure, then |fm| → |f | in measure; and if fm → f inmeasure and gm → g in measure, then αfm + βgm → αf + βg in measure for all α, β ∈ C. Moreover, if fm → fin measure, gm → g in measure and µk(D) <∞, then fmgm → fg in measure.

The next lemma provide a relation between the convergence in measure and the L1-convergence. Since theresult is not so popular, for the reader’s convenience we include the details of the proof.

Lemma 2.1. Let K be either R or C, let F : K → C be a uniformly continuous bounded function, and letgm, g : D ⊆ Rd → K be measurable functions such that gm → g in measure over D. Then, F gm → F g inL1(D).

Proof. For every m and every ε > 0,∫D

∣∣F (gm(x))− F (g(x))∣∣dx =

∫|gm−g|≥ε

∣∣F (gm(x))− F (g(x))∣∣dx +

∫|gm−g|<ε

∣∣F (gm(x))− F (g(x))∣∣dx

≤ 2‖F‖∞ µd |gm − g| ≥ εµd(D)

+ ωF (ε), (2.3)

where ωF is the modulus of continuity of F . Since

limm→∞

µd |gm − g| ≥ ε = limε→0

ωF (ε) = 0

(because gm → g in measure and F is uniformly continuous), passing first to the lim supm→∞

and then to the limε→0

in

(2.3), we conclude that F gm → F g in L1(D).

15

Lemma 2.2 is the last result we need about the convergence in measure; it will play a crucial role in the proofof Theorem 5.7. For the proof of Lemma 2.2, we recall that the space generated by the monomials

ei( 2πb1−a1 j1θ1 + . . .+ 2π

bk−ak jkθk) : j = (j1, . . . , jk) ∈ Zk,

that is the set of all finite linear combinations of such monomials (we may call it the space of ‘scaled’ k-variatetrigonometric polynomials), is dense in L1([a1, b1]× · · · × [ak, bk]).

Lemma 2.2. Let κ : [0, 1]d × [−π, π]d → C be a measurable function. Then, there exists a sequence κmm, withκm : [0, 1]d × [−π, π]d → C a function of the form

κm(x,θ) =Nm∑

j=−Nm

a(m)j (x)eij·θ, aj ∈ C∞([0, 1]d), Nm ∈ Nd, (2.4)

such that κm → κ in measure.

Proof. The space generated by the monomialsei2π`·xeij·θ : `, j ∈ Zd

(2.5)

is dense in L1([0, 1]d× [−π, π]d). The function κm = κχ|κ|≤1/m belongs to L∞([0, 1]d× [−π, π]d) ⊂ L1([0, 1]d×[−π, π]d) and κm → κ in measure. Choose κm in the space generated by the monomials (2.5) such that ‖κm −κm‖L1 ≤ 1/m. Note that κm is of the form (2.4). Then, for all ε > 0,

µ2d|κm − κ| > ε ≤ µ2d|κm − κm| > ε/2+ µ2d|κm − κ| > ε/2 ≤ ‖κm − κm‖L1

(ε/2)+ µ2d|κm − κ| > ε/2,

which converges to 0 as m→∞. Hence, κm → κ in measure.

2.3 Preliminaries on Linear Algebra and Matrix AnalysisGivenX ∈ Cm×m, we know from the Singular Value Decomposition (SVD) that rank(X) is the number of nonzerosingular values of X . As a consequence, recalling that ‖X‖ = σmax(X), we obtain

‖X‖1 =m∑i=1

σi(X) ≤ rank(X)‖X‖ ≤ m‖X‖, X ∈ Cm×m. (2.6)

Let 1 ≤ p, q ≤ ∞ be conjugate exponents (1p

+ 1q

= 1). As a consequence of the general Holder-typeinequalities for unitarily invariant norms, see [7, Corollary IV.2.6], the Schatten norms satisfy

‖XY ‖1 ≤ ‖X‖p‖Y ‖q, X, Y ∈ Cm×m. (2.7)

If X ∈ Cm×m is a normal matrix, i.e. XX∗ = X∗X , then X is unitarily diagonalizable, meaning that thereexist a unitary matrixU and a diagonal matrixD such thatX = UDU∗. Using this, it can be shown that the singularvalues of X coincide with the moduli of the eigenvalues, |λj(X)|, j = 1, . . . ,m. Consequently, ‖X‖ = ρ(X) and‖X‖1 =

∑mj=1 |λj(X)|. Note that, ifX is Hermitian (X∗ = X) or skew-Hermitian (X∗ = −X), thenX is normal.

We now recall some important interlacing and perturbation theorems for both singular values and eigenvalues.In the statement of Theorems 2.1–2.2, we use the convention introduced in Section 2.1 about the meaning of λj(X)and σj(X) when j lies outside the range of indices 1, . . . ,m, being m the size of X .

16

Theorem 2.1 (interlacing theorem for singular values). Let Y = X+E, whereX,E ∈ Cm×m and rank(E) ≤ k.Then

σj−k(X) ≥ σj(Y ) ≥ σj+k(X), j = 1, . . . ,m. (2.8)

Theorem 2.2 (interlacing theorem for eigenvalues). Let Y = X + E, where X,E ∈ Cm×m are Hermitian. Letk+, k− ≥ 0 be respectively the number of positive and the number of negative eigenvalues of E, i.e.,

k+ = #j ∈ 1, . . . ,m : λj(E) > 0, k− = #j ∈ 1, . . . ,m : λj(E) < 0.

Thenλj−k+(X) ≥ λj(Y ) ≥ λj+k−(X), j = 1, . . . ,m.

In particular, if rank(E) ≤ k then

λj−k(X) ≥ λj(Y ) ≥ λj+k(X), j = 1, . . . ,m. (2.9)

Theorem 2.1 can be obtained as a corollary of Theorem 2.2, as follows. For any A ∈ Cm×m, the eigenvaluesof the (2m)× (2m) Hermitian matrix

A =

[O AA∗ O

]are σj(A), −σj(A), j = 1, . . . ,m; see [7, Exercise II.1.15]. Therefore, applying Theorem 2.2 with Y , X, Ein place of Y, X, E, we obtain Theorem 2.1. The proof of Theorem 2.2 can be derived from the result in [7,Exercise III.2.4].

Theorem 2.3 (perturbation theorem for singular values). Let X, Y ∈ Cm×m, then

|σj(X)− σj(Y )| ≤ ‖X − Y ‖, j = 1, . . . ,m.

Theorem 2.4 (perturbation theorem for eigenvalues). Let X, Y ∈ Cm×m be Hermitian, then

|λj(X)− λj(Y )| ≤ ‖X − Y ‖, j = 1, . . . ,m.

Theorem 2.4 is Weyl’s perturbation theorem [7, Corollary III.2.6]. Theorem 2.3 can be obtained as a corollaryof Theorem 2.4 by considering again the matrices X and Y . Alternatively, Theorem 2.3 (resp. Theorem 2.4) canbe proved by using the minimax principle for singular values [7, Problem III.6.1] (resp. the minimax principle foreigenvalues [7, Corollary III.1.2]). We also refer the reader to [7, Problem II.6.13] for a more general perturbationtheorem for singular values, which extends Theorem 2.3.

2.3.1 Tensor products and direct sums

If X, Y are matrices of any dimension, say X ∈ Cm1×m2 and Y ∈ C`1×`2 , the tensor (Kronecker) product of Xand Y is the m1`1 ×m2`2 matrix

X ⊗ Y = [xijY ]i=1,...,m1j=1,...,m2

=

x11Y · · · x1m2Y...

...xm11Y · · · xm1m2Y

;

and the direct sum of X and Y is the (m1 + `1)× (m2 + `2) matrix

X ⊕ Y = diag(X, Y ) =

[X OO Y

].

17

Tensor products and direct sums possess a lot of nice algebraic properties; see [22, Section 1.2.1] or [26, Sec-tion 2.4]. Here, we mention the bilinearity of tensor products; the associativity, which allows us to omit parenthesesin expressions likeX1⊗X2⊗· · ·⊗Xd orX1⊕X2⊕· · ·⊕Xd; the relations (X1⊗Y1)(X2⊗Y2) = (X1X2)⊗(Y1Y2)and (X1 ⊕ Y1)(X2 ⊕ Y2) = (X1X2) ⊕ (Y1Y2), which hold whenever X1, X2 can be multiplied and Y1, Y2 can bemultiplied; the identities (X ⊗ Y )∗ = X∗⊗ Y ∗, (X ⊕ Y )∗ = X∗⊕ Y ∗ and (X ⊗ Y )T = XT ⊗ Y T , (X ⊕ Y )T =XT ⊕ Y T , which hold for all matrices X, Y ; and the (very important) multi-index formula for tensor products: ifXk ∈ Cmk×mk , k = 1, . . . , d, then

(X1 ⊗ · · · ⊗Xd)ij = (X1)i1j1 · · · (Xd)idjd , i, j = 1, . . . ,m = (m1, . . . ,md). (2.10)

From these basic properties, a lot of other interesting results follow. For example, if X, Y are normal (resp.Hermitian, symmetric, unitary) then X ⊗ Y is also normal (resp. Hermitian, symmetric, unitary). If X ∈ Cm×m

and Y ∈ C`×`, the eigenvalues and singular values of X ⊗ Y (resp. X ⊕ Y ) are λi(X)λj(Y ), i = 1, . . . ,m, j =1, . . . , ` and σi(X)σj(Y ), i = 1, . . . ,m, j = 1, . . . , ` (resp. λi(X), λj(Y ), i = 1, . . . ,m, j = 1, . . . , ` andσi(X), σj(Y ), i = 1, . . . ,m, j = 1, . . . , `). In particular, for all matrices X ∈ Cm×m and Y ∈ Cn×n,

‖X ⊕ Y ‖ = max(‖X‖, ‖Y ‖), ‖X ⊗ Y ‖ = ‖X‖ ‖Y ‖, (2.11)

‖X ⊕ Y ‖p =(‖X‖pp + ‖Y ‖pp

)1/p, ‖X ⊗ Y ‖p = ‖X1‖p ‖Y ‖p, 1 ≤ p <∞, (2.12)

rank(X ⊕ Y ) = rank(X) + rank(Y ) rank(X ⊗ Y ) = rank(X) rank(Y ), (2.13)

and if X, Y are Hermitian positive definite (HPD), then X ⊗ Y is HPD as well, with

λmin(X ⊗ Y ) = λmin(X)λmin(Y ), λmax(X ⊗ Y ) = λmax(X)λmax(Y ). (2.14)

Moreover,

X ⊗ Y ≥ X ′ ⊗ Y ′, for all HPD matrices X, Y,X ′, Y ′ such that X ≥ X ′ and Y ≥ Y ′, (2.15)

because X⊗Y −X ′⊗Y ′ = (X−X ′)⊗Y +X ′⊗ (Y −Y ′) is a sum of two HPD matrices. We also highlight thefollowing property [26, p. 7]: let X1, . . . , Xd, Y1, . . . , Yd be matrices with Xi, Yi ∈ Cmi×mi for all i = 1, . . . , d,then

rank(X1 ⊗ · · · ⊗Xd − Y1 ⊗ · · · ⊗ Yd) ≤ N(m)d∑i=1

rank(Xi − Yi)mi

. (2.16)

A property of tensor products, which is not as popular as the previous ones, is given in Lemma 2.3. For theproof we refer the reader to [26, Lemma 2].

Lemma 2.3. For all m ∈ Nd and all permutations σ of the set 1, . . . , d, there exists a permutation matrixΠm;σ ∈ CN(m)×N(m) such that

Xσ(1) ⊗ · · · ⊗Xσ(d) = Πm;σ(X1 ⊗ · · · ⊗Xd)ΠTm;σ,

for all matrices X1 ∈ Cm1×m1 , . . . , Xd ∈ Cmd×md .

Lemma 2.3 says that the tensor product is ‘almost’ commutative. It is important to notice that the permutationmatrix Πm;σ depends only on m and σ, and not on the specific matrices X1, . . . , Xd. Concerning the ‘distributiveproperties’ of tensor products with respect to direct sums, a result analogous to Lemma 2.3 holds: these propertieshold modulo permutation transformations which depend only on the dimensions of the involved matrices. For theproof of Lemma 2.4, see [26, Lemma 4].

18

Lemma 2.4. For all ` ∈ N, m ∈ Nd there exists a permutation matrix Q`,m ∈ C`(m1+...+md)×`(m1+...+md) suchthat

X ⊗ (Y1 ⊕ · · · ⊕ Yd) = Q`,m[(X ⊗ Y1)⊕ · · · ⊕ (X ⊗ Yd)]QT`,m,

for all matrices X ∈ C`×`, Y1 ∈ Cm1×m1 , . . . , Yd ∈ Cmd×md .

Lemma 2.4 gives the distributive law on the left; the distributive law on the right holds without permutationmatrices, as stated in the following remark.

Remark 2.1. From the definition of tensor products and direct sums, for all matrices X1, . . . , Xd, Y we have

(X1 ⊕X2 ⊕ · · · ⊕Xd)⊗ Y = (X1 ⊗ Y )⊕ (X2 ⊗ Y )⊕ · · · ⊕ (Xd ⊗ Y ).

We end this section with a result about direct sums, which is completely analogous to Lemma 2.3: it showsthat the direct sum operation is ‘almost’ commutative. The proof is rather technical and may be skipped in a firstreading; take also into account that Lemma 2.5 will be used only in Theorem 4.1, whose proof may be skipped aswell.

Lemma 2.5. For all m ∈ Nd and all permutations σ of the set 1, . . . , d, there exists a permutation matrixVm;σ ∈ C(m1+...+md)×(m1+...+md) such that

Xσ(1) ⊕ · · · ⊕Xσ(d) = Vm;σ(X1 ⊕ · · · ⊕Xd)VTm;σ,

for all matrices X1 ∈ Cm1×m1 , . . . , Xd ∈ Cmd×md .

Proof. The proof is done by induction on d. For d = 1, the only possible permutation is σ = [1] and we can takeVm;[1] = Im. For d = 2, the only possible permutations are the identity σ = [1, 2] and the transposition σ = [2, 1],and we can take

Vm;[1,2] = Im1+m2 , Vm;[2,1] =

[O Im2

Im1 O

].

For d ≥ 3, let i be the index such that σ(i) = d. Define the permutation τ of 1, . . . , d− 1 by setting τ(j) = σ(j)for j = 1, . . . , i− 1 and τ(j) = σ(j + 1) for j = i, . . . , d− 1. If i = d, then, by induction hypothesis,

Xσ(1) ⊕ · · · ⊕Xσ(d) = Xτ(1) ⊕ · · · ⊕Xτ(d−1) ⊕Xd

= V(m1,...,md−1);τ (X1 ⊕ · · · ⊕Xd−1)V T(m1,...,md−1);τ ⊕Xd

= (V(m1,...,md−1);τ ⊕ Imd)(X1 ⊕ · · · ⊕Xd)(V(m1,...,md−1);τ ⊕ Imd)T

and the proof is over, with Vm;σ = V(m1,...,md−1);τ ⊕ Imd . If i < d, then, by induction hypothesis,

Xσ(1) ⊕ · · · ⊕Xσ(d) = Xσ(1) ⊕ · · ·Xσ(i−1) ⊕Xd ⊕Xσ(i+1) ⊕ · · · ⊕Xσ(d)

= Xσ(1) ⊕ · · ·Xσ(i−1) ⊕[V(mσ(i+1)+...+mσ(d),md);[2,1](Xσ(i+1) ⊕ · · · ⊕Xσ(d) ⊕Xd)(V(mσ(i+1)+...+mσ(d),md);[2,1])

T]

= (Imσ(1)+...+mσ(i−1)⊕ V(mσ(i+1)+...+mσ(d),md);[2,1])

(Xτ(1) ⊕ · · · ⊕Xτ(d−1) ⊕Xd

)· (Imσ(1)+...+mσ(i−1)

⊕ V(mσ(i+1)+...+mσ(d),md);[2,1])T

= Um;σ

(Xτ(1) ⊕ · · · ⊕Xτ(d−1) ⊕Xd

)UTm;σ,

where Um;σ = Imσ(1)+...+mσ(i−1)⊕ V(mσ(i+1)+...+mσ(d),md);[2,1]. Again by induction hypothesis,

Xτ(1) ⊕ · · · ⊕Xτ(d−1) = V(m1,...,md−1);τ (X1 ⊕ · · · ⊕Xd−1)V T(m1,...,md−1);τ ,

and the thesis is proved with Vm;σ = Um;σ(V(m1,...,md−1);τ ⊕ Imd).

19

2.4 Singular value and eigenvalue distributions of matrix-sequences: the symbolDefinition 2.2 (spectral distribution of a matrix-sequence, spectral symbol). Let Ann be a matrix-sequence,and let f : D → C be a measurable function, defined on a measurable set D ⊂ Rd with 0 < µd(D) <∞.

• We say that Ann has an asymptotic eigenvalue (or spectral) distribution described by f , in symbolsAnn ∼λ f , if, for all F ∈ Cc(C),

limn→∞

1

N(n)

N(n)∑j=1

F (λj(An)) =1

µd(D)

∫D

F (f(x))dx. (2.17)

In this case, f is referred to as the eigenvalue (or spectral) symbol of the matrix-sequence Ann.

• We say that Ann has an asymptotic singular value distribution described by f , in symbols Ann ∼σ f ,if, for all F ∈ Cc(R),

limn→∞

1

N(n)

N(n)∑j=1

F (σj(An)) =1

µd(D)

∫D

F (|f(x)|)dx. (2.18)

In this case, f is referred to as the singular value symbol of the matrix-sequence Ann.

It is clear that Ann ∼σ f is equivalent to Ann ∼σ |f |. Moreover, if every An is normal and Ann ∼λ f ,then Ann ∼σ f . Indeed, since An is normal, its singular values coincide with the moduli of the eigenvalues.Therefore, for any fixed F ∈ Cc(R), by applying the eigenvalue distribution relation with the test function F (|·|) ∈Cc(C), we get

limn→∞

N(n)∑j=1

F (σj(An)) = limn→∞

N(n)∑j=1

F (|λj(An)|) =1

µd(D)

∫D

F (|f(x)|)dx.

Hence, Ann ∼σ f .

2.5 Multilevel Toeplitz matricesIn this section we recall the definition and some properties of multilevel Toeplitz matrices. Of course, we do notpretend to cover here, in a couple of pages, all the details of this extensive topic; we just report the results that willbe used hereinafter.

Given n ∈ Nd, a matrix of the form

[ai−j ]ni,j=1 ∈ CN(n)×N(n), (2.19)

whose (i, j) depends only on the difference between the multi-indices i and j, is called a multilevel Toeplitzmatrix, or, more precisely, a d-level Toeplitz matrix, being d the length of n. Given a function f : [−π, π]d → Cin L1([−π, π]d), we denote its Fourier coefficients by

fk =1

(2π)d

∫[−π,π]d

f(θ)e−ik·θdθ, k ∈ Zd, (2.20)

where k · θ = k1θ1 + . . .+ kdθd. For every n ∈ Nd, the n-th Toeplitz matrix associated with f is defined as

Tn(f) = [fi−j ]ni,j=1. (2.21)

20

We call Tn(f)n∈Nd the family of (multilevel) Toeplitz matrices associated with f , which, in turn, is called thegenerating function of Tn(f)n∈Nd .

For each fixed n ∈ Nd, the application Tn(·) : L1([−π, π]d) → CN(n)×N(n) is linear: for all α, β ∈ C andf, g ∈ L1([−π, π]d),

Tn(αf + βg) = αTn(f) + βTn(g).

This follows from the relation (αf + βg)k = αfk + βgk, k ∈ Zd, which is a consequence of the linearity of theintegral in (2.20). For every f ∈ L1([−π, π]d), the Fourier coefficients of f are related to those of f by

fj =1

(2π)d

∫[−π,π]d

f(θ)e−ij·θdθ =1

(2π)d

∫[−π,π]d

f(θ)eij·θdθ = (f)−j , j ∈ Zd.

Therefore, for all i, j = 1, . . . ,n,

[Tn(f)]ij = (f)i−j = fj−i = [Tn(f)∗]ij ,

i.e.,Tn(f)∗ = Tn(f).

From this identity, which holds for all n ∈ Nd and f ∈ L1([−π, π]d), we infer that, if f is real-valued, or if f isreal a.e.,1 then all the matrices Tn(f) are Hermitian. Another nice property of the Toeplitz operator Tn(·) is thatTn(1) = IN(n).

Theorem 2.5 is a fundamental result concerning multilevel Toeplitz matrices. It is known in the literature asthe Szego–Tilli theorem. We refer the reader to [8] for a rich account concerning the history of the Szego theorem,originally appeared in [31]. Tilli’s proof of Theorem 2.5 can be found in [53]. We also refer the reader to [28]for a proof of Theorem 2.5 based on the notion of a.c.s. (see Section 3); the proof in [28] is made only in thecase of eigenvalues for d = 1, but the argument is general and can be extended to singular values and to higherdimensionalities d.

Theorem 2.5. Let f ∈ L1([−π, π]d), then Tn(f)n ∼σ f . If moreover f is real a.e., then Tn(f)n ∼λ f .

In Theorem 2.5, Tn(f)n is any sequence of Toeplitz matrices extracted from the family Tn(f)n∈Nd andsuch that n = n(n) → ∞ when n → ∞. In this regard, we recall that, for all matrix-sequences Ann, it isalways understood that the multi-index n tends to∞ when n→∞; see Section 2.1.2.

Important inequalities involving Toeplitz matrices and Schatten p-norms can be found in [44, Corollary 3.5].We report them in the next theorem for future use.

Theorem 2.6. Let f ∈ Lp([−π, π]d), 1 ≤ p ≤ ∞, and let n ∈ Nd. Then,

‖Tn(f)‖ = ‖Tn(f)‖∞ ≤ ‖f‖L∞ , if p =∞, (2.22)

‖Tn(f)‖p ≤1

(2π)d‖f‖Lp N(n)1/p, if 1 ≤ p <∞. (2.23)

In particular, using the convention that N(n)1/∞ = 1, the inequality

‖Tn(f)‖p ≤ N(n)1/p‖f‖Lp (2.24)

holds for all p ∈ [1,∞].

Lemma 2.6 relates tensor products and Toeplitz matrices.1Note that two functions f, g ∈ L1([−π, π]d) which coincide a.e. give rise to the same multilevel Toeplitz matrices Tn(f) =

Tn(g), n ∈ Nd, because the Fourier coefficients of f and g coincide.

21

Lemma 2.6. Let f1, . . . , fd ∈ L1([−π, π]) and n ∈ Nd. Then,

Tn1(f1)⊗ · · · ⊗ Tnd(fd) = Tn(f1 ⊗ · · · ⊗ fd) (2.25)

(note that the tensor-product function f1⊗ · · · ⊗ fd : [−π, π]d → C belongs to L1([−π, π]d) by Fubini’s theorem).

Proof. The proof is simple if we use the fundamental property (2.10). The Fourier coefficients of f1⊗ · · ·⊗ fd aregiven by

(f1 ⊗ · · · ⊗ fd)k = (f1)k1 · · · (fd)kd , k ∈ Zd.

Hence, for all i, j = 1, . . . ,n,

[Tn1(f1)⊗ · · · ⊗ Tnd(fd)]ij = [Tn1(f1)]i1j1 · · · [Tnd(fd)]idjd = (f1)i1−j1 · · · (fd)id−jd = (f1 ⊗ · · · ⊗ fd)i−j= [Tn(f1 ⊗ · · · ⊗ fd)]ij ,

and (2.25) follows.

Lemma 2.7 will be used in Section 4.1.1 to study the Locally Toeplitz operator. The result of this lemmageneralizes [16, Proposition 2] in the case where the involved generating functions f, g are scalar-valued. ByHolder’s inequality [37], if f ∈ Lp([−π, π]d) and g ∈ Lq([−π, π]d), where 1 ≤ p, q ≤ ∞ are conjugate exponents(1p

+ 1q

= 1), then fg ∈ L1([−π, π]d). In this case, we can consider the three matrices Tk(f), Tk(g) and Tk(fg).

Lemma 2.7. Let f ∈ Lp([−π, π]d) and g ∈ Lq([−π, π]d), where 1 ≤ p, q ≤ ∞ are conjugate exponents. Then,

limk→∞

‖Tk(f)Tk(g)− Tk(fg)‖1

N(k)= 0. (2.26)

Proof. If f, g were in L∞([−π, π]d), then (2.26) holds by [16, Proposition 2] and the proof is over. In the generalcase where f ∈ Lp([−π, π]d) and g ∈ Lq([−π, π]d), the proof requires a little more effort.

Take two sequences fm and gm such that fm, gm ∈ L∞([−π, π]d) for all m, fm → f in Lp([−π, π]d) andgm → g in Lq([−π, π]d); for example, one can choose fm = f χ[−m,m] and gm = g χ[−m,m]. By the linearity ofTk(·) and Eqs. (2.7), (2.24), for every m and every k ∈ Nd we have

‖Tk(f)Tk(g)− Tk(fg)‖1

≤ ‖Tk(f − fm)Tk(g)‖1 + ‖Tk(fm)Tk(g − gm)‖1 + ‖Tk(fm)Tk(gm)− Tk(fmgm)‖1 + ‖Tk(fmgm − fg)‖1

≤ N(k)1/p‖f − fm‖LpN(k)1/q‖g‖Lq +N(k)1/p‖fm‖LpN(k)1/q‖g − gm‖Lq+ ‖Tk(fm)Tk(gm)− Tk(fmgm)‖1 +N(k)‖fmgm − fg‖L1

≤ N(k)

[‖f − fm‖Lp‖g‖Lq + sup

h

‖fh‖Lp‖g − gm‖Lq +‖Tk(fm)Tk(gm)− Tk(fmgm)‖1

N(k)+ ‖fmgm − fg‖L1

].

(2.27)

By [16, Proposition 2],

limk→∞

‖Tk(fm)Tk(gm)− Tk(fmgm)‖1

N(k)= 0,

so, dividing (2.27) by N(k) and passing to the limit as k→∞, we get

lim supk→∞

‖Tk(f)Tk(g)− Tk(fg)‖1

N(k)≤ ‖f − fm‖Lp‖g‖Lq + sup

h

‖fh‖Lp‖g − gm‖Lq + ‖fmgm − fg‖L1 .

This relation holds for every m. Passing to the limit as m → ∞ and observing that fmgm → fg in L1([−π, π]d)by Holder’s inequality, we get the thesis.

22

3 Approximating classes of sequences (a.c.s.)In this section, we introduce the fundamental definition on which the theory of GLT sequences is based: the notionof approximating class of sequences, first introduced in [41]. This notion lays the foundations for a (spectral)approximation theory for matrix-sequences and provides general tools (Theorems 3.3 and 3.5) for computing theasymptotic spectral (or singular value) distribution of a ‘difficult’ matrix-sequence Ann from the one of ‘simpler’matrix-sequences Bn,mnm that approximate Ann in a suitable sense when m → ∞; we refer the reader tothe introduction of [28] for a deeper insight on this subject.

Definition 3.1 (approximating class of sequences). Let Ann be a matrix-sequence. An approximating classof sequences (a.c.s.) for Ann is a sequence of matrix-sequences Bn,mnm with the following property: forevery m there exists nm such that, for n ≥ nm,

An = Bn,m +Rn,m +Nn,m, (3.1)

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

where the quantities nm, c(m), ω(m) depend only on m, and

limm→∞

c(m) = limm→∞

ω(m) = 0.

In this definition, it is understood that the matrices Bn,m, Rn,m, Nn,m have size N(n), like the matrix An,otherwise (3.1) would have no meaning. Roughly speaking, Bn,mnm is an a.c.s. for Ann if An is equal toBn,m plus a small-rank matrix (with respect to the matrix size N(n)), plus a small-norm matrix.

Remark 3.1. An equivalent definition of a.c.s. is obtained by replacing, in Definition 3.1, ‘for every m’ with ‘forevery sufficiently large m’ (i.e., ‘for every m greater than or equal to some number M ’). Indeed, suppose that thesplitting (3.1) and the related conditions on Rn,m and Nn,m hold for m ≥ M ; then, defining nm = 1, c(m) =1, ω(m) = 0 and Rn,m = An,m −Bn,m, Nn,m = O for m < M , we see that they actually hold for every m.

3.1 The a.c.s. machinery as a tool for computing singular value and eigenvalue distribu-tions

The importance of the a.c.s. notion lies in Theorems 3.3 and 3.5, whose proofs appeared in [41, 28]. Theorem 3.3provides a general tool for determining the singular value distribution of a ‘difficult’ matrix-sequence Ann,starting from the knowledge of the singular value distribution of simpler matrix-sequences Bn,mn, m ∈ N. Forits proof, some intermediate results are needed.

Theorem 3.1. Let Ann be a matrix-sequence. Assume that:

1. Bn,mnm is an a.c.s. for Ann;

2. for every m and every F ∈ C1c (R), there exists lim

n→∞

1

N(n)

N(n)∑j=1

F (σj(Bn,m)) = φm(F ) ∈ C;

3. for every F ∈ C1c (R), there exists lim

m→∞φm(F ) = φ(F ) ∈ C.

Then, for all F ∈ C1c (R),

limn→∞

1

N(n)

N(n)∑j=1

F (σj(An)) = φ(F ). (3.2)

23

Proof. We first observe that it suffices to prove (3.2) for those test functions F ∈ C1c (R) that are real-valued.

Indeed, any (complex-valued) F ∈ C1c (R) can be decomposed as F = <(F ) + i=(F ), where <(F ), =(F ) ∈

C1c (R). Thus, once we have proved (3.2) for all real-valued functions in C1

c (R), we have

limn→∞

1

N(n)

N(n)∑j=1

F (σj(An)) = limn→∞

1

N(n)

N(n)∑j=1

[<(F (σj(An))) + i=(F (σj(An)))]

= φ(<(F )) + iφ(=(F )) = φ(F ),

where the last equality holds by the linearity of the functional φ, which follows from its definition.Let F ∈ C1

c (R) be real-valued. For all n,m we have∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− φ(F )

∣∣∣∣∣∣ ≤∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(Bn,m))− φm(F )

∣∣∣∣∣∣+ |φm(F )− φ(F )|. (3.3)

By hypothesis, the second term in the right-hand side tends to 0 for n → ∞, while the third one tends to 0 form→∞. Therefore, if we prove that

limm→∞

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣ = 0, (3.4)

then, passing first to the lim supn→∞

and then to the limm→∞

in (3.3), we get the thesis.

In conclusion, we only have to prove (3.4). To this end, we recall that Bn,mnm is an a.c.s. for Ann.Hence, for every m there exists nm such that, for n ≥ nm,

An = Bn,m +Rn,m +Nn,m, (3.5)

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

where limm→∞

c(m) = limm→∞

ω(m) = 0. We can then write, for every m and every n ≥ nm,∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣≤

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m +Rn,m))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(Bn,m +Rn,m))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣ . (3.6)

We will consider separately the two terms in the right-hand side of (3.6), and we will show that each of them isbounded from above by a quantity depending only on m and tending to 0 as m → ∞. After this, (3.4) is provedand the thesis follows.

24

In order to estimate the first term in the right-hand side of (3.6), we use the the perturbation theorem for singularvalues (Theorem 2.3). We have∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m +Rn,m))

∣∣∣∣∣∣ ≤ 1

N(n)

N(n)∑j=1

|F (σj(An))− F (σj(Bn,m +Rn,m))|

≤ 1

N(n)

N(n)∑j=1

‖F ′‖∞ |σj(An)− σj(Bn,m +Rn,m)| ≤ ‖F ′‖∞‖An −Bn,m −Rn,m‖ = ‖F ′‖∞‖Nn,m‖

≤ ‖F ′‖∞ω(m),

which tends to 0 as m→∞.In order to estimate the second term in the right-hand side of (3.6), we use the interlacing theorem for singular

values (Theorem 2.1). We first observe that F can be expressed as the difference between two non-negative,non-decreasing, bounded functions:

F = H −K, H(x) =

∫ x

−∞(F ′)+(t)dt, K(x) =

∫ x

−∞(F ′)−(t)dt,

where (F ′)+ = max(F ′, 0) and (F ′)− = max(−F ′, 0). Hence, for the second term in the right-hand side of (3.6)we have ∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(Bn,m +Rn,m))− 1

N(n)

N(n)∑j=1

F (σj(Bn,m))

∣∣∣∣∣∣≤

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

H(σj(Bn,m +Rn,m))− 1

N(n)

N(n)∑j=1

H(σj(Bn,m))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

K(σj(Bn,m +Rn,m))− 1

N(n)

N(n)∑j=1

K(σj(Bn,m))

∣∣∣∣∣∣ . (3.7)

Defining rn,m = rank(Rn,m) ≤ c(m)N(n), Theorem 2.1 gives

σj−rn,m(Bn,m) ≥ σj(Bn,m +Rn,m) ≥ σj+rn,m(Bn,m), j = 1, . . . , N(n),

and, moreover, it is clear from our notation that

σj−rn,m(Bn,m) ≥ σj(Bn,m) ≥ σj+rn,m(Bn,m), j = 1, . . . , N(n).

Recalling the monotonicity and non-negativity of H , we get∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

H(σj(Bn,m +Rn,m))− 1

N(n)

N(n)∑j=1

H(σj(Bn,m))

∣∣∣∣∣∣≤ 1

N(n)

N(n)∑j=1

|H(σj(Bn,m +Rn,m))−H(σj(Bn,m))|

≤ 1

N(n)

N(n)∑j=1

∣∣H(σj−rn,m(Bn,m))−H(σj+rn,m(Bn,m))∣∣

25

=1

N(n)

N(n)∑j=1

H(σj−rn,m(Bn,m))− 1

N(n)

N(n)∑j=1

H(σj+rn,m(Bn,m))

=1

N(n)

N(n)−rn,m∑j=1−rn,m

H(σj(Bn,m))− 1

N(n)

N(n)+rn,m∑j=1+rn,m

H(σj(Bn,m))

=1

N(n)

rn,m∑j=1−rn,m

H(σj(Bn,m))− 1

N(n)

N(n)+rn,m∑j=N(n)−rn,m+1

H(σj(Bn,m))

≤ 1

N(n)

rn,m∑j=1−rn,m

H(σj(Bn,m)) ≤ 2rn,mH(∞)

N(n)≤ 2c(m)‖H‖∞.

Similarly, one can show that the second term in the right-hand side of (3.7) is bounded from above by 2c(m)‖K‖∞,implying that the quantity in (3.7), namely the second term in the right-hand side of (3.6), is less than or equal to2(‖H‖∞ + ‖K‖∞)c(m). Since the latter tends to 0 as m→∞, the thesis is proved.

The only unpleasant point about Theorem 3.1 is that, in traditional formulations of asymptotic singular valuedistribution results, the usual set of test functions F is Cc(R) and not C1

c (R); see Definition 2.2. However, thispoint is readily settled in Theorem 3.3. For the proof of Theorem 3.3, we shall use the following corollary of theBanach-Steinhaus theorem [37].

Theorem 3.2. Let E ,F be normed vector spaces, with E a Banach space, and let Tn : E → F be a sequence ofcontinuous linear operators. Assume that, for all x ∈ E , there exists lim

n→∞Tnx = Tx ∈ F . Then,

• sup ‖Tn‖ <∞;

• T : E → F is a continuous linear operator with ‖T‖ ≤ lim infn→∞

‖Tn‖.

Theorem 3.3. Let Ann be a sequence of matrices. Assume that:

1. Bn,mnm is an a.c.s. for Ann;

2. for every m and every F ∈ Cc(R), there exists limn→∞

1

N(n)

N(n)∑j=1

F (σj(Bn,m)) = φm(F ) ∈ C;

3. for every F ∈ Cc(R), there exists limm→∞

φm(F ) = φ(F ) ∈ C.

Then φ : (Cc(R), ‖ · ‖∞)→ C is a continuous linear functional with ‖φ‖ ≤ 1, and, for all F ∈ Cc(R),

limn→∞

1

N(n)

N(n)∑j=1

F (σj(An)) = φ(F ). (3.8)

Proof. For fixed n,m, let

φn,m(F ) =1

N(n)

N(n)∑j=1

F (σj(Bn,m)) : (Cc(R), ‖ · ‖∞)→ C.

It is clear that each φn,m is a continuous linear functional on (Cc(R), ‖ · ‖∞) with ‖φn,m‖ ≤ 1. Indeed, thelinearity of φn,m is obvious and the inequality |φn,m(F )| ≤ ‖F‖∞, which is satisfied for all F ∈ Cc(R), yields the

26

continuity of φn,m as well as the bound ‖φn,m‖ ≤ 1. The functional φm is the pointwise limit of φn,m as n → ∞.Hence, by Theorem 3.2, φm is a continuous linear functional on (Cc(R), ‖ · ‖∞) with ‖φm‖ ≤ 1. The functionalφ is the pointwise limit of φm as m → ∞. Hence, again by Theorem 3.2, φ is a continuous linear functional on(Cc(R), ‖ · ‖∞) with ‖φ‖ ≤ 1.

Now, fix F ∈ Cc(R). For all ε > 0 we can find Fε ∈ C1c (R) such that ‖F − Fε‖∞ ≤ ε. As a consequence, for

all ε > 0 and for all n we have∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− φ(F )

∣∣∣∣∣∣≤

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− 1

N(n)

N(n)∑j=1

Fε(σj(An))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

Fε(σj(An))− φ(Fε)

∣∣∣∣∣∣+ |φ(Fε)− φ(F )|

≤ ‖F − Fε‖∞ +

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

Fε(σj(An))− φ(Fε)

∣∣∣∣∣∣+ |φ(Fε)− φ(F )|.

Considering that (3.8) holds for Fε by Theorem 3.1, we have

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− φ(F )

∣∣∣∣∣∣ ≤ ε+ |φ(Fε)− φ(F )|.

Passing to the limit as ε→ 0 and taking into account the continuity of φ, we obtain

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (σj(An))− φ(F )

∣∣∣∣∣∣ = 0,

which means that (3.8) holds for every F ∈ Cc(R).

In Theorems 3.4 and 3.5 we prove analogous versions of Theorems 3.1 and 3.3 for the case of the eigenvalues,but we need to add the assumption that An and Bn,m are Hermitian. Theorem 3.5 is then a general tool fordetermining the spectral distribution of a ‘difficult’ matrix-sequence Ann formed by Hermitian matrices, startingfrom the spectral distribution of simpler matrix-sequences Bn,mn, m ∈ N, again formed by Hermitian matrices.

The next lemma shows that, whenever the matrices An and Bn,m are Hermitian, the small-rank matrix Rn,mand the small-norm matrix Nn,m in the splitting (3.1) may be supposed to be Hermitian.

Lemma 3.1. Let Ann be a sequence of Hermitian matrices, and let Bn,mnm be an a.c.s. for Ann formedby Hermitian matrices (i.e. every Bn,m is Hermitian). Then, for every m there exists nm such that, for n ≥ nm,

An = Bn,m +Rn,m +Nn,m,

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

where the quantities nm, c(m), ω(m) depend only on m, the matrices Rn,m, Nn,m are Hermitian, and

limm→∞

c(m) = limm→∞

ω(m) = 0.

Proof. Take the real part in (3.1) and use the inequalities rank(<(X)) ≤ 2 rank(X) and ‖<(X)‖ ≤ ‖X‖ toconclude that, by replacing Rn,m, Nn,m with <(Rn,m), <(Nn,m) (if necessary), we can assume Rn,m, Nn,m to beHermitian.

27

Theorem 3.4. Let Ann be a sequence of Hermitian matrices. Assume that:

1. Bn,mnm is an a.c.s. for Ann formed by Hermitian matrices;

2. for every m and every F ∈ C1c (R), there exists lim

n→∞

1

N(n)

N(n)∑j=1

F (λj(Bn,m)) = φm(F ) ∈ C;

3. for every F ∈ C1c (R), there exists lim

m→∞φm(F ) = φ(F ) ∈ C.

Then, for all F ∈ C1c (R),

limn→∞

1

N(n)

N(n)∑j=1

F (λj(An)) = φ(F ). (3.9)

Proof. The proof is essentially the same as the proof of Theorem 3.1; we just outline the main steps, emphasizingthe analogies with Theorem 3.1 and pointing out where we use the assumption that An and Bn,m are Hermitian.

As in the proof of Theorem 3.1, one can show that it suffices to prove (3.9) for all real-valued test functionsF ∈ C1

c (R). We fix a real-valued function F ∈ C1c (R) and we bound the quantity∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− φ(F )

∣∣∣∣∣∣as in (3.3), with ‘σj’ replaced by ‘λj’. By hypothesis, the second term in the right-hand side tends to 0 for n→∞,while the third one tends to 0 for m→∞. Therefore, the thesis is proved if we show that

limm→∞

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− 1

N(n)

N(n)∑j=1

F (λj(Bn,m))

∣∣∣∣∣∣ = 0. (3.10)

To prove (3.10), we recall that Bn,mnm is a.c.s. for Ann and that An, Bn,m are Hermitian. Hence,by Lemma 3.1, for every m there exists nm such that, for n ≥ nm, the splitting (3.5) holds with HermitianRn,m, Nn,m. Following the proof of Theorem 3.1, we arrive at the inequality (3.6), with ‘σj’ replaced by ‘λj’, andthe thesis is proved if we are able to bound the two terms in the right-hand side by a quantity depending only on mand tending to 0 as m→∞.

The first term is bounded exactly as in Theorem 3.1, using the perturbation theorem for eigenvalues (Theorem2.4) instead of the perturbation theorem for singular values (Theorem 2.3). Note that the perturbation theorem foreigenvalues, contrary to the perturbation theorem for singular values, applies only to Hermitian matrices.

Also the second term is bounded exactly as in Theorem 3.1, using the interlacing theorem for eigenvalues(Theorem 2.2) instead of the interlacing theorem for singular values (Theorem 2.1). Even in this case, the inter-lacing theorem for eigenvalues, contrary to the interlacing theorem for singular values, applies only to Hermitianmatrices.

The only unpleasant point about Theorem 3.4 is that, in traditional formulations of asymptotic spectral distri-bution results, the usual set of test functions F is Cc(C) and not C1

c (R); see Definition 2.2. This point is readilysettled in Theorem 3.5, which is analogous to Theorem 3.3.

Theorem 3.5. Let Ann be a sequence of Hermitian matrices. Assume that:

1. Bn,mnm is an a.c.s. for Ann formed by Hermitian matrices;

28

2. for every m and every F ∈ Cc(C), there exists limn→∞

1

N(n)

N(n)∑j=1

F (λj(Bn,m)) = φm(F ) ∈ C;

3. for every F ∈ Cc(C), there exists limm→∞

φm(F ) = φ(F ) ∈ C.

Then φ : (Cc(C), ‖ · ‖∞)→ C is a continuous linear functional with ‖φ‖ ≤ 1, and, for all F ∈ Cc(C),

limn→∞

1

N(n)

N(n)∑j=1

F (λj(An)) = φ(F ). (3.11)

Proof. The proof is similar to the one used for proving Theorem 3.3. For fixed n,m, let

φn,m(F ) =1

N(n)

N(n)∑j=1

F (λj(Bn,m)) : (Cc(C), ‖ · ‖∞)→ C.

It is clear that each φn,m is a continuous linear functional on (Cc(C), ‖ · ‖∞) with ‖φn,m‖ ≤ 1. Indeed, the linearityof φn,m is obvious and the inequality

|φn,m(F )| ≤ ‖F‖∞,R, (3.12)

which is satisfied for all F ∈ Cc(C), yields the continuity of φn,m as well as the bound ‖φn,m‖ ≤ 1. The functionalφm is the pointwise limit of φn,m as n → ∞. Hence, by Theorem 3.2, φm is a continuous linear functional on(Cc(C), ‖ · ‖∞) with ‖φm‖ ≤ 1; moreover, by (3.12), for all F ∈ Cc(C) we have

|φm(F )| ≤ ‖F‖∞,R. (3.13)

The functional φ is the pointwise limit of φm as m → ∞. Hence, again by Theorem 3.2, φ is a continuous linearfunctional on (Cc(C), ‖ · ‖∞) with ‖φ‖ ≤ 1; moreover, by (3.13), for all F ∈ Cc(C) we have

|φ(F )| ≤ ‖F‖∞,R. (3.14)

Now, fix F ∈ Cc(C). For all ε > 0 we can find Fε ∈ Cc(C) such that Fε restricted to R belongs to C1c (R) and

‖F − Fε‖∞,R ≤ ε. Then, for all ε > 0 and for all n we have∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− φ(F )

∣∣∣∣∣∣≤

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− 1

N(n)

N(n)∑j=1

Fε(λj(An))

∣∣∣∣∣∣+

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

Fε(λj(An))− φ(Fε)

∣∣∣∣∣∣+ |φ(Fε)− φ(F )|

≤ ‖F − Fε‖∞,R +

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

Fε(λj(An))− φ(Fε)

∣∣∣∣∣∣+ |φ(Fε − F )|.

Considering that (3.11) holds for Fε by Theorem 3.4 and using (3.14), we have

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− φ(F )

∣∣∣∣∣∣ ≤ ε+ ε.

29

Passing to the limit as ε→ 0, we obtain

lim supn→∞

∣∣∣∣∣∣ 1

N(n)

N(n)∑j=1

F (λj(An))− φ(F )

∣∣∣∣∣∣ = 0,

which means that (3.11) holds for every F ∈ Cc(C).

Two important corollaries of Theorems 3.3 and 3.5 are given in the following. They will be used in Section 5.3to prove the asymptotic spectral properties of GLT sequences.

Corollary 3.1. Let Ann be a sequence of matrices. Assume that:

1. Bn,mnm is an a.c.s. for Ann;

2. for every m, Bn,mn ∼σ fm for some measurable function fm : D ⊂ Rd → C;

3. |fm| → |f | in measure over D when m→∞, being f : D → C another measurable function.2

Then Ann ∼σ f .

Proof. Apply Theorem 3.3 with

φm(F ) =1

µd(D)

∫D

F (|fm(x)|)dx, φ(F ) =1

µd(D)

∫D

F (|f(x)|)dx.

Note that φm(F )→ φ(F ) for all F ∈ Cc(R) by Lemma 2.1.

Corollary 3.2. Let Ann be a sequence of Hermitian matrices. Assume that:

1. Bn,mnm is an a.c.s. for Ann formed by Hermitian matrices;

2. for every m, Bn,mn ∼λ fm for some measurable function fm : D ⊂ Rd → C;

3. fm → f in measure over D when m→∞, being f : D → C another measurable function.

Then Ann ∼λ f .

Proof. Apply Theorem 3.5 with

φm(F ) =1

µd(D)

∫D

F (fm(x))dx, φ(F ) =1

µd(D)

∫D

F (f(x))dx,

and use again Lemma 2.1 to see that φm(F )→ φ(F ) for all F ∈ Cc(C).

2Note that ‘fm → f in measure’ implies ‘|fm| → |f | in measure’.

30

3.2 The a.c.s. algebraIn this section, we investigate the numerous algebraic properties possessed by the approximating classes of se-quences. These properties form the basis of the so-called GLT algebra, which will be studied later on, in Sec-tion 5.4. We begin with the following observation, whose proof is very simple and is left to the reader.

Remark 3.2. Let Bn,mnm be an a.c.s. for Ann. Then B∗n,mnm is an a.c.s. for A∗nn.

Proposition 3.1. Let Ann, A′nn be matrix-sequences, and let

• Bn,mnm an a.c.s. for Ann,

• B′n,mnm an a.c.s. for A′nn.

Then αBn,m + βB′n,mnm is an a.c.s. for αAn + βA′nn, for all α, β ∈ C.

Proof. By hypothesis,

• for every m there exists nm such that, for n ≥ nm,

An = Bn,m +Rn,m +Nn,m,

where rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m) and limm→∞ c(m) = limm→∞ ω(m) = 0;

• for every m there exists n′m such that, for n ≥ n′m,

A′n = B′n,m +R′n,m +N ′n,m,

where rank(R′n,m) ≤ c′(m)N(n), ‖N ′n,m‖ ≤ ω′(m) and limm→∞ c′(m) = limm→∞ ω

′(m) = 0.

Hence, for every m and every n ≥ max(nm, n′m),

αAn + βA′n = (αBn,m + βB′n,m) + (αRn,m + βR′n,m) + (αNn,m + βN ′n,m),

rank(αRn,m + βR′n,m) ≤ [c(m) + c′(m)]N(n), ‖αNn,m + βN ′n,m‖ ≤ |α|ω(m) + |β|ω′(m),

wherelimm→∞

(c(m) + c′(m)

)= lim

m→∞

(|α|ω(m) + |β|ω′(m)

)= 0.

Hence, by Definition 3.1, αBn,m + βB′n,mnm is an a.c.s. for αAn + βA′nn.

Proposition 3.1 addresses the case of a linear combination αAn+βA′nn of two matrix-sequences Ann andA′nn. We would like to prove an analogous result for the product AnA′nn. To this end, an additional (mild)assumption on Ann and A′nn is required, namely that Ann and A′nn are sparsely unbounded.

Definition 3.2 (sparsely unbounded matrix-sequence). Let Ann be a matrix-sequence. We say that Ann issparsely unbounded (s.u.) if for every M > 0 there exists nM such that, for n ≥ nM ,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M),

where limM→∞ r(M) = 0.

The following proposition provides equivalent characterizations of sparsely unbounded matrix-sequences.

Proposition 3.2. Let Ann be a matrix-sequence. The following conditions are equivalent.

31

1. Ann is s.u.

2. limM→∞

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

= 0.

3. For every M > 0 there exists nM such that, for n ≥ nM ,

An = An,M + An,M ,

rank(An,M) ≤ r(M)N(n), ‖An,M‖ ≤M,

where limM→∞ r(M) = 0.

Note that condition 2 can be rewritten as

limM→∞

lim supn→∞

1

N(n)

N(n)∑i=1

χ(M,∞)(σi(An)) = 0.

Proof. (1⇒ 2) Suppose that Ann is s.u. Then, for every M > 0 there exists nM such that, for n ≥ nM ,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M),

where limM→∞ r(M) = 0. Therefore,

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M)

and, consequently,

limM→∞

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

= 0.

(2⇒ 1) Suppose that condition 2 is met. For every M > 0, define

δ(M) = lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

∈ [0, 1]

and note that (obviously)

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

< δ(M) +1

M.

Hence, by definition of lim sup, for every M > 0 the sequence #i∈1,...,N(n):σi(An)>MN(n)

is eventually less thanr(M) = δ(M) + 1

M, i.e., there exists nM such that, for n ≥ nM ,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M).

Since r(M)→ 0 as M →∞, item 1 is proved.(1⇒ 3) Suppose that Ann is s.u.: for every M > 0 there exists nM such that, for n ≥ nM ,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M),

32

where limM→∞ r(M) = 0. Let An = UnΣnV∗n be a SVD of An. Let Σn,M be the matrix obtained from Σn by

setting to 0 all the singular values of An that are less than or equal to M , and let Σn,M = Σn− Σn,M be the matrixobtained from Σn by setting to 0 all the singular values of An that exceed M . Then,

An = UnΣnV∗n = UnΣn,MV

∗n + UnΣn,MV

∗n = An,M + An,M ,

where An,M = UnΣn,MV∗n and An,M = UnΣn,MV

∗n satisfy, for n ≥ nM ,

rank(An,M) = #i ∈ 1, . . . , N(n) : σi(An) > M ≤ r(M)N(n), ‖An,M‖ = σmax(An,M) ≤M.

(3⇒ 1) Suppose that condition 3 holds. Then, for every M > 0 there exists nM such that, for n ≥ nM ,

An = An,M + An,M ,

where rank(An,M) ≤ r(M)N(n), ‖An,M‖ ≤ M and limM→∞ r(M) = 0. By the minimax principle for singularvalues [7, Problem III.6.1], for all i = 1, . . . , N(n) we have

σi(An) = maxdimV=i

minx∈V, ‖x‖=1

‖Anx‖ ≤ maxdimV=i

minx∈V, ‖x‖=1

(‖An,Mx‖+ ‖An,Mx‖

)≤ max

dimV=imin

x∈V, ‖x‖=1

(‖An,Mx‖+ ‖An,M‖

)≤ σi(An,M) +M.

Since rank(An,M) ≤ r(M)N(n), An,M has at most r(M)N(n) nonzero singular values. Therefore, An has atmost r(M)N(n) singular values greater than M , i.e., #i ∈ 1, . . . , N(n) : σi(An) > M ≤ r(M)N(n), or,equivalently,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M).

This implies that Ann is s.u.

We now show that any matrix-sequence enjoying an asymptotic singular value distribution is s.u.; cf. [51,Proposition 2.7].

Proposition 3.3. Let Ann be a matrix-sequence such that Ann ∼σ f for some measurable f : D ⊂ Rd → C.Then Ann is s.u.

Proof. If we could choose F = χ(M,∞) as a test function in (2.18), then we would obtain

limn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

= limn→∞

1

N(n)

N(n)∑i=1

χ(M,∞)(σi(An))

=1

µd(D)

∫D

χ(M,∞)(|f(x)|)dx =µd|f | > M

µd(D);

since µd|f | > M → 0 as M → ∞ (by the dominated convergence theorem [37]), the proof would be over,thanks to Proposition 3.2. Nevertheless, χ(M,∞) cannot be chosen as a test function in (2.18), since it does notbelong to Cc(R). Therefore, to obtain the thesis, we need a little more work.

Fix M > 0 and take FM ∈ Cc(R) such that FM = 1 over [−M/2,M/2], FM = 0 outside [−M,M ] and0 ≤ FM ≤ 1 over R. Note that FM ≤ χ[−M,M ] over R. Then,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

= 1− #i ∈ 1, . . . , N(n) : σi(An) ≤MN(n)

= 1− 1

N(n)

N(n)∑i=1

χ[−M,M ](σi(An))

≤ 1− 1

N(n)

N(n)∑i=1

FM(σi(An))n→∞−→ 1− 1

µd(D)

∫D

FM(|f(x)|)dx

33

and

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ 1− 1

µd(D)

∫D

FM(|f(x)|)dx.

Since FM(|f(x)|)→ 1 a.e. and |F (|f(x)|)| ≤ 1, by the dominated convergence theorem we get

limM→∞

∫D

FM(|f(x)|)dx = µd(D),

and so

limM→∞

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

= 0.

By Proposition 3.2, Ann is s.u.

We now prove the analogue of Proposition 3.1 for the case of the product of two a.c.s. This important resultappeared for the first time in [41].

Proposition 3.4. Let Ann, A′nn be s.u. matrix-sequences, and let

• Bn,mnm an a.c.s. for Ann,

• B′n,mnm an a.c.s. for A′nn.

Then, Bn,mB′n,mnm is an a.c.s. for AnA′nn.

Proof. By hypothesis, for every m there exists nm such that, for n ≥ nm,

An = Bn,m +Rn,m +Nn,m,

A′n = B′n,m +R′n,m +N ′n,m,

whererank(Rn,m), rank(R′n,m) ≤ c(m)N(n), ‖Nn,m‖, ‖N ′n,m‖ ≤ ω(m)

and limm→∞ c(m) = limm→∞ ω(m) = 0. Hence,

AnA′n = Bn,mB

′n,m +Bn,mR

′n,m + Bn,mN

′n,m +Rn,mA

′n + Nn,mA

′n .

Since Ann and A′nn are s.u., for every M > 0 there exists n(M) such that, for n ≥ n(M),

An = An,M + An,M ,

A′n = A′n,M + A′n,M ,

whererank(An,M), rank(A′n,M) ≤ r(M)N(n), ‖An,M‖, ‖A′n,M‖ ≤M

and limM→∞ r(M) = 0. Setting Mm = [ω(m)]−1/2, for every m and every n ≥ max(nm, n(Mm)) we have

Bn,mN′n,m +Nn,mA

′n = (An −Rn,m −Nn,m)N ′n,m +Nn,m(A′n,Mm

+ A′n,Mm)

= (An,Mm + An,Mm −Rn,m −Nn,m)N ′n,m +Nn,mA′n,Mm

+Nn,mA′n,Mm

= An,MmN′n,m + An,MmN

′n,m −Rn,mN ′n,m −Nn,mN ′n,m +Nn,mA

′n,Mm

+Nn,mA′n,Mm

,

34

and so

AnA′n = Bn,mB

′n,m +Bn,mR

′n,m + Bn,mN

′n,m +Rn,mA

′n + Nn,mA

′n

= Bn,mB′n,m +Bn,mR

′n,m +Rn,mA

′n + An,MmN

′n,m + An,MmN

′n,m −Rn,mN ′n,m −Nn,mN ′n,m

+Nn,mA′n,Mm

+Nn,mA′n,Mm

,

where

rank(Bn,mR′n,m +Rn,mA

′n + An,MmN

′n,m −Rn,mN ′n,m +Nn,mA

′n,Mm

) ≤ [3c(m) + 2r(Mm)]N(n),

‖An,MmN′n,m −Nn,mN ′n,m +Nn,mA

′n,Mm

‖ ≤ 2[ω(m)]1/2 + [ω(m)]2.

Thus, Bn,mB′n,mnm is an a.c.s. for AnA′nn.

3.3 Some criterions to identify a.c.s.In practical applications, it sometimes happens that a matrix-sequence Ann is given together with a sequenceof matrix-sequences Bn,mnm, and one would like to show that Bn,mnm is an a.c.s. for Ann, withoutconstructing the splitting of Definition 3.1. In this section, we provide two useful criterions to solve this problem.

The first criterion, expressed in Theorem 3.6 and Corollary 3.3, is formulated in terms of Schatten p-norms; itwill be mainly applied with p = 1.

Lemma 3.2. Let C be a square matrix of size s. Suppose that

‖C‖pp ≤ εs

for some p ∈ [1,∞). ThenC = R +N,

withrank(R) ≤ ε

1p+1 s, ‖N‖ ≤ ε

1p+1 .

Proof. Since ‖C‖pp =∑s

i=1[σi(C)]p ≤ εs, the number of singular values of C that exceed ε1p+1 cannot be larger

than ε1p+1 s. Let C = UΣV ∗ be a SVD of C and write

C = UΣV ∗ = UΣ(1)V ∗ + UΣ(2)V ∗,

where Σ(1) is obtained from Σ by setting to 0 all the singular values that are less than or equal to ε1p+1 , while

Σ(2) = Σ− Σ(1) is obtained from Σ by setting to 0 all the singular values that exceed ε1p+1 . Then

C = R +N,

where R = UΣ(1)V ∗ and N = UΣ(2)V ∗ satisfy rank(R) ≤ ε1p+1 s and ‖N‖ ≤ ε

1p+1 .

Theorem 3.6. Let Cn,mnm be a sequence of matrix-sequences and let 1 ≤ p <∞. Suppose that for every mthere exists nm such that, for n ≥ nm,

‖Cn,m‖pp ≤ ε(m,n)N(n),

where limm→∞

lim supn→∞

ε(m,n) = 0. Then, for every m there exists nm such that, for n ≥ nm,

Cn,m = Rn,m +Nn,m,

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

where limm→∞

c(m) = limm→∞

ω(m) = 0.

35

Proof. By Lemma 3.2, for every m and every n ≥ nm we have

Cn,m = Rn,m +Nn,m,

rank(Rn,m) ≤ [ε(m,n)]1p+1N(n), ‖Nn,m‖ ≤ [ε(m,n)]

1p+1 .

Letε(m) = lim sup

n→∞ε(m,n).

By definition of lim sup, for every m there exists n′m such that, for n ≥ n′m,

ε(m,n) ≤ ε(m) +1

m.

Setting nm = max(nm, n′m), for every m and every n ≥ nm we have

Cn,m = Rn,m +Nn,m,

rank(Rn,m) ≤(ε(m) +

1

m

) 1p+1N(n), ‖Nn,m‖ ≤

(ε(m) +

1

m

) 1p+1.

Since ε(m)→ 0 by assumption, the thesis is proved with c(m) = ω(m) =(ε(m) + 1

m

) 1p+1 .

Corollary 3.3. Let Ann be a matrix-sequence, let Bn,mnm be a sequence of matrix-sequences, and let1 ≤ p <∞. Suppose that for every m there exists nm such that, for n ≥ nm,

‖An −Bn,m‖pp ≤ ε(m,n)N(n),

where limm→∞

lim supn→∞

ε(m,n) = 0. Then Bn,mnm is an a.c.s. for Ann.

The second criterion, expressed in Theorem 3.7 and Corollary 3.4, is formulated in terms of the singular valuedistribution.

Theorem 3.7. Let Cn,mnm be a sequence of matrix-sequences. Suppose that Cn,mn ∼σ gm for each m,and gm → 0 in measure. Then, for every m there exists nm such that, for n ≥ nm,

Cn,m = Rn,m +Nn,m,

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

where limm→∞

c(m) = limm→∞

ω(m) = 0.

Proof. For any fixed k ∈ N, let Fk ∈ Cc(R) such that Fk = 1 over [0, 1/2k], F = 0 outside [−1/k, 1/k], and0 ≤ Fk ≤ 1 over R. For every m, k, we have

#i ∈ 1, . . . , N(n) : σi(Cn,m) > 1/kN(n)

= 1− #i ∈ 1, . . . , N(n) : σi(Cn,m) ≤ 1/kN(n)

= 1− 1

N(n)

N(n)∑i=1

χ[0,1/k](σi(Cn,m))

≤ 1− 1

N(n)

N(n)∑i=1

Fk(σi(Cn,m))n→∞−→ c(m, k), (3.15)

36

wherec(m, k) = 1− 1

µd(D)

∫D

Fk(|gm(x)|)dx.

By Lemma 2.1, c(m, k)→ 0 as m→∞ and so there exists a sequence kmm of natural numbers such that

limm→∞

km =∞, limm→∞

c(m, km) = 0.

From (3.15),

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(Cn,m) > 1/kmN(n)

≤ c(m, km). (3.16)

Let Cn,m = Un,mΣn,mV∗n,m be a SVD of Cn,m. Let Σ

(1)n,m be the matrix obtained from Σn,m by setting to 0 all the

singular vaules that are less than or equal to 1/km, and let Σ(2)n,m = Σn,m−Σ

(1)n,m be the matrix obtained from Σn,m

by setting to 0 all the singular values that exceed 1/km. Then we can write

Cn,m = Rn,m +Nn,m,

where Rn,m = Un,mΣ(1)n,mV ∗n,m and Nn,m = Un,mΣ

(2)n,mV ∗n,m. By definition, ‖Nn,m‖ ≤ 1/km. Moreover, Eq. (3.16)

says that

lim supn→∞

rank(Rn,m)

N(n)≤ c(m, km),

implying the existence of a nm such that, for n ≥ nm,

rank(Rn,m) ≤(c(m, km) + 1/m

)N(n).

This completes the proof.

Corollary 3.4. Let Ann be a matrix-sequence and let Bn,mnm be a sequence of matrix-sequences. Supposethat An −Bn,mn ∼σ gm for each m, and gm → 0 in measure. Then Bn,mnm is an a.c.s. for Ann.

4 LT and sLT sequencesIn this section, we introduce and analyze the so-called Locally Toeplitz operator. Then, we define the LocallyToeplitz (LT) and separable Locally Toeplitz (sLT) sequences, and we study their properties. The results containedin this section are of fundamental importance for developing the theory of Generalized Locally Toeplitz (GLT)sequences, which will be the subject of Section 5.

4.1 The Locally Toeplitz operator LTmn (a, f)

Definition 4.1.

• Let m,n ∈ N, let a : [0, 1]→ C, and let f : [−π, π]→ C in L1([−π, π]). Then, we define the n× n matrix

LTmn (a, f) = Dm(a)⊗ Tbn/mc(f) ⊕ Onmodm = diagj=1,...,m

a( jm

)Tbn/mc(f) ⊕ Onmodm.

It is understood that LTmn (a, f) = On when n < m and that the term Onmodm is not present when n is amultiple of m. Moreover, here and in the following, the tensor product operation ⊗ is always applied beforethe direct sum ⊕, exactly as in the case of numbers, where multiplication is always applied before addition.

37

• Let m,n ∈ Nd, let a : [0, 1]d → C, and let f1, . . . , fd : [−π, π] → C in L1([−π, π]). Then, we define theN(n)×N(n) matrix

LTmn (a, f1 ⊗ · · · ⊗ fd) = LTm1,...,mdn1,...,nd

(a(x1, . . . , xd), f1 ⊗ · · · ⊗ fd)

= diagj1=1,...,m1

Tbn1/m1c(f1)⊗ LTm2,...,mdn2,...,nd

(a( j1

m1

, x2, . . . , xd

), f2 ⊗ · · · ⊗ fd

)⊕ O(n1 modm1)n2···nd .

This is a recursive definition, whose base case has been considered in the previous item. For example, in thecase d = 2 we have

LTm1,m2n1,n2

(a, f1 ⊗ f2)

= diagj1=1,...,m1

Tbn1/m1c(f1)⊗[

diagj2=1,...,m2

a( j1

m1

,j2

m2

)Tbn2/m2c(f2) ⊕ On2 modm2

]⊕ O(n1 modm1)n2 .

In this section, especially in Subsection 4.1.1, we investigate the properties of LTmn (a, f) that will be of interestlater on. We write LTmn (a, f) instead of LTmn (a, f1⊗· · ·⊗fd) because we are going to see that LTmn (a, f) is well-defined (in a unique way) for any function f ∈ L1([−π, π]d); see Definition 4.2. In particular, if f is separable,the definition is independent of the factorization of f as a tensor-product of the form f1 ⊗ · · · ⊗ fd, f1, . . . , fd ∈L1([−π, π]).

The main property of the Locally Toeplitz operator is stated in Theorem 4.1: it shows that LTmn (a, f1⊗· · ·⊗fd)coincides with Dm(a)⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ O up to a permutation transformation which only depends onm, n and not on the specific functions a, f1, . . . , fd. This result allows us to extend the definition of the LocallyToeplitz operator as in Definition 4.2. The proof of Theorem 4.1 is rather technical and may be skipped in a firstreading; only the statement of the theorem is relevant for what follows.

Theorem 4.1. For anym,n ∈ Nd there exists a permutation matrix Πmn , depending only onm and n, such that

LTmn (a, f1 ⊗ · · · ⊗ fd) = Πmn

[Dm(a)⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(n)−N(m)N(bn/mc)

](Πmn )T

for every a : [0, 1]d → C and every f1, . . . , fd ∈ L1([−π, π]).

Proof. The proof is done by induction on d. For d = 1 the result holds with Πmn = In. For d ≥ 2, let n′ =

(n2, . . . , nd) andm′ = (m2, . . . ,md). By definition,

LTmn (a, f1 ⊗ · · · ⊗ fd) = diagj1=1,...,m1

Tbn1/m1c(f1)⊗ LTm′n′

(a( j1

m1

, ·), f2 ⊗ · · · ⊗ fd

)⊕ O(n1 modm1)n2···nd , (4.1)

where a(j1/m1, ·) : [0, 1]d−1 → C is the function (x2, . . . , xd) 7→ a(j1/m1, x2, . . . , xd). By induction hypothesis,setting N(n′,m′) = N(n′)−N(m′)N(bn′/m′c), we have

LTm′

n′

(a( j1

m1

, ·), f2 ⊗ · · · ⊗ fd

)= Πm

′

n′

[Dm′

(a( j1

m1

, ·))⊗ Tbn′/m′c(f2 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)

](Πm

′

n′ )T .

(4.2)Let us now work on the argument of the ‘diag operator’ in (4.1). By Lemma 2.3, Eq. (4.2) and the properties of

38

tensor products (see Section 2.3.1), we get

Tbn1/m1c(f1)⊗ LTm′n′

(a( j1

m1

, ·), f2 ⊗ · · · ⊗ fd

)= Π(bn1/m1c,N(n′));[2,1]

LTm

′

n′

(a( j1

m1

, ·), f2 ⊗ · · · ⊗ fd

)⊗ Tbn1/m1c(f1)

(Π(bn1/m1c,N(n′));[2,1])

T

= Π(bn1/m1c,N(n′));[2,1]

Πm

′

n′

[Dm′

(a( j1

m1

, ·))⊗ Tbn′/m′c(f2 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)

](Πm

′

n′ )T

⊗ Tbn1/m1c(f1)

(Π(bn1/m1c,N(n′));[2,1])

T

= Π(bn1/m1c,N(n′));[2,1](Πm′

n′ ⊗ Ibn1/m1c)

[Dm′

(a( j1

m1

, ·))⊗ Tbn′/m′c(f2 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)

]⊗ Tbn1/m1c(f1)

(Πm

′

n′ ⊗ Ibn1/m1c)T (Π(bn1/m1c,N(n′));[2,1])

T . (4.3)

Using Remark 2.1, Lemma 2.3, Lemma 2.6 and the properties of tensor products and direct sums (see Sec-tion 2.3.1), we obtain[

Dm′(a( j1

m1

, ·))⊗ Tbn′/m′c(f2 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)

]⊗ Tbn1/m1c(f1)

= Dm′(a( j1

m1

, ·))⊗ Tbn′/m′c(f2 ⊗ · · · ⊗ fd)⊗ Tbn1/m1c(f1) ⊕ ON(n′,m′)bn1/m1c

= Π(N(m′),bn1/m1c,N(bn′/m′c));[1,3,2]

[Dm′

(a( j1m1

, ·))⊗ Tbn1/m1c(f1)⊗ Tbn′/m′c(f2 ⊗ · · · ⊗ fd)

]· (Π(N(m′),bn1/m1c,N(bn′/m′c));[1,3,2])

T ⊕ ON(n′,m′)bn1/m1c

= Π(N(m′),bn1/m1c,N(bn′/m′c));[1,3,2]

[Dm′

(a( j1

m1

, ·))⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd)

]· (Π(N(m′),bn1/m1c,N(bn′/m′c));[1,3,2])

T ⊕ ON(n′,m′)bn1/m1c

= (Π(N(m′),bn1/m1c,N(bn′/m′c));[1,3,2] ⊕ IN(n′,m′)bn1/m1c)

·[Dm′

(a( j1

m1

, ·))⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)bn1/m1c

]· (Π(N(m′),bn1/m1c,N(bn′/m′c));[1,3,2] ⊕ IN(n′,m′)bn1/m1c)

T . (4.4)

Substituting (4.4) into (4.3), we arrive at

Tbn1/m1c(f1)⊗ LTm′n′

(a( j1

m1

, ·), f2 ⊗ · · · ⊗ fd

)= Pmn

[Dm′

(a( j1

m1

, ·))⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)bn1/m1c

](Pmn )T , (4.5)

where Pmn = Π(bn1/m1c,N(n′));[2,1](Πm′

n′ ⊗Ibn1/m1c)(Π(N(m′),bn1/m1c,N(bn′/m′c));[1,3,2]⊕IN(n′,m′)bn1/m1c). Combining(4.5) and (4.1), we obtain

LTmn (a, f1 ⊗ · · · ⊗ fd)

=( m1⊕j1=1

Pmn

)diag

j1=1,...,m1

[Dm′

(a( j1

m1

, ·))⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)bn1/m1c

]( m1⊕j1=1

Pmn

)T⊕ O(n1 modm1)n2···nd .

39

From Lemma 2.5,

diagj1=1,...,m1

[Dm′

(a( j1

m1

, ·))⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)bn1/m1c

]=

m1⊕j1=1

[Dm′

(a( j1

m1

, ·))⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)bn1/m1c

]

= Vmn

[m1⊕j1=1

[Dm′

(a( j1

m1

, ·))⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd)

]⊕ ON(n′,m′)bn1/m1cm1

](Vmn )T

= Vmn[Dm(a)⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)bn1/m1cm1

](Vmn )T ,

where

Vmn = Vh(m,n);σ,

σ = [1,m1 + 1, 2,m1 + 2, . . . ,m1, 2m1],

h(m,n) = (N(m′)N(bn/mc), N(n′,m′)bn1/m1c︸ ︷︷ ︸1

, . . . , N(m′)N(bn/mc), N(n′,m′)bn1/m1c︸ ︷︷ ︸m1

).

Thus,

LTmn (a, f1 ⊗ · · · ⊗ fd)

=( m1⊕j1=1

Pmn

)Vmn

[Dm(a)⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)bn1/m1cm1

](Vmn )T

( m1⊕j1=1

Pmn

)T⊕ O(n1 modm1)n2···nd

=

[( m1⊕j1=1

Pmn

)Vmn ⊕ I(n1 modm1)n2···nd

]·[Dm(a)⊗ Tbn/mc(f1 ⊗ · · · ⊗ fd) ⊕ ON(n′,m′)bn1/m1cm1+(n1 modm1)n2···nd

]·

[(Vmn )T

( m1⊕j1=1

Pmn

)T⊕ I(n1 modm1)n2···nd

].

This concludes the proof; note that the permutation matrix Πmn is given by

Πmn =( m1⊕j1=1

Pmn

)Vmn ⊕ I(n1 modm1)n2···nd

and, moreover, N(n′,m′)bn1/m1cm1 + (n1 modm1)n2 · · ·nd = N(n)−N(m)N(bn/mc).

As a consequence of Theorem 4.1, we can extend Definition 4.1 in the following way.

Definition 4.2. Letm,n ∈ Nd, let a : [0, 1]d → C and let f ∈ L1([−π, π]d). Then, we define

LTmn (a, f) = Πmn[Dm(a)⊗ Tbn/mc(f) ⊕ ON(n)−N(m)N(bn/mc)

](Πmn )T ,

where Πmn is the permutation matrix appearing in Theorem 4.1.

Remark 4.1. Suppose that f = f1 ⊗ · · · ⊗ fd a.e., with f1, . . . , fd ∈ L1([−π, π]); then LTmn (a, f) is equal toLTmn (a, f1⊗ · · · ⊗ fd), as defined by Definition 4.1. Note also that LTmn (a, f) = LTmn (a, g) whenever f = g a.e.

40

4.1.1 Properties of LTmn (a, f)

We now derive a lot of interesting properties of LTmn (a, f) that we shall use in the study of LT, sLT and GLTsequences. We first not that, for any n,m ∈ Nd and any pair of functions a : [0, 1]d → C and f ∈ L1([−π, π]d),

[LTmn (a, f)]∗ = LTmn (a, f). (4.6)

This follows from Definition 4.2, from the relations (X⊗Y )∗ = X∗⊗Y ∗, (X⊕Y )∗ = X∗⊕Y ∗ (see Section 2.3.1),and from the equality [Tk(f)]∗ = Tk(f) (see Section 2.5).

Proposition 4.1. Letm,n ∈ Nd, let a : [0, 1]d → C and let f ∈ L1([−π, π]d). Then,

‖LTmn (a, f)‖ = ‖Dm(a)‖ ‖Tbn/mc(f)‖ = maxj=1,...,m

∣∣∣a( jm

)∣∣∣‖Tbn/mc(f)‖, (4.7)

‖LTmn (a, f)‖p = ‖Dm(a)‖p ‖Tbn/mc(f)‖p =

(m∑j=1

∣∣∣∣a( jm)∣∣∣∣p)1/p

‖Tbn/mc(f)‖p, 1 ≤ p <∞. (4.8)

Proof. Use the definition of LTmn (a, f), the invariance of ‖ · ‖ and ‖ · ‖p by unitary transformations (such aspermutations), and the equalities (2.11)–(2.12).

We denote by C[0,1]d the vector space of all functions a : [0, 1]d → C.

Proposition 4.2. Letm,n ∈ Nd. Then, the operator

LTmn (a, ·) : L1([−π, π]d)→ CN(n)×N(n)

is linear for any a : [0, 1]d → C, and the operator

LTmn (·, f) : C[0,1]d → CN(n)×N(n)

is linear for any f ∈ L1([−π, π]d).

By Holder’s inequality [37], if f ∈ Lp([−π, π]d) and f ∈ Lq([−π, π]d), where 1 ≤ p, q ≤ ∞ are conjugateexponents, then ff ∈ L1([−π, π]d). In this case, for any a, a : [0, 1]d → C, we can consider the three matricesLTmn (a, f), LTmn (a, f) and LTmn (aa, f f). In Proposition 4.3 we show that LTmn (a, f)LTmn (a, f) is ‘close’ toLTmn (aa, f f).

Proposition 4.3. Let n,m ∈ Nd, let a, a : [0, 1]d → C be bounded, and let f ∈ Lp([−π, π]d) and f ∈Lq([−π, π]d), where 1 ≤ p, q ≤ ∞ are conjugate exponents. Then

‖LTmn (a, f)LTmn (a, f)− LTmn (aa, f f)‖1 ≤ ε(bn/mc)N(n), (4.9)

where

ε(k) = ‖aa‖∞‖Tk(f)Tk(f)− Tk(ff)‖1

N(k)

and limk→∞

ε(k) = 0 by Lemma 2.7. In particular, for everym ∈ Nd there exists nm ∈ Nd such that, for n ≥ nm,

‖LTmn (a, f)LTmn (a, f)− LTmn (aa, f f)‖1 ≤N(n)

N(m), (4.10)

LTmn (a, f)LTmn (a, f) = LTmn (aa, f f) +Rn,m +Nn,m, rank(Rn,m) ≤ N(n)√N(m)

, ‖Nn,m‖ ≤1√N(m)

.

(4.11)

41

Proof. By Definition 4.2 and the properties of tensor products and direct sums,

LTmn (a, f)LTmn (a, f)− LTmn (aa, f f)

= Πmn

[Dm(aa)⊗

(Tbn/mc(f)Tbn/mc(f)− Tbn/mc(ff)

)⊕ O

](Πmn )T .

Hence,

‖LTmn (a, f)LTmn (a, f)− LTmn (aa, f f)‖1 = ‖Dm(aa)‖1 ‖Tbn/mc(f)Tbn/mc(f)− Tbn/mc(ff)‖1

≤ N(n)‖aa‖∞‖Tbn/mc(f)Tbn/mc(f)− Tbn/mc(ff)‖1

N(bn/mc),

and (4.9) is proved. Since ε(k) → 0 when k → ∞, for every m ∈ Nd there exists nm ∈ Nd such that, forn ≥ nm, (4.10) holds; (4.11) follows from (4.10) and Lemma 3.2.

The next two theorems provide information about the asymptotic singular value and eigenvalue distribution ofa finite sum of the form

∑pi=1 LT

mn (ai, fi). Together with Theorems 3.3 and 3.5, they play a central role in the

computation of the singular value and eigenvalue distribution of GLT sequences.

Theorem 4.2. Let m ∈ Nd, let a1, . . . , ap : [0, 1]d → C and let f1, . . . , fp ∈ L1([−π, π]d). Then, for everyF ∈ Cc(R),

limn→∞

1

N(n)

N(n)∑r=1

F(σr

( p∑i=1

LTmn (ai, fi)))

= φm(F ) =1

N(m)

m∑j=1

1

(2π)d

∫[−π,π]d

F(∣∣∣ p∑

i=1

ai

( jm

)fi(θ)

∣∣∣)dθ.(4.12)

Moreover, if a1, . . . , ap are Riemann-integrable, then, for every F ∈ Cc(R),

limm→∞

φm(F ) = φ(F ) =1

(2π)d

∫[0,1]d×[−π,π]d

F(∣∣∣ p∑

i=1

ai(x)fi(θ)∣∣∣)dxdθ. (4.13)

Proof. By Definition 4.2,

(Πmn )T

(p∑i=1

LTmn (ai, fi)

)Πmn =

(p∑i=1

Dm(ai)⊗ Tbn/mc(fi)

)⊕ ON(n)−N(m)N(bn/mc).

The j-th block of this matrix, 1 ≤ j ≤m, is given by

p∑i=1

ai

( jm

)Tbn/mc(fi) = Tbn/mc

( p∑i=1

ai

( jm

)fi

).

It follows that the singular values of∑p

i=1 LTmn (ai, fi) are

σk

(Tbn/mc

( p∑i=1

ai

( jm

)fi

)), k = 1, . . . , N(bn/mc), j = 1, . . . ,m,

42

plus further N(n)−N(m)N(bn/mc) singular values equal to 0. Note that N(n)−N(m)N(bn/mc)N(n)

→ 0 as n→∞.Therefore, for any F ∈ Cc(R),

limn→∞

1

N(n)

N(n)∑r=1

F(σr

( p∑i=1

LTmn (ai, fi)))

= limn→∞

N(m)N(bn/mc)N(n)

1

N(m)

m∑j=1

1

N(bn/mc)

N(bn/mc)∑k=1

F(σk

(Tbn/mc

( p∑i=1

ai

( jm

)fi

)))=

1

N(m)

m∑j=1

1

(2π)d

∫[−π,π]d

F(∣∣∣ p∑

i=1

ai

( jm

)fi(θ)

∣∣∣)dθ, (4.14)

where the latter equality is due to Theorem 2.5. This proves (4.12).If a1, . . . , ap are Riemann-integrable, then the function x 7→ F (|

∑pi=1 ai (x) fi(θ)|) is Riemann-integrable for

each fixed θ ∈ [−π, π]d, and so

limm→∞

1

N(m)

m∑j=1

F(∣∣∣ p∑

i=1

ai

( jm

)fi(θ)

∣∣∣) =

∫[0,1]d

F(∣∣∣ p∑

i=1

ai(x)fi(θ)∣∣∣)dx.

Passing to the limit form→∞ in (4.14) and using the dominated convergence theorem, we get (4.13).

Theorem 4.3. Let m ∈ Nd, let a1, . . . , ap : [0, 1]d → C and let f1, . . . , fp ∈ L1([−π, π]d). Then, for everyF ∈ Cc(C),

limn→∞

1

N(n)

N(n)∑r=1

F(λr

(<( p∑i=1

LTmn (ai, fi))))

= φm(F )

=1

N(m)

m∑j=1

1

(2π)d

∫[−π,π]d

F(<( p∑i=1

ai

( jm

)fi(θ)

))dθ. (4.15)

Moreover, if a1, . . . , ap are Riemann-integrable, then, for every F ∈ Cc(C),

limm→∞

φm(F ) = φ(F ) =1

(2π)d

∫[0,1]d×[−π,π]d

F(<( p∑i=1

ai(x)fi(θ)))dxdθ. (4.16)

Proof. The proof follows the same pattern as the proof of Theorem 4.2. By (4.6) and Definition 4.2,

(Πmn )T

(<( p∑i=1

LTmn (ai, fi)))

Πmn = (Πmn )T

(1

2

(p∑i=1

LTmn (ai, fi) +

p∑i=1

LTmn (ai, fi)

))Πmn

=1

2

(p∑i=1

Dm(ai)⊗ Tbn/mc(fi) +

p∑i=1

Dm(ai)⊗ Tbn/mc(fi)

)⊕ ON(n)−N(m)N(bn/mc).

The j-th block of this matrix, 1 ≤ j ≤m, is given by

1

2

(p∑i=1

ai

( jm

)Tbn/mc(fi) +

p∑i=1

ai

( jm

)Tbn/mc(fi)

)= Tbn/mc

(<( p∑i=1

ai

( jm

)fi

)).

43

It follows that the eigenvalues of < (∑p

i=1 LTmn (ai, fi)) are

λk

(Tbn/mc

(<( p∑i=1

ai

( jm

)fi

))), k = 1, . . . , N(bn/mc), j = 1, . . . ,m,

plus further N(n)−N(m)N(bn/mc) eigenvalues equal to 0. Therefore, for any F ∈ Cc(C),

limn→∞

1

N(n)

N(n)∑r=1

F(λr

(<( p∑i=1

LTmn (ai, fi))))

limn→∞

=N(m)N(bn/mc)

N(n)

1

N(m)

m∑j=1

1

N(bn/mc)

N(bn/mc)∑k=1

F(λk

(Tbn/mc

(<( p∑i=1

ai

( jm

)fi

))))=

1

N(m)

m∑j=1

1

(2π)d

∫[−π,π]d

F(<( p∑i=1

ai

( jm

)fi(θ)

))dθ, (4.17)

where the latter equality follows from Theorem 2.5. This proves (4.15).If a1, . . . , ap are Riemann-integrable, then the function x 7→ F (< (

∑pi=1 ai (x) fi(θ))) is Riemann-integrable

for each fixed θ ∈ [−π, π]d, and so

limm→∞

1

N(m)

m∑j=1

F(<( p∑i=1

ai

( jm

)fi(θ)

))=

∫[0,1]d

F(<( p∑i=1

ai(x)fi(θ)))dx.

Passing to the limit asm→∞ in (4.17) and using the dominated convergence theorem, we get (4.16).

4.2 LT and sLT sequencesFirst we start with the definitions and basic examples, including Toeplitz and diagonal sampling matrix-sequences,which represent the building blocks for approximating PDE matrix-sequences. Then we illustrate the main prop-erties of LT and sLT sequences.

4.2.1 Definition and basic examples: Toeplitz and diagonal sampling matrix-sequences

Definition 4.3 (LT sequence). Let Ann be a matrix-sequence, with n ∈ Nd. We say that Ann is a LocallyToeplitz (LT) sequence if there exist

- a Riemann-integrable function a : [0, 1]d → C,

- a function f ∈ L1([−π, π]d),

such that LTmn (a, f)nm is an a.c.s. for Ann, in the following sense: for allm ∈ Nd there is nm ∈ Nd suchthat, for n ≥ nm,

An = LTmn (a, f) +Rn,m +Nn,m, (4.18)

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

where the quantities nm, c(m), ω(m) are independent of n, and limm→∞ c(m) = limm→∞ ω(m) = 0. In thiscase, we write Ann ∼LT a⊗ f . The function a⊗ f is referred to as the symbol of the sequence Ann, a is theweight function and f is the generating function.3

3We refer the reder to the introduction of Tilli’s paper [54] for the origin and the meaning of this terminology.

44

Remark 4.2. An equivalent definition of LT sequence is obtained by replacing, in Definition 4.3, ‘for allm ∈ Nd’with ‘for all sufficiently large m ∈ Nd’ (i.e., ‘for every m greater than or equal to some m ∈ Nd’). Indeed,suppose that the splitting (4.18) and the related conditions on Rn,m and Nn,m hold for m ≥ m; then, definingnm = 1, c(m) = 1, ω(m) = 0 and Rn,m = An,m−Bn,m, Nn,m = O for all the other values ofm, we see thatthey actually hold for everym ∈ Nd.

Remark 4.3. If LTmn (a, f)nm is an a.c.s. for Ann in the sense of Definition 4.3, then LTmn (a, f)nmis an a.c.s. for Ann (in the sense of the classical Definition 3.1) for all sequences m = m(m)m such thatm→∞ when m→∞.

Remark 4.4. Suppose that Bn,mnm is an a.c.s. for Ann (in the sense of Definition 4.3) and B′n,mnmis an a.c.s. for A′nn (in the sense of Definition 4.3). Then:

1. B∗n,mnm is an a.c.s. for A∗nn (in the sense of Definition 4.3);

2. αBn,m + βB′n,mnm is an a.c.s. for αAn + βA′nn (in the sense of Definition 4.3) for all α, β ∈ C;

3. if Ann and A′nn are s.u., then Bn,mB′n,mnm is an a.c.s. for AnA′nn (in the sense of Defini-tion 4.3).

The proof of these results is omitted, because it is essentially the same as the proof of the analogous results forstandard a.c.s.; see Remark 3.2 and Propositions 3.1–3.4.

Definition 4.4 (sLT sequence). Let Ann be a matrix-sequence, with n ∈ Nd. We say that Ann is a separableLocally Toeplitz (sLT) sequence if there exist

- a Riemann-integrable function a : [0, 1]d → C,

- a separable function f ∈ L1([−π, π]d),

such that Ann ∼LT a⊗ f . In this case, we write Ann ∼sLT a⊗ f .

It is clear from the definition that a sLT sequence is just a LT sequence with separable generating functionf . From now on, if a matrix-sequence Ann is given (with n ∈ Nd) and if we write Ann ∼LT a ⊗ f (resp.Ann ∼sLT a ⊗ f ), it is understood that a : [0, 1]d → C is Riemann-integrable and f ∈ L1([−π, π]d) (resp.f ∈ L1([−π, π]d) is separable).

Remark 4.5. Suppose that A(i)n n ∼LT a ⊗ fi, i = 1, . . . , r. Then

∑ri=1 A

(i)n n ∼LT a ⊗ (

∑ri=1 fi). Suppose

that A(i)n n ∼LT ai ⊗ f, i = 1, . . . , r. Then

∑ri=1A

(i)n n ∼LT (

∑ri=1 ai) ⊗ f . The proof of these results relies

on the linearity of the Locally Toeplitz operator with respect to both its arguments (see Proposition 4.2); we leaveit as an exercise for the reader.

Let us now provide basic examples of LT sequences: sequences distributed in the singular value sense as thezero function, sequences formed by multilevel diagonal sampling matrices, and sequences of multilevel Toeplitzmatrices. These may be regarded as the building blocks of the theory of GLT sequences, because, starting fromthem, we can construct through algebraic operations a lot of other matrix-sequences which will be seen in Section 5to be GLT sequences.

Theorem 4.4. Assume that a sequence Ann is given, An of size dn. Then the following are equivalent

• Ann ∼σ 0;

• Onnm is an a.c.s. for Ann.

45

Under the additional assumption that dn = N(n), the previous statements are equivalent to the following

• Ann ∼sLT a⊗ f with a⊗ f ≡ 0;

• Ann ∼LT a⊗ f with a⊗ f ≡ 0;

Proof. Let m > 0 and let φ(m,n) the cardinality of the singular values of An which are bounded by 1/m. SinceAnn ∼σ 0 there exist a function c(m) such that limm→0 c(m) = 0 and φ(n,m) ≤ c(m)dn where dn the size ofAn. Now take the singular value decomposition [30] of An. We have

An = UnΣnVn

with Un, Vn unitary and Σn diagonal containing the singular values. We split Σn as Σn,m,<1/m + Σn,m,≥1/m,where the diagonal matrix Σn,m,<1/m contains the singular values less than 1/m in the same position as Σn andzero otherwise. Because Un, Vn are unitary we deduce

An = Nn,m +Rn,m, Nn,m = UnΣn,m,<1/mVn, Rn,m = UnΣn,m,≥1/mVn

with‖Nn,m‖ < 1/m, rank(Rn,m) ≤ c(m)dn.

By definition of a.c.s., the latter shows that Onnm is an a.c.s. for Ann. Now consider dn = N(n). Byobserving that a ⊗ f ≡ 0 is equivalent to the fact that LTmn (a, f) is the null matrix, the equivalence of latter twostatements to the previous is plain.

Remark 4.6. Taking into consideration the standard notion of clustering [56], the statements in Lemma 4.4 arealso equivalent to write that the sequence Ann is clustered at zero.

For proving Theorems 4.5–4.6, we will need the following technical lemma, whose proof may be skipped in afirst reading.

Lemma 4.1. Let N be an infinite subset of N. Let x(·, ·) : N× Nd → R be any function satisfying

limm→∞

limh→∞

x(m,h) = ξ ∈ R.

Then, there exists a function m(·) : Nd → N such that m(h)→∞ and x(m(h),h)→ ξ when h→∞.

Proof. Let N = m1,m2, . . .; we denote by m+ (m−) the successor (predecessor) of m in N. Set

x(m) = limh→∞

x(m,h), m ∈ N.

Since x(m) → ξ by assumption, x(m) is eventually a real number (different from −∞ or +∞); suppose, forinstance, that x(m) ∈ R for all m ≥ mr. We construct an injective function h(·) : mr,mr+1, . . . → Nd asfollows: we set h(mr) = 1 and, for m ∈ N\mr, we choose h(m) > h(m−) such that, for h ≥ h(m),

|x(m,h)− x(m)| ≤ 1

m⇒ |x(m,h)− ξ| ≤ |x(m)− ξ|+ 1

m.

Hence, we have a sequence

1 = h(mr) < h(mr+1) < h(mr+2) < h(mr+3) < . . .

and, if m ≥ mr and h ≥ h(m),

|x(m,h)− ξ| ≤ |x(m)− ξ|+ 1

m. (4.19)

46

In view of this, we define m(·) : Nd → N as follows:

m(h) =

mr if h ∈ j ∈ Nd : j ≥ h(mr) = 1\j ∈ Nd : j ≥ h(mr+1),mr+1 if h ∈ j ∈ Nd : j ≥ h(mr+1)\j ∈ Nd : j ≥ h(mr+2),mr+2 if h ∈ j ∈ Nd : j ≥ h(mr+2)\j ∈ Nd : j ≥ h(mr+3),...

...

Then, when h→∞, we have m(h)→∞. Moreover, noting that m(h) ≥ mr and h ≥ h(m(h)) for all h ∈ Nd,by (4.19) we also have |x(m(h),h)− ξ| ≤ |x(m(h))− ξ|+ 1/m(h)→ 0.

Theorem 4.5. Let a : [0, 1]d → C be Riemann-integrable and consider the sequence of matrices Dn(a)n, wheren ∈ Nd and, of course, n→∞ as n→∞. Then Dn(a)n ∼sLT a⊗ 1.

Proof. The proof is organized in two steps: we first show by induction on d that the thesis holds if a is continuous;then, by using an approximation argument, we show that it holds even in the case where a is any Riemann-integrable function. As we shall see, the approximation argument heavily relies on the Riemann-integrability of a.

1. We prove by induction on d that, if a ∈ C([0, 1]d), then

Dn(a) = LTmn (a, 1) +Rn,m +Nn,m, rank(Rn,m) ≤ N(n)d∑i=1

mi

ni, ‖Nn,m‖ ≤

d∑i=1

ωa

( 1

mi

+mi

ni

), (4.20)

where ωa is the modulus of continuity of a:

ωa : (0,∞)→ R, ωa(δ) = maxx,y∈[0,1]d

‖x−y‖≤δ

|a(x)− a(y)|.

Since ωa(δ) → 0 as δ → 0, Eq. (4.20) implies that the thesis holds for any continuous function a ∈ C([0, 1]d); itsuffices to take, in Definition 4.3, nm = m2, c(m) =

∑di=1(1/mi) and ω(m) =

∑di=1 ωa(2/mi).

For the case d = 1, LTmn (a, 1) is a n× n diagonal matrix given by

LTmn (a, 1) = Dm(a)⊗ Ibn/mc ⊕ Onmodm = a(1/m)Ibn/mc ⊕ a(2/m)Ibn/mc ⊕ · · · ⊕ a(1)Ibn/mc ⊕Onmodm.

For every i = 1, . . . ,mbn/mc, let j = j(i) be the index in 1, . . . ,m such that (j−1)bn/mc+1 ≤ i ≤ jbn/mc.We have

|[LTmn (a, 1)]ii − [Dn(a)]ii| = |a(j/m)− a(i/n)| ≤ ωa(1/m+m/n),

because ∣∣∣∣ jm − i

n

∣∣∣∣ ≤ j

m− (j − 1)bn/mc

n≤ j

m− (j − 1)(n/m− 1)

n=

1

m+j − 1

n≤ 1

m+m

n. (4.21)

Define the following n× n diagonal matrices:

Dn,m(a) = diagi=1,...,mbn/mc

a(i/n) ⊕ Onmodm, Dn,m(a) = Ombn/mc ⊕ diagi=mbn/mc+1,...,n

a(i/n).

Then,Dn(a)− LTmn (a, 1) = Dn,m(a) + Dn,m(a)− LTmn (a, 1) = Rn,m +Nn,m,

where Rn,m = Dn,m(a) and Nn,m = Dn,m(a)− LTmn (a, 1) satisfy

rank(Rn,m) ≤ nmodm ≤ m, ‖Nn,m‖ = maxi=1,...,mbn/mc

|[LTmn (a, 1)]ii − [Dn(a)]ii| ≤ ωa(1/m+m/n).

47

This shows that (4.20) holds for d = 1.For the case d > 1, LTmn (a, 1) is a N(n)×N(n) diagonal matrix given by

LTmn (a, 1) = diagj1=1,...,m1

Ibn1/m1c ⊗ LTm2,...,mdn2,...,nd

(a( j1

m1

, ·), 1)⊕ O(n1 modm1)n2···nd , (4.22)

where, for any x1 ∈ [0, 1], a(x1, ·) : [0, 1]d−1 → C is the function (x2, . . . , xd) 7→ a(x1, x2, . . . , xd). For everyj1 = 1, . . . ,m1 and every i1 = (j1 − 1)bn1/m1c+ 1, . . . , j1bn1/m1c, by induction hypothesis we have

LTm2,...,mdn2,...,nd

(a( j1

m1

, ·), 1)−Dn2,...,nd

(a( i1n1

, ·))

=

[Dn2,...,nd

(a( j1

m1

, ·))−Dn2,...,nd

(a( i1n1

, ·))]

+R[j1/m1]n2,...,nd,m2,...,md

+N [j1/m1]n2,...,nd,m2,...,md

,

where

rank(R[j1/m1]n2,...,nd,m2,...,md

) ≤ n2 · · ·ndd∑

k=2

mk

nk, ‖N [j1/m1]

n2,...,nd,m2,...,md‖ ≤

d∑k=2

ωa

( 1

mk

+mk

nk

).

Moreover, ∥∥∥∥Dn2,...,nd

(a( j1

m1

, ·))−Dn2,...,nd

(a( i1n1

, ·))∥∥∥∥ ≤ ωa

( 1

m1

+m1

n1

),

because, as in (4.21), one can show that ∣∣∣∣ j1m1

− i1n1

∣∣∣∣ ≤ 1

m1

+m1

n1

.

Thus,

LTm2,...,mdn2,...,nd

(a( j1m1

, ·), 1)−Dn2,...,nd

(a( i1n1

, ·))

= R[j1/m1]n2,...,nd,m2,...,md

+N [j1/m1, i1/n1]n,m

rank(R[j1/m1]n2,...,nd,m2,...,md

) ≤ n2 · · ·ndd∑

k=2

mk

nk, ‖N [j1/m1, i1/n1]

n,m ‖ ≤d∑

k=1

ωa

( 1

mk

+mk

nk

). (4.23)

Now we observe that the diagonal matrices LTmn (a, 1) and Dn(a) can be written as

LTmn (a, 1) = diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1cLTm2,...,md

n2,...,nd

(a( j1

m1

, ·), 1)]⊕ O(n1 modm1)n2···nd ,

Dn(a) = diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1cDn2,...,nd

(a( i1n1

, ·))]

⊕ diagi1=m1bn1/m1c+1,...,n1

Dn2,...,nd

(a( i1n1

, ·))

;

48

see (4.22) and (2.2). Hence,

Dn(a)− LTmn (a, 1)

= diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1c

[Dn2,...,nd

(a( i1n1

, ·))− LTm2,...,md

n2,...,nd

(a( j1

m1

, ·), 1)]]

⊕ diagi1=m1bn1/m1c+1,...,n1

Dn2,...,nd

(a( i1n1

, ·))

= diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1c

[R[j1/m1]n2,...,nd,m2,...,md

+N [j1/m1, i1/n1]n,m

]]⊕ diag

i1=m1bn1/m1c+1,...,n1

Dn2,...,nd

(a( i1n1

, ·))

= Rn,m +Nn,m,

where

Rn,m = diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1cR[j1/m1]n2,...,nd,m2,...,md

]⊕ diag

i1=m1bn1/m1c+1,...,n1

Dn2,...,nd

(a( i1n1

, ·)),

Nn,m = diagj1=1,...,m1

[diag

i1=(j1−1)bn1/m1c+1,...,j1bn1/m1cN [j1/m1, i1/n1]n,m

]⊕ O(n1 modm1)n2···nd .

By (4.23) and (2.11), (2.13), we have

rank(Rn,m) ≤ m1

⌊n1

m1

⌋n2 · · ·nd

d∑k=2

mk

nk+ (n1 modm1)n2 · · ·nd

≤ n1n2 · · ·ndd∑

k=2

mk

nk+m1n2 · · ·nd = N(n)

d∑k=1

mk

nk,

‖Nn,m‖ ≤d∑

k=1

ωa

( 1

mk

+mk

nk

),

and (4.20) is proved.

2. Let a : [0, 1]d → C be any Riemann-integrable function. Take any sequence of continuous functionsam : [0, 1]d → C such that am → a in L1([0, 1]d). Note that such a sequence exists because C([0, 1]d) is dense inL1([0, 1]d); see [37]. By the first part of the proof, Dn(am)n ∼sLT am ⊗ 1. Hence, for each m and each h ∈ Nd

there is nm,h such that, for n ≥ nm,h,

Dn(am) = LThn (am, 1) +Rn,m,h +Nn,m,h,

rank(Rn,m,h) ≤ c(m,h)N(n), ‖Nn,m,h‖ ≤ ω(m,h),

wherelimh→∞

c(m,h) = limh→∞

ω(m,h) = 0.

Moreover, Dn(am)nm is an a.c.s. for Dn(a)n. Indeed,

‖Dn(a)−Dn(am)‖1 = N(n)1

N(n)

n∑j=1

∣∣∣∣a( jn)− am( jn)∣∣∣∣ .

49

By the Riemann-integrability of |a − am|, which follows from the Riemann-integrability of a and am, and by thefact that am → a in L1([0, 1]d), the quantity

ε(m,n) =1

N(n)

n∑j=1

∣∣∣∣a( jn)− am( jn)∣∣∣∣

satisfieslimm→∞

limn→∞

ε(m,n) = limm→∞

∫[0,1]d|a(x)− am(x)|dx = lim

m→∞‖a− am‖L1 = 0.

By Corollary 3.3, this implies that Dn(am)nm is an a.c.s. for Dn(a)n. Thus, for every m there exists nmsuch that, for n ≥ nm,

Dn(a) = Dn(am) +Rn,m +Nn,m,

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

wherelimm→∞

c(m) = limm→∞

ω(m) = 0.

It follows that, for every m, every h ∈ Nd and every n ≥ max(nm,nm,h),

Dn(a) = LThn (a, 1) +[LThn (am, 1)− LThn (a, 1)

]+ (Rn,m +Rn,m,h) + (Nn,m +Nn,m,h),

rank(Rn,m +Rn,m,h) ≤ (c(m) + c(m,h))N(n),

‖Nn,m +Nn,m,h‖ ≤ ω(m) + ω(m,h),∥∥LThn (am, 1)− LThn (a, 1)∥∥

1≤ N(n)

N(h)

h∑j=1

∣∣∣∣a( jh)− am( jh)∣∣∣∣ = ε(m,h)N(n).

Choose, for every h ∈ Nd, a m(h) such that m(h)→∞ when h→∞ and

limh→∞

ε(m(h),h) = limh→∞

c(m(h),h) = limh→∞

ω(m(h),h) = 0.

An explicit construction of such a function m(h) has been given in Lemma 4.1; apply the lemma with x(m,h) =ε(m,h) + c(m,h) + ω(m,h). Then, for every h ∈ Nd and every n ≥ max(nm(h),nm(h),h),

Dn(a) = LThn (a, 1) +[LThn (am(h), 1)− LThn (a, 1)

]+ (Rn,m(h) +Rn,m(h),h) + (Nn,m(h) +Nn,m(h),h),

rank(Rn,m(h) +Rn,m(h),h) ≤ (c(m(h)) + c(m(h),h))N(n),

‖Nn,m(h) +Nn,m(h),h‖ ≤ ω(m(h)) + ω(m(h),h),∥∥LThn (am(h), 1)− LThn (a, 1)∥∥

1≤ ε(m(h),h)N(n).

The application of Lemma 3.2 allows one to decompose LThn (am(h), 1)−LThn (a, 1) as the sum of a small-rank termRn,h, with rank bounded from above by

√ε(m(h),h)N(n), plus a small-norm term Nn,h, with norm bounded

from above by√ε(m(h),h). This concludes the proof.

Theorem 4.6. Let f ∈ L1([−π, π]d) and consider the sequence of matrices Tn(f)n, where n ∈ Nd and, ofcourse, n→∞ as n→∞. Then Tn(f)n ∼LT 1⊗ f .

Proof. The proof is organized in three steps: we first show by induction on d that the thesis holds if f is a separabled-variate trigonometric polynomial; then, by linearity, we show that it also holds if f is an arbitrary d-variatetrigonometric polynomial; finally, using an approximation argument, we prove the theorem in its full generality,when f is only assumed to be in L1([−π, π]d).

50

1. We show by induction on d that, if f is a separable d-variate trigonometric polynomial, say f = f1⊗· · ·⊗fdwith f1, . . . , fd univariate trigonometric polynomials, then

Tn(f) = LTmn (1, f) +Rn,m, rank(Rn,m) ≤ N(n)d∑i=1

(2ri + 1)mi

ni, (4.24)

where ri is the degree of fi. From (4.24), it follows that the theorem holds for any separable trigonometricpolynomial f ; it suffices to take, in Definition 4.3, nm = m2, c(m) =

∑di=1(2ri + 1)/mi and ω(m) = 0.

For the case d = 1, let f =∑r

j=−r fjeijθ. Then

LTmn (1, f) = Im ⊗ Tbn/mc(f) ⊕ Onmodm = Tbn/mc(f)⊕ · · · ⊕ Tbn/mc(f)⊕Onmodm.

Looking carefully at the structure of Tn(f) and LTmn (1, f), we see that the number of nonzero rows of the differ-ence Tn(f)− LTmn (1, f) is at most 2rm− r + (nmodm). Hence,

Tn(f) = LTmn (1, f) +Rn,m, rank(Rn,m) ≤ 2rm− r + (nmodm) ≤ (2r + 1)m, (4.25)

and so (4.24) holds for d = 1.For the case d > 1, by induction hypothesis we have

LTm2,...,mdn2,...,nd

(1, f2 ⊗ · · · ⊗ fd) = Tn2,...,nd(f2 ⊗ · · · ⊗ fd) +Rn2,...,nd,m2,...,md ,

rank(Rn2,...,nd,m2,...,md) ≤ n2 · · ·ndd∑i=2

(2ri + 1)mi

ni.

From the definition of LTmn (1, f) and the properties of tensor products and direct sums (see Section 2.3.1), weobtain

LTmn (1, f) = diagj1=1,...,m1

Tbn1/m1c(f1)⊗ LTm2,...,mdn2,...,nd

(1, f2 ⊗ · · · ⊗ fd) ⊕ O(n1 modm1)n2···nd

=

[diag

j1=1,...,m1

Tbn1/m1c(f1)

]⊗ [Tn2,...,nd(f2 ⊗ · · · ⊗ fd) +Rn2,...,nd,m2,...,md ] ⊕ O(n1 modm1)n2···nd

=

[diag

j1=1,...,m1

Tbn1/m1c(f1) ⊕ On1 modm1

]⊗ [Tn2,...,nd(f2 ⊗ · · · ⊗ fd) +Rn2,...,nd,m2,...,md ]

= LTm1n1

(1, f1)⊗ [Tn2,...,nd(f2 ⊗ · · · ⊗ fd) +Rn2,...,nd,m2,...,md ]

= LTm1n1

(1, f1)⊗ Tn2,...,nd(f2 ⊗ · · · ⊗ fd) +Rn1,...,nd,m1,...,md ,

where Rn1,...,nd,m1,...,md = Lm1n1

(1, f1)⊗Rn1,...,nd,m1,...,md satisfies

rank(Rn1,...,nd,m1,...,md) ≤ N(n)d∑i=2

(2ri + 1)mi

ni.

Using (4.25), we can decompose LTm1n1

(1, f1) as the sum of Tn1(f1) plus a small-rank matrix Rn1,m1 , whose rankis bounded by (2r1 + 1)m1. Thus, recalling Lemma 2.6, we arrive at

LTmn (1, f) = (Tn1(f1) +Rn1,m1)⊗ Tn2,...,nd(f2 ⊗ · · · ⊗ fd) +Rn1,...,nd,m1,...,md = Tn(f) +Rn,m,

where Rn,m = Rm1n1⊗ Tn2,...,nd(f2 ⊗ · · · ⊗ fd) +Rn1,...,nd,m1,...,md satisfies

rank(Rn,m) ≤ (2r1 + 1)m1n2 · · ·nd +N(n)d∑i=2

(2ri + 1)mi

ni= N(n)

d∑i=1

(2ri + 1)mi

ni.

51

This completes the proof of (4.24).

2. Let f be any d-variate trigonometric polynomial. By definition, f is a finite linear combination of the Fourierfrequencies eij·θ, j ∈ Z, and so we can write f =

∑rj=−r fje

ij·θ for some separable trigonometric polynomialsfjeij·θ. By linearity,

Tn(f) =r∑

j=−r

fjTn(eij·θ), LTmn (1, f) =r∑

j=−r

fjLTmn (1, eij·θ).

By the first part of the proof, Tn(eij·θ)n ∼LT 1⊗ eij·θ, hence LTmn (1, eij·θ)nm is an a.c.s. for Tn(eij·θ)nin the sense of Definition 4.3. It follows that LTmn (1, f)nm is an a.c.s. for Tn(f)n in the sense of Defini-tion 4.3; see Remark 4.4. Thus, Tn(f)n ∼LT 1⊗ f for every trigonometric polynomial f .

3. Let f ∈ L1([−π, π]d). Since the set of d-variate trigonometric polynomials is dense in L1([−π, π]d), thereis a sequence fm of d-variate trigonometric polynomials such that fm → f in L1([−π, π]d). By the second partof the proof, Tn(fm)n ∼LT 1⊗ fm. Hence, for each m and each h ∈ Nd there is nm,h such that, for n ≥ nm,h,

Tn(fm) = LThn (1, fm) +Rn,m,h +Nn,m,h,

rank(Rn,m,h) ≤ c(m,h)N(n), ‖Nn,m,h‖ ≤ ω(m,h),

wherelimh→∞

c(m,h) = limh→∞

ω(m,h) = 0.

Moreover, by Theorem 2.6,

‖Tn(f)− Tn(fm)‖1 = ‖Tn(f − fm)‖1 ≤ N(n)‖f − fm‖L1

and so Tn(fm)nm is an a.c.s. for Tn(f)n by Corollary 3.3. Thus, for every m there exists nm such that, forn ≥ nm,

Tn(f) = Tn(fm) +Rn,m +Nn,m,

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

wherelimm→∞

c(m) = limm→∞

ω(m) = 0.

It follows that, for every m, every h ∈ Nd and every n ≥ max(nm,nm,h),

Tn(f) = LThn (1, f) +[LThn (1, fm)− LThn (1, f)

]+ (Rn,m +Rn,m,h) + (Nn,m +Nn,m,h),

rank(Rn,m +Rn,m,h) ≤ (c(m) + c(m,h))N(n),

‖Nn,m +Nn,m,h‖ ≤ ω(m) + ω(m,h),∥∥LThn (1, fm)− LThn (1, f)∥∥

1=∥∥LThn (1, fm − f)

∥∥1≤ N(n)‖f − fm‖L1 .

Choose, for every h ∈ Nd, a m(h) such that m(h)→∞ when h→∞ and

limh→∞

c(m(h),h) = limh→∞

ω(m(h),h) = 0.

An explicit construction of such a function m(h) is given in Lemma 4.1; apply the lemma with x(m,h) =c(m,h) + ω(m,h). Then, for every h ∈ Nd and every n ≥ max(nm(h),nm(h),h),

Tn(f) = LThn (1, f) +[LThn (1, fm(h))− LThn (1, f)

]+ (Rn,m(h) +Rn,m(h),h) + (Nn,m(h) +Nn,m(h),h),

rank(Rn,m(h) +Rn,m(h),h) ≤ (c(m(h)) + c(m(h),h))N(n),

‖Nn,m(h) +Nn,m(h),h‖ ≤ ω(m(h)) + ω(m(h),h),∥∥LThn (1, fm(h))− LThn (1, f)∥∥

1≤ ‖fm(h) − f‖L1N(n).

52

The application of Lemma 3.2 allows one to decompose LThn (1, fm(h)) − LThn (1, f) as the sum of a small-rankterm Rn,h, with rank bounded from above by

√‖fm(h) − f‖L1 N(n), plus a small-norm term Nn,h, with norm

bounded from above by√‖fm(h) − f‖L1 . This concludes the proof.

It follows from Theorem 4.6 that Tn(f)n ∼sLT 1⊗ f whenever f ∈ L1([−π, π]d) is separable.

4.2.2 Properties of LT and sLT sequences

We begin with a basic spectral result for sLT sequences. It is stated as a lemma, both because it is needed for theproof of Theorem 4.7 and because it will be generalized afterwards, in the more general context of GLT sequences(so, we do not need to remember it). To simplify the presentation, from now on, until the end of this section, it isunderstood that multi-indices are d-indices (i.e., they have length d).

Lemma 4.2. If Ann ∼LT a⊗ f then Ann ∼σ a⊗ f.

Proof. Take any sequence of multi-indices m = m(m)m such that m → ∞ as m → ∞. By Definition 4.3,LTmn (a, f)nm is an a.c.s. for Ann. By Theorem 4.2, for all F ∈ Cc(R),

limn→∞

1

N(n)

N(n)∑r=1

F (σr(LTmn (a, f))) = φm(F ),

wherelimm→∞

φm(F ) = φ(F ) =1

(2π)d

∫[0,1]d×[−π,π]d

F (a(x)f(θ))dxdθ.

Therefore, by Theorem 3.3, Ann ∼σ a⊗ f .

As a consequence of Lemma 4.2 and Proposition 3.3, every LT sequence is s.u. in the sense of Definition 3.2.We now show, under mild assumptions, that the product of LT sequences is again a LT sequence with symbol givenby the product of the symbols.

Theorem 4.7. Suppose thatAnn ∼LT a⊗ f, Ann ∼LT a⊗ f ,

where f ∈ Lp([−π, π]d), f ∈ Lq([−π, π]d), and p, q are conjugate exponents (1 ≤ p, q ≤ ∞). Then

AnAnn ∼LT aa⊗ ff .

Proof. By Lemma 4.2 and Proposition 3.3, every LT sequence is s.u., so in particular Ann and Ann are s.u.Since LTmn (a, f)nm is an a.c.s. for Ann (in the sense of Definition 4.3) and LTmn (a, f)nm is an a.c.s.for Ann (in the sense of Definition 4.3), the product LTmn (a, f)LTmn (a, f)nm is an a.c.s. for AnAnn (inthe sense of Definition 4.3); see Remark 4.4. The thesis now follows from Definition 4.3 and Proposition 4.3.

As a consequence of Theorem 4.7 and Theorems 4.5–4.6, we immediately obtain the following result.

Theorem 4.8. Let a : [0, 1]d → C be Riemann-integrable, let f ∈ L1([−π, π]d), and consider the sequence ofmatrices Dn(a)Tn(f)n, where n ∈ Nd and, of course, n→∞ as n→∞. Then

Dn(a)Tn(f)n ∼LT a⊗ f.

53

Theorem 4.8 shows that, for any a, f as in Definition 4.3, there always exists a matrix-sequence Ann such thatAnn ∼LT a⊗f . Indeed, it suffices to takeAn = Dn(a)Tn(f). Theorems 4.9–4.10 show that the sequences of theform Dn(a)Tn(f)n play a special role in the world of LT sequences. Indeed, Ann ∼LT a⊗f if and only if Anequals Dn(a)Tn(f) up to a small-rank plus small-norm correction. More precisely, any LT sequence Ann ∼LT

a ⊗ f admits Dn(a)Tn(f)n as an a.c.s., and, vice versa, any sequence Ann admitting Dn(a)Tn(f)n as ana.c.s. is a LT sequence with symbol a⊗ f . Moreover, any LT sequence Ann ∼LT a⊗ f also admits an a.c.s. ofthe form Dn(am)Tn(fm)nm, with am continuous and fm trigonometric polynomial; as one could guess, amwill be chosen as an approximation of a, converging to a form→∞, while fm will be chosen as an approximationof f , converging to f for m→∞.

Theorem 4.9. Let Ann ∼LT a⊗ f . Then, for any am, fm, A(m)n nm with the following properties:

∗ am : [0, 1]d → C is Riemann-integrable and am → a in L1([0, 1]d);

∗ fm ∈ L1([−π, π]d) and fm → f in L1([−π, π]d);

∗ A(m)n n ∼LT am ⊗ fm;

it holds that A(m)n nm is an a.c.s. for Ann. In particular, for any am, fm with the above properties,

Dn(am)Tn(fm)nm is an a.c.s. for Ann. Taking am = a and fm = f , we see that Dn(a)Tn(f)nm is ana.c.s. for Ann.

Proof. Let am, fm, A(m)n nm as in the statement of the theorem. Then, for each m and each h ∈ Nd there

is nm,h such that, for n ≥ nm,h,

A(m)n = LThn (am, fm) +Rn,m,h +Nn,m,h,

rank(Rn,m,h) ≤ c(m,h)N(n), ‖Nn,m,h‖ ≤ ω(m,h),

wherelimh→∞

c(m,h) = limh→∞

ω(m,h) = 0.

Moreover, since Ann ∼LT a⊗ f , for every h ∈ Nd there is nh such that, for n ≥ nh,

An = LThn (a, f) +Rn,h +Nn,h,

rank(Rn,h) ≤ c(h)N(n), ‖Nn,h‖ ≤ ω(h),

wherelimh→∞

c(h) = limh→∞

ω(h) = 0.

Hence, for every m, every h ∈ Nd and every n ≥ max(nm,h,nh),

An = A(m)n +

[LThn (a, f)− LThn (am, fm)

]+ (Rn,h −Rn,m,h) + (Nn,h −Nn,m,h), (4.26)

rank(Rn,h −Rn,m,h) ≤ (c(h) + c(m,h))N(n), ‖Nn,h −Nn,m,h‖ ≤ ω(h) + ω(m,h). (4.27)Thanks to Propositions 4.1–4.2 and to Theorem 2.6, we have

‖LThn (a, f)− LThn (am, fm)‖1 ≤ ‖LThn (a, f − fm)‖1 + ‖LThn (a− am, fm)‖1

=h∑j=1

∣∣∣a( jh

)∣∣∣‖Tbn/hc(f − fm)‖1 +h∑j=1

∣∣∣a( jh

)− am

( jh

)∣∣∣‖Tbn/hc(fm)‖1

≤ N(n)‖a‖∞‖f − fm‖L1 + ‖fm‖L1

N(n)

N(h)

h∑j=1

∣∣∣a( jh

)− am

( jh

)∣∣∣≤

[‖a‖∞‖f − fm‖L1 + sup

k

‖fk‖L1

1

N(h)

h∑j=1

∣∣∣a( jh

)− am

( jh

)∣∣∣]N(n); (4.28)

54

note that ‖fk‖L1 is uniformly bounded with respect to k, because fk → f in L1([−π, π]d). By the Riemann-integrability of |a− am|, which follows from the Riemann-integrability of a and am, and by the fact that am → ain L1([0, 1]d) and fm → f in L1([−π, π]d), the quantity

ε(m,h) = ‖a‖∞‖f − fm‖L1 + supk

‖fk‖L1

1

N(h)

h∑j=1

∣∣∣a( jh

)− am

( jh

)∣∣∣ (4.29)

satisfies

limm→∞

limh→∞

ε(m,h) = limm→∞

(‖a‖∞‖f − fm‖L1 + sup

k

‖fk‖L1

∫[0,1]d|a(x)− am(x)|dx

)= 0. (4.30)

Choose any sequence of multi-indices h(m)m such that h(m)→∞ for m→∞ and

limm→∞

c(m,h(m)) = limm→∞

ω(m,h(m)) = limm→∞

ε(m,h(m)) = 0.

Then, by (4.26)–(4.28), for every m and every n ≥ max(nm,h(m),nh(m)),

An = A(m)n +

[LTh(m)

n (a, f)− LTh(m)n (am, fm)

]+ (Rn,h(m) −Rn,m,h(m)) + (Nn,h(m) −Nn,m,h(m)),

rank(Rn,h(m) −Rn,m,h(m)) ≤ [c(h(m)) + c(m,h(m))]N(n),

‖Nn,h(m) −Nn,m,h(m)‖ ≤ ω(h(m)) + ω(m,h(m)),∥∥LTh(m)n (a, f)− LTh(m)

n (am, fm)∥∥

1≤ ε(m,h(m))N(n).

Using Lemma 3.2, we can decompose LTh(m)n (a, f)−LTh(m)

n (am, fm) as the sum of a small-rank term Rn,m, withrank bounded from above by

√ε(m,h(m))N(n), plus a small-norm term Nn,m, with norm bounded from above

by√ε(m,h(m)). This concludes the proof.

Theorem 4.10. Let Ann be a matrix-sequence, let a : [0, 1]d → C be a Riemann-integrable function and letf ∈ L1([−π, π]d). Then, the following are equivalent.

1. Ann ∼LT a⊗ f .

2. There exist sequences am, fm such that:

∗ am : [0, 1]d → C is continuous, ‖am‖∞ ≤ ‖a‖L∞ for all m and am → a a.e.;

∗ fm : [−π, π]d → C is a trigonometric polynomial and fm → f a.e. and in L1([−π, π]d);

∗ Dn(am)Tn(fm)nm is an a.c.s. for Ann.

3. There exist am, fm, A(m)n nm such that:

∗ am : [0, 1]d → C is Riemann-integrable and am → a in L1([0, 1]d);

∗ fm ∈ L1([−π, π]d) and fm → f in L1([−π, π]d);

∗ A(m)n n ∼LT am ⊗ fm and A(m)

n nm is an a.c.s. for Ann.

4. Dn(a)Tn(f)nm is an a.c.s. for Ann.

55

Proof. (1⇒ 2) Since any Riemann-integrable function is bounded, we have a ∈ L∞([0, 1]d). Hence, by the Lusintheorem [37], there exists a sequence of continuous functions am : [0, 1]d → C such that ‖am‖∞ ≤ ‖a‖L∞ for allm and am → a in measure. This implies that am → a also in L1([0, 1]d), because of the uniform boundednessof ‖am‖∞. Thus, there exists a subsequence of am, say am, which converges to a a.e. in [0, 1]d. Thesequence am satisfies all the properties required in item 2; note also that am → a in L1([0, 1]d) by the dominatedconvergence theorem.

Since f ∈ L1([−π, π]d) and the set of d-variate trigonometric polynomials is dense in L1([−π, π]d), thereexists a sequence fm of d-variate trigonometric polynomials such that fm → f in L1([−π, π]d). Choosing asubsequence fm of fm which converges to f a.e., fm satisfies all the properties required in item 2.

The application of Theorem 4.9 concludes the proof.(2⇒ 3) Obvious; we just recall that, under the assumptions in item 2, am → a in L1([0, 1]d) by the dominated

convergence theorem; moreover, Dn(am)Tn(fm)nm ∼LT am ⊗ fm by Theorem 4.8.(3⇒ 1) By assumption, for each m and each h ∈ Nd there is nm,h such that, for n ≥ nm,h,

A(m)n = LThn (am, fm) +Rn,m,h +Nn,m,h,

rank(Rn,m,h) ≤ c(m,h)N(n), ‖Nn,m,h‖ ≤ ω(m,h),

wherelimh→∞

c(m,h) = limh→∞

ω(m,h) = 0.

Moreover, since A(m)n nm is an a.c.s. for Ann, for every m there exists nm such that, for n ≥ nm,

An = A(m)n +Rn,m +Nn,m,

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

wherelimm→∞

c(m) = limm→∞

ω(m) = 0.

Thus, for every m, every h ∈ Nd and every n ≥ max(nm,nm,h),

An = LThn (a, f) +[LThn (am, fm)− LThn (a, f)

]+ (Rn,m +Rn,m,h) + (Nn,m +Nn,m,h),

rank(Rn,m +Rn,m,h) ≤ (c(m) + c(m,h))N(n), ‖Nn,m +Nn,m,h‖ ≤ ω(m) + ω(m,h),∥∥LThn (am, fm)− LThn (a, f)∥∥

1≤ ε(m,h)N(n),

where in the last inequalities we used (4.28); the quantity ε(m,h) is defined in (4.29) and satisfies (4.30). Choose,for every h ∈ Nd, a m(h) such that m(h)→∞ when h→∞ and

limh→∞

ε(m(h),h) = limh→∞

c(m(h),h) = limh→∞

ω(m(h),h) = 0.

A construction of such a function m(h) is provided in Lemma 4.1; apply the lemma with x(m,h) = ε(m,h) +c(m,h) + ω(m,h). Then, for every h ∈ Nd and every n ≥ max(nm(h),nm(h),h),

An = LThn (a, f) +[LThn (am(h), fm(h))− LThn (a, f)

]+ (Rn,m(h) +Rn,m(h),h) + (Nn,m(h) +Nn,m(h),h),

rank(Rn,m(h) +Rn,m(h),h) ≤ (c(m(h)) + c(m(h),h))N(n),

‖Nn,m(h) +Nn,m(h),h‖ ≤ ω(m(h)) + ω(m(h),h),∥∥LThn (am(h), fm(h))− LThn (a, f)∥∥

1≤ ε(m(h),h)N(n).

The application of Lemma 3.2 allows one to decompose LThn (am(h), fm(h)) − LThn (a, f) as the sum of a small-rank term Rn,h, with rank bounded from above by

√ε(m(h),h)N(n), plus a small-norm term Nn,h, with norm

bounded from above by√ε(m(h),h). This concludes the proof of the implication 3⇒ 1.

To conclude the proof of the theorem, we note that 1⇒ 4 (by Theorem 4.9) and 4⇒ 3 (obviously).

56

Remark 4.7. Theorem 4.10 continues to hold if f is assumed to be separable and we add in item 2 the requirementthat each fm is separable. The proof is left as an exercise for the reader. Note that, if f is separable, we can replaceitem 1 with ‘ Ann ∼sLT a⊗ f ’.

We end with a result that provides a relation between LT and sLT sequences. This result will be used inthe next section to show that any LT sequence is a GLT sequence, and, implicitly, that the definition of GLTsequences, originally formulated in [45, 46] in terms of sLT sequences, can be equivalently formulated in terms ofLT sequences.

Proposition 4.4. Let Ann ∼LT a ⊗ f . Then, for any m ∈ N there exist matrix-sequences A(i,m)n n ∼sLT

a ⊗ fi,m, i = 1, . . . , Nm, such that∑Nm

i=1 fi,m → f in L1([−π, π]d) when m → ∞ and ∑Nm

i=1 A(i,m)n nm is an

a.c.s. for Ann.

Proof. Take any sequence of d-variate trigonometric polynomials fm such that fm → f in L1([−π, π]d). We recallthat such a sequence exists because the set of d-variate trigonometric polynomials is dense in L1([−π, π]d). Bydefinition, any d-variate trigonometric polynomial is a finite sum of separable d-variate trigonometric polynomials.Hence, we can write

fm =Nm∑i=1

fi,m,

for some separable d-variate trigonometric polynomials fi,m, i = 1, . . . , Nm. Take arbitrary matrix-sequencesA(i,m)

n n ∼sLT a ⊗ fi,m, i = 1, . . . , Nm. For example, in view of Theorem 4.8, one can choose A(i,m)n =

Dn(a)Tn(fi,m). By Remark 4.5, ∑Nm

i=1 A(i,m)n n ∼LT a ⊗ (

∑Nmi=1 fi,m) = a ⊗ fm. Hence, by Theorem 4.9,

∑Nm

i=1 A(i,m)n nm is an a.c.s. for Ann.

5 GLT sequencesIn this section we develop the GLT theory. In summary we prove all the statements contained in items GLT1-GLT5and, in Theorem 5.6, we prove that f(Hermitian GLT) =Hermitian GLT, under mild assumptions of f .

5.1 Definition and characterizationsWe first report a ‘corrected’ version of the original definition of GLT sequences; cf. [45, Definition 2.3] and [46,Definition 1.5]. Then, we will show that such a definition admits some useful equivalent characterizations.

Definition 5.1 (GLT sequence). Let Ann be a matrix-sequence, withn ∈ Nd, and let κ : [0, 1]d × [−π, π]d → Cbe a measurable function. We say that Ann is a GLT sequence with symbol κ, and we write Ann ∼GLT κ, if:

• for any ε > 0 there exist matrix-sequences A(i,ε)n n ∼sLT ai,ε ⊗ fi,ε, i = 1, . . . , Nε;

•∑Nε

i=1 ai,ε ⊗ fi,ε → κ in measure over [0, 1]d × [−π, π]d when ε→ 0;

•∑Nε

i=1A(i,ε)n n

m

, with ε = (m+ 1)−1, is an a.c.s. for Ann.

From now on, until the end of Section 5, it will be always understood that multi-indices are actually d-indices(as in Section 4.2.2). Moreover, if a matrix-sequence Ann is given and if we write Ann ∼GLT κ, it is implicitlyassumed that κ : [0, 1]d × [−π, π]d → C is measurable.

It is clear that any sLT sequence is a GLT sequence. More precisely, if Ann ∼sLT a ⊗ f then Ann ∼GLT

a ⊗ f . To see this, it suffices to take, in Definition 5.1, Nε = 1, A(1,ε)n n = Ann, a1,ε = a and fi,ε = f , for

57

all ε > 0. Proposition 4.4 and the first characterization of GLT sequences (Proposition 5.1) imply that any LTsequence is a GLT sequence; more precisely,

Ann ∼LT a⊗ f ⇒ Ann ∼GLT a⊗ f. (5.1)

Proposition 5.1. We have Ann ∼GLT κ if and only if:

• there exist matrix-sequences A(i,m)n n ∼LT ai,m ⊗ fi,m, i = 1, . . . , Nm;

•∑Nm

i=1 ai,m ⊗ fi,m → κ in measure over [0, 1]d × [−π, π]d when m→∞;

•∑Nm

i=1 A(i,m)n n

m

is an a.c.s. for Ann.

Proof. If Ann ∼GLT κ, then the three conditions of the proposition hold with

ai,m = ai,ε(m), fi,m = fi,ε(m), A(i,m)n n = A(i,ε(m))

n n, Nm = Nε(m),

where ai,ε, fi,ε, A(i,ε)n n, ε(m) = (m+1)−1 are as in Definition 5.1. Conversely, suppose that the three conditions

of the proposition hold. Then, the three conditions of Definition 5.1 hold with

ai,ε = ai,m(ε), fi,ε = fi,m(ε), A(i,ε)n n = A(i,m(ε))

n n, Nε = Nm(ε),

where m(ε)ε>0 is any family of indices such that m(ε) → ∞ when ε → 0, and ai,m, fi,m, A(i,m)n n are as in

the statement of the proposition. Thus, Ann ∼GLT κ.

Corollary 5.1. If Ann ∼LT a⊗ f then Ann ∼GLT a⊗ f .

Proof. It follows from Proposition 5.1 and Proposition 4.4.

The next proposition shows that the functions ai,m in Proposition 5.1 may be supposed to be continuous, thefunctions fi,m may be supposed to be separable trigonometric polynomials, and the matrix-sequences A(i,m)

n nmay be choosen as A(i,m)

n = Dn(ai,m)Tn(fi,m).

Proposition 5.2. We have Ann ∼GLT κ if and only if:

• there exist functions ai,m, fi,m, i = 1, . . . , Nm, where each ai,m : [0, 1]d → C is continuous and eachfi,m : [−π, π]d → C is a separable trigonometric polynomial;

•∑Nm

i=1 ai,m ⊗ fi,m → κ in measure over [0, 1]d × [−π, π]d when m→∞;

•∑Nm

i=1 Dn(ai,m)Tn(fi,m)nm

is an a.c.s. for Ann.

Proof. If the three conditions of the proposition are met, then also the three conditions of Proposition 5.1 are met,because Dn(ai,m)Tn(fi,m)n ∼sLT ai,m ⊗ fi,m (Theorem 4.8). Hence, Ann ∼GLT κ.

Conversely, suppose that Ann ∼GLT κ. We show that the three conditions of Proposition 5.2 are met. FromProposition 5.1 we know that:

• there exist A(i,m)n nm, i = 1, . . . , Nm, such that A(i,m)

n n ∼sLT ai,m ⊗ fi,m;

•∑Nm

i=1 ai,m ⊗ fi,m → κ in measure over [0, 1]d × [−π, π]d when m→∞;

•∑Nm

i=1 A(i,m)n n

m

is an a.c.s. for Ann.

Let:

58

∗ a(k)i,m : [0, 1]d → C a continuous function such that a(k)

i,m → ai,m in L1([0, 1]d) and a.e. when k →∞;

∗ f (k)i,m : [−π, π]d → C a separable trigonometric polynomial such that f (k)

i,m → fi,m in L1([−π, π]d) anda.e. when k → ∞. To construct f (k)

i,m, take into account that fi,m is separable, say fi,m = fi,m,1 ⊗ · · · ⊗fi,m,d, fi,m,j ∈ L1([−π, π]d) for all j = 1, . . . , d; choose f (k)

i,m = f(k)i,m,1 ⊗ · · · ⊗ f

(k)i,m,d, where each f (k)

i,m,j is aunivariate trigonometric polynomial that converges to fi,m,j in L1([−π, π]) and a.e. when k →∞.

Since A(i,m)n n ∼sLT ai,m ⊗ fi,m and Dn(a

(k)i,m)Tn(f

(k)i,m)nk ∼sLT a

(k)i,m ⊗ f

(k)i,m by Theorem 4.8, it follows from

Theorem 4.9 that Dn(a(k)i,m)Tn(f

(k)i,m)nk is an a.c.s. for A(i,m)

n n for each fixed i and m. Hence, for all k thereexists n(k)

i,m such that, for n ≥ n(k)i,m,

A(i,m)n = Dn(a

(k)i,m)Tn(f

(k)i,m) +R

(i,m)n,k +N

(i,m)n,k , rank(R

(i,m)n,k ) ≤ c(i,m, k)N(n), ‖N (i,m)

n,k ‖ ≤ ω(i,m, k), (5.2)

where limk→∞ c(i,m, k) = limk→∞ ω(i,m, k) = 0. For every δ > 0,

µ(m, k, δ) = µ2d

∣∣∣∣∣Nm∑i=1

a(k)i,m ⊗ f

(k)i,m − κ

∣∣∣∣∣ ≥ δ

→ µ(m, δ) = µ2d

∣∣∣∣∣Nm∑i=1

ai,m ⊗ fi,m − κ

∣∣∣∣∣ ≥ δ

as k →∞;

this can be seen by writing the measures of sets as integrals of indicator functions and by applying the dominatedconvergence theorem, taking into account that

∑Nmi=1 a

(k)i,m ⊗ f

(k)i,m →

∑Nmi=1 ai,m ⊗ fi,m a.e. when k → ∞. Since

µ(m, δ) → 0 as m → ∞, for every δ > 0, we can choose δm 0 such that µ(m, δm) → 0 as m → ∞.Thus, µ(m, k, δm) → µ(m, δm) as k → ∞ and µ(m, δm) → 0 as m → ∞. In view of the limit relationslimk→∞ c(i,m, k) = limk→∞ ω(i,m, k) = 0, limk→∞ µ(m, k, δm) = µ(m, δm), it is possible to choose km ∞such that the following conditions are satisfied for every m:

maxi=1,...,Nm

c(i,m, km) ≤ 1

mNm

, maxi=1,...,Nm

ω(i,m, km) ≤ 1

mNm

, µ(m, km, δm) ≤ 1

m+ µ(m, δm).

Now, ∑Nm

i=1 A(i,m)n nm is an a.c.s. for Ann: for all m there exists nm such that, for n ≥ nm,

An =Nm∑i=1

A(i,m)n +Rn,m +Nn,m, rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m), (5.3)

and limm→∞ c(m) = limm→∞ ω(m) = 0. Combining (5.2)–(5.3), for all m and n ≥ max(nm,maxi=1,...,Nm n(km)i,m )

we have

An =Nm∑i=1

Dn(a(km)i,m )Tn(f

(km)i,m ) +Rn,m +

Nm∑i=1

R(i,m)n,km

+Nn,m +Nm∑i=1

N(i,m)n,km

,

rank(Rn,m +

Nm∑i=1

R(i,m)n,km

)≤(c(m) +

1

m

)N(n),

∥∥∥Nn,m +Nm∑i=1

N(i,m)n,km

∥∥∥ ≤ ω(m) +1

m.

Therefore, ∑Nm

i=1 Dn(a(km)i,m )Tn(f

(km)i,m )nm is an a.c.s. for Ann. To finish the proof, we note that

Nm∑i=1

a(km)i,m ⊗ f

(km)i,m → κ

in measure when m→∞; this is true because, for any δ > 0, we have

µ2d

∣∣∣∣∣Nm∑i=1

a(km)i,m ⊗ f

(km)i,m − κ

∣∣∣∣∣ ≥ δ

= µ(m, km, δ),

which tends to 0, because it is eventually less than µ(m, km, δm)→ 0.

59

5.2 Approximation results for GLT sequencesThe following result is analogous to Corollaries 3.1–3.2, where ‘∼λ’ and ‘∼σ’ are replaced by ‘∼GLT’.

Theorem 5.1. Let Ann be a matrix-sequence. Suppose that:

• Bn,mnm is an a.c.s. for Ann;

• Bn,mn ∼GLT κm for every m;

• κm → κ in measure over [0, 1]d × [−π, π]d when m→∞, being κ some measurable function.

Then Ann ∼GLT κ.

Proof. For every m we have Bn,mn ∼GLT κm, hence:

- for every k there exist matrix-sequences A(i,k)n,mn ∼sLT ai,k,m ⊗ fi,k,m, i = 1, . . . , Nk,m;

-∑Nk,m

i=1 ai,k,m ⊗ fi,k,m → κm in measure over [0, 1]d × [−π, π]d when k →∞;

-∑Nk,m

i=1 A(i,k)n,mn

k

is an a.c.s. for Bn,mn: for every k there exists nk,m such that, for n ≥ nk,m,

Bn,m =

Nk,m∑i=1

A(i,k)n,m +Rn,k,m +Nn,k,m, rank(Rn,k,m) ≤ c(k,m)N(n), ‖Nn,k,m‖ ≤ ω(k,m),

where limk→∞ c(k,m) = limk→∞ ω(k,m) = 0.

Let δm 0. Since∑Nk,m

i=1 ai,k,m ⊗ fi,k,m → κm in measure when k →∞,

µ(m, k, δm) = µ2d

∣∣∣∣∣∣Nk,m∑i=1

ai,k,m ⊗ fi,k,m − κm

∣∣∣∣∣∣ ≥ δm

→ 0 as k →∞.

Now we recall that Bn,mnm is an a.c.s. for Ann: for every m there exists nm such that, for n ≥ nm,

An = Bn,m +Rn,m +Nn,m, rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

where limm→∞ c(m) = limm→∞ ω(m) = 0. It follows that, for every m, every k, and every n ≥ max(nm, nk,m),

An =

Nk,m∑i=1

A(i,k)n,m + (Rn,k,m +Rn,m) + (Nn,k,m +Nn,m),

rank(Rn,k,m +Rn,m) ≤ (c(k,m) + c(m))N(n), ‖Nn,k,m +Nn,m‖ ≤ ω(k,m) + ω(m).

Choose km ∞ such that

limm→∞

c(km,m) = limm→∞

ω(km,m) = limm→∞

µ(m, km, δm) = 0.

Then, for every m and every n ≥ max(nm, nkm,m),

An =

Nkm,m∑i=1

A(i,km)n,m + (Rn,km,m +Rn,m) + (Nn,km,m +Nn,m),

60

rank(Rn,km,m +Rn,m) ≤ c(km,m) + c(m), ‖Nn,km,m +Nn,m‖ ≤ ω(km,m) + ω(m).

It follows that ∑Nkm,m

i=1 A(i,km)n,m nm is an a.c.s. for Ann. Moreover, A(i,km)

n,m n ∼sLT ai,km,m ⊗ fi,km,m fori = 1, . . . , Nkm,m, and

Nkm,m∑i=1

ai,km,m ⊗ fi,km,m → κ

in measure over [0, 1]d × [−π, π]d when m→∞. Indeed, for any δ > 0,

µ2d

∣∣∣∣∣∣Nkm,m∑i=1

ai,km,m ⊗ fi,km,m − κ

∣∣∣∣∣∣ ≥ δ

≤ µ2d

∣∣∣∣∣∣Nkm,m∑i=1

ai,km,m ⊗ fi,km,m − κm

∣∣∣∣∣∣ ≥ δ/2

+ µ2d |κm − κ| ≥ δ/2 ,

µ2d |κm − κ| ≥ δ/2 → 0 by assumption (since κm → κ in measure), and

µ2d

∣∣∣∣∣∣Nkm,m∑i=1

ai,km,m ⊗ fi,km,m − κm

∣∣∣∣∣∣ ≥ δ/2

= µ(m, km, δ/2)

tends to 0, because it is eventually less than µ(m, km, δm)→ 0. Thus, Ann ∼GLT κ by Proposition 5.1.

As a first application of Theorem 5.1, we show in Proposition 5.4 that GLT sequences could be defined interms of LT sequences instead of sLT sequences. In particular, Proposition 5.4 is the same as Proposition 5.1 with‘sLT’ replaced by ‘LT’. For the proof of Proposition 5.4 we need to point out that any linear combination of GLTsequences is again a GLT sequence with symbol given by the same linear combination of the symbols. This isone of the most elementary results in the world of the algebraic properties possessed by GLT sequences; suchproperties will be investigated in Section 5.4 and give rise to the so-called GLT algebra.

Proposition 5.3. IfAnn ∼GLT κ, Bnn ∼GLT ξ,

thenαAn + βBnn ∼GLT ακ+ βξ

for all α, β ∈ C.

The proof of Proposition 5.3 is easy: it suffices to write the meaning of Ann ∼GLT κ and Bnn ∼GLT ξ(using the characterization of Proposition 5.1), and to apply Proposition 3.1; the details are left to the reader.

Proposition 5.4. We have Ann ∼GLT κ if and only if:

• there exist matrix-sequences A(i,m)n n ∼LT ai,m ⊗ fi,m, i = 1, . . . , Nm;

•∑Nm

i=1 ai,m ⊗ fi,m → κ in measure over [0, 1]d × [−π, π]d when m→∞;

•∑Nm

i=1 A(i,m)n n

m

is an a.c.s. for Ann.

Proof. It is clear that, if Ann ∼GLT κ, then the three conditions hold by Proposition 5.1. Conversely, supposethe three conditions hold. Then,

∑Nmi=1 A

(i,m)n nm is an a.c.s. for Ann by hypothesis,

∑Nmi=1 A

(i,m)n n ∼GLT∑Nm

i=1 ai,m ⊗ fi,m by Corollary 5.1 and Proposition 5.3, and∑Nm

i=1 ai,m ⊗ fi,m → κ in measure, by hypothesis. Thethesis follows from Theorem 5.1.

61

The approximation result for GLT sequences stated in Theorem 5.1 admits the following converse, which canbe seen as another approximation result for GLT sequences. It looks like a characterization of GLT sequences interms of a.c.s.

Theorem 5.2. Let Ann ∼GLT κ. Suppose that:

• Bn,mn ∼GLT κm for each m;

• κm → κ in measure.

Then Bn,mnm is an a.c.s. for Ann.

Proof. By Proposition 5.3, An − Bn,mn ∼GLT κ − κm for each m. Hence, by Theorem 5.3 below, An −Bn,mn ∼σ κ− κm, with κ− κm tending to 0 in measure by hypothesis. Hence, by Corollary 3.4, Bn,mn is ana.c.s. for Ann.

Corollary 5.2. Let Ann ∼GLT κ. Then, for any ai,mm, fi,m, i = 1, . . . , Nm, with the following properties:

∗ ai,m : [0, 1]d → C is Riemann-integrable and fi,m ∈ L1([−π, π]d);

∗∑Nm

i=1 ai,m ⊗ fi,m → κ in measure over [0, 1]d × [−π, π]d;

it holds that ∑Nm

i=1 Dn(ai,m)Tn(fi,m)nm is an a.c.s. for Ann. In particular, Ann admits an a.c.s. of theform Nm∑

j=−Nm

Dn(a(m)j )Tn(eij·θ)

n

m

, a(m)j ∈ C∞([0, 1]d), Nm ∈ Nd. (5.4)

Proof. By Theorem 4.8 and Corollary 5.1, we have Dn(ai,m)Tn(fi,m)n ∼GLT ai,m ⊗ fi,m. Hence, by Proposi-tion 5.3,

∑Nmi=1 Dn(ai,m)Tn(fi,m)n ∼GLT

∑Nmi=1 ai,m ⊗ fi,m. Therefore, the thesis follows from Theorem 5.2 and

Lemma 2.2.

5.3 Singular value and eigenvalue distribution of GLT sequencesWe begin with a lemma concerning the singular value distribution of a finite sum of LT sequences. The lemmawill be used in the proof of the singular value distribution result for GLT sequences (Theorem 5.3).

Lemma 5.1. Let A(i)n n ∼LT ai ⊗ fi, i = 1, . . . , p. Then

∑pi=1A

(i)n n ∼σ

∑pi=1 ai ⊗ fi.

Proof. Choose any sequence m = m(m)m such that m → ∞ when m → ∞. From the properties of a.c.s.,see Proposition 3.1, and from the definition of LT sequences, we know that

∑pi=1 LT

mn (ai, fi)nm is an a.c.s.

for ∑p

i=1A(i)n n. By Theorem 4.2, for every F ∈ Cc(R),

limn→∞

1

N(n)

N(n)∑r=1

F(σr

( p∑i=1

LTmn (ai, fi)))

= φm(F )

and

limm→∞

φm(F ) = φ(F ) =1

(2π)d

∫[0,1]d×[−π,π]d

F(∣∣∣ p∑

i=1

ai(x)fi(θ)∣∣∣)dxdθ.

Hence, by Theorem 3.3 and Definition 2.2, ∑p

i=1A(i)n n ∼σ

∑pi=1 ai ⊗ fi.

Theorem 5.3. If Ann ∼GLT κ then Ann ∼σ κ.

62

Proof. By Proposition 5.4, there exist matrix-sequences A(i,m)n n ∼LT ai,m ⊗ fi,m, i = 1, . . . , Nm, such that∑Nm

i=1 ai,m ⊗ fi,m → κ in measure and ∑Nm

i=1 A(i,m)n nm is an a.c.s. for Ann. By Lemma 5.1, we have

∑Nm

i=1 A(i,m)n n ∼σ

∑Nmi=1 ai,m⊗fi,m. Since

∑Nmi=1 ai,m⊗fi,m → κ in measure, all the assumptions of Corollary 3.1

are satisfied and so Ann ∼σ κ.

As a consequence of Theorem 5.3, every GLT sequence is s.u. in the sense of Definition 3.2 (see Proposi-tion 3.3). Using Theorem 5.3, we show in Proposition 5.5 that the symbol of a GLT sequence is unique.

Proposition 5.5. Assume that Ann ∼GLT κ and Ann ∼GLT ξ. Then κ = ξ a.e. in [0, 1]d × [−π, π]d.

Proof. By Proposition 5.3, ON(n)n ∼GLT κ − ξ. Therefore, by Theorem 5.3, for all test functions F ∈ Cc(R)we have

F (0) =1

(2π)d

∫[0,1]d×[−π,π]d

F (|κ(x,θ)− ξ(x,θ)|)dxdθ. (5.5)

If we assume by contradiction that κ is not a.e. equal to ξ, then |κ − ξ| is not a.e. equal to 0 and we can find aninterval [a, b] ⊂ (0,∞) such that µ2d|κ− ξ| ∈ [a, b] > 0. Choosing F ∈ Cc(R) such that F ≥ 0 over R, F = 1over [a, b] and F = 0 over (−∞, 0], it is clear that (5.5) does not hold for this test function F . This contradictionconcludes the proof.

Remark 5.1. If Ann ∼GLT κ then A∗nn ∼GLT κ. The proof is easy and is left as an exercise for the reader.

Proposition 5.6. Let Ann ∼GLT κ and assume that the matrices An are Hermitian. Then κ ∈ R a.e.

Proof. Since the matrices An are Hermitian, by Remark 5.1 we have Ann ∼GLT κ and Ann ∼GLT κ. Thus,by Proposition 5.5, κ = κ a.e., i.e., κ ∈ R a.e.

The next lemma concerns the eigenvalue distribution of (the real part of) a finite sum of LT sequences. Thelemma will be used in the proof of the eigenvalue distribution result for (Hermitian) GLT sequences (Theorem 5.4).

Lemma 5.2. Let A(i)n n ∼LT ai ⊗ fi, i = 1, . . . , p. Then <(

∑pi=1 A

(i)n )n ∼λ <(

∑pi=1 ai ⊗ fi).

Proof. Choose any sequence m = m(m)m such that m → ∞ when m → ∞. From the properties of a.c.s.,see Remark 3.2 and Proposition 3.1, and from the definition of LT sequences, <(

∑pi=1 LT

mn (ai, fi))nm is an

a.c.s. for <(∑p

i=1A(i)n )n. By Theorem 4.3, for every F ∈ Cc(C),

limn→∞

1

N(n)

N(n)∑r=1

F(λr

(<( p∑i=1

LTmn (ai, fi))))

= φm(F )

and

limm→∞

φm(F ) = φ(F ) =1

(2π)d

∫[0,1]d×[−π,π]d

F(<( p∑i=1

ai(x)fi(θ)))dxdθ.

Hence, by Theorem 3.5 and Definition 2.2, <(∑p

i=1A(i)n )n ∼λ <(

∑pi=1 ai ⊗ fi).

Theorem 5.4. Let Ann ∼GLT κ and suppose that the matrices An are Hermitian. Then Ann ∼λ κ.

Proof. By Proposition 5.4, there exist matrix-sequences A(i,m)n n ∼LT ai,m ⊗ fi,m, i = 1, . . . , Nm, such that∑Nm

i=1 ai,m ⊗ fi,m → κ in measure and ∑Nm

i=1 A(i,m)n nm is an a.c.s. for Ann. Since the matrices An

are Hermitian, <(∑Nm

i=1 A(i,m)n )nm is another a.c.s. for An = <(An), formed by Hermitian matrices. By

Lemma 5.2, <(∑Nm

i=1 A(i,m)n )n ∼λ <(

∑Nmi=1 ai,m ⊗ fi,m). The function κ is real a.e. by Proposition 5.6, and so

from∑Nm

i=1 ai,m ⊗ fi,m → κ (in measure) we get <(∑Nm

i=1 ai,m ⊗ fi,m) → κ (in measure). All the assumptions ofCorollary 3.2 are satisfied, and it follows that Ann ∼λ κ.

Remark 5.2. By Proposition 5.3 and Remark 5.1, Lemmas 4.2 and 5.1 are particular cases of Theorem 5.3, andLemma 5.2 is a particular case of Theorem 5.4.

63

5.4 The GLT algebraWe investigate in this section the important algebraic properties possessed by GLT sequences, which give rise to theso-called GLT algebra. In short, these properties establish that, if A(1)

n n, . . . , A(r)n n are given GLT sequences

with symbols κ1, . . . , κr, respectively, and ifAn = ops(A(1)n , . . . , A

(r)n ) is obtained fromA

(1)n , . . . , A

(r)n by means of

certain operations ‘ops’, then Ann is a GLT sequence with symbol κ = ops(κ1, . . . , κr) obtained by performingthe same operations on the symbols κ1, . . . , κr.

Theorem 5.5. Suppose that Ann ∼GLT κ and Bnn ∼GLT ξ. Then:

1. A∗nn ∼GLT κ;

2. αAn + βBnn ∼GLT ακ+ βξ, for all α, β ∈ C;

3. AnBnn ∼GLT κξ.

Proof. The first two statements have already been settled before; see Remark 5.1 and Proposition 5.3. We prove thethird statement. By assumption and Proposition 5.4, there exist matrix-sequences A(i,m)

n n ∼LT ai,m ⊗ fi,m, i =

1, . . . , Nm, and B(j,m)n n ∼sLT bj,m ⊗ gj,m, j = 1, . . . ,Mm, such that:

•∑Nm

i=1 ai,m ⊗ fi,m → κ in measure and∑Mm

j=1 bj,m ⊗ gj,m → ξ in measure;

• ∑Nm

i=1 A(i,m)n nm is an a.c.s. for Ann and

∑Mm

j=1 B(j,m)n nm is an a.c.s. for Bnn.

Thanks to Proposition 5.2, the functions fi,m, gj,m may be supposed to be in L∞([−π, π]d); actually, they might besupposed to be separable trigonometric polynomials, the functions ai,m, bj,m might be supposed to be continuous,and A(i,m)

n n, B(j,m)n n might be chosen of the form Dn(ai,m)Tn(fi,m)n, Dn(bj,m)Tn(gj,m)n. By Theo-

rem 5.3, Ann ∼σ κ and Bnn ∼σ ξ, which implies, by Proposition 3.3, that Ann and Bnn are s.u. Thus,by Proposition 3.4,

(Nm∑i=1

A(i,m)n

)(Mm∑j=1

B(j,m)n

)n

m

=

Nm∑i=1

Mm∑j=1

A(i,m)n B(j,m)

n

n

m

is an a.c.s. for AnBnn. By Theorem 4.7 and by the fact that the functions fi,m, gj,m belong to L∞([−π, π]d),we have A(i,m)

n B(j,m)n n ∼LT ai,mbj,m ⊗ fi,mgj,m, i = 1, . . . , Nm, j = 1, . . . ,Mm. Finally, it is clear that

Nm∑i=1

Mm∑j=1

ai,mbj,m ⊗ fi,mgj,m =

(Nm∑i=1

ai,m ⊗ fi,m

)(Mm∑j=1

bj,m ⊗ gj,m

)→ κξ

in measure, and the proof is over.

Corollary 5.3. Let r, q1, . . . , qr ∈ N and, for i = 1, . . . , r and j = 1, . . . , qi, let A(ij)n n ∼GLT κij . Then,

r∑i=1

qi∏j=1

A(ij)n

n

∼GLT

r∑i=1

qi∏j=1

κij.

The results we have seen so far are enough to conclude that the set of GLT sequences is an algebra over thecomplex field C. More precisely, fix any sequence of d-indices n = n(n)n such that n → ∞ when n → ∞;then,

A =Ann : Ann ∼GLT κ for some measurable function κ : [0, 1]d × [−π, π]d → C

(5.6)

64

is an algebra over C, with respect to the natural operations of addition, scalar-multiplication and product of matrix-sequences. We call A the GLT algebra. We are going to see in Theorems 5.6–5.7 that the GLT algebra enjoysother nice properties, in addition to those of Theorem 5.5, which make it look like a ‘big container’, closed underany type of ‘regular’ operation.

Theorem 5.6 provides a positive answer to a question raised in [47]. Incidentally, we note that in [47] theauthors proved that f(Bn,m)nm is an a.c.s. for f(An)n whenever Bn,mnm is an a.c.s. for Ann andf : R → R is continuous (and other mild assumptions are met); this result enlarges the algebraic properties ofa.c.s. studied in Section 3.2.

Theorem 5.6. Let Ann ∼GLT κ and suppose that the matrices An are Hermitian. Then

f(An)n ∼GLT f(κ)

for any continuous function f : R→ R.4

Proof. For eachM > 0, let pm,Mm be a sequence of polynomials that converges uniformly to f over the compactinterval [−M,M ]:

limm→∞

‖f − pm,M‖∞,[−M,M ] = 0.

For every M > 0 and every m,n, write

f(An) = pm,M(An) + f(An)− pm,M(An). (5.7)

Since any GLT sequence is s.u. (by Theorem 5.3 and Proposition 3.3), the sequence Ann is s.u. Hence, byProposition 3.2, for all M > 0 there exists nM such that, for n ≥ nM ,

An = An,M + An,M , rank(An,M) ≤ r(M)N(n), ‖An,M‖ ≤M, (5.8)

where limM→∞ r(M) = 0. However, for the purpose of this proof we need a splitting of the form (5.8) such thatg(An,M + An,M) = g(An,M) + g(An,M) for all functions g : R → R. Luckily, the matrices An are Hermitianand, consequently, such a splitting can be constructed by following the same argument used in the proof of Propo-sition 3.2. For the reader’s convenience, we include the details of the construction. By definition, since Ann iss.u., for every M > 0 there exists nM such that, for n ≥ nM ,

#i ∈ 1, . . . , N(n) : σi(An) > MN(n)

≤ r(M),

where limM→∞ r(M) = 0. Let An = UnΛnU∗n be a spectral decomposition of An. Let Λn,M be the matrix

obtained from Λn by setting to 0 all the eigenvalues of An whose absolute value is less than or equal to M , andlet Λn,M = Λn − Λn,M be the matrix obtained from Λn by setting to 0 all the eigenvalues of An whose absolutevalue is greater than M . Then, for M > 0 and n ≥ nM ,

An = UnΛnU∗n = UnΛn,MU

∗n + UnΛn,MU

∗n = An,M + An,M ,

where An,M = UnΛn,MU∗n and An,M = UnΛn,MU

∗n. The matrices An,M , An,M constructed in this way are

Hermitian, satisfy the properties in (5.8) and, moreover,

g(An,M + An,M) = g(An,M) + g(An,M) = Un g(Λn,M)U∗n + Un g(Λn,M)U∗n

for all functions g : R→ R.4Recall from Proposition 5.6 that κ ∈ R a.e., because every An is Hermitian. Hence, f(κ) is well-defined.

65

Going back to (5.7), for every M > 0, every m and every n ≥ nM we can write

f(An) = pm,M(An) + f(An,M) + f(An,M)− pm,M(An,M)− pm,M(An,M)

= pm,M(An) + (f − pm,M)(An,M) + (f − pm,M)(An,M). (5.9)

The term (f − pm,M)(An,M) can be split in the sum of two terms Rn,m,M + N ′n,m,M : Rn,m,M is obtained from(f − pm,M)(An,M) by setting to 0 all the eigenvalues that are equal to (f − pm,Mm)(0), so that rank(Rn,m,M) =

rank(An,M); while N ′n,m,M is obtained from (f −pm,M)(An,M) by setting to 0 all the eigenvalues that are differentfrom (f − pm,Mm)(0). Let N ′′n,m,M = (f − pm,M)(An,M) and Nn,m,M = N ′n,m,M +N ′′n,m,M . From (5.9), for everyM > 0, every m and every n ≥ nM we have

f(An) = pm,M(An) +Rn,m,M +Nn,m,M , (5.10)

and, by our construction,

rank(Rn,m,M) = rank(An,M) ≤ r(M)N(n),

‖Nn,m,M‖ ≤ |f(0)− pm,M(0)|+ ‖f − pm,M‖∞,[−M,M ] ≤ 2‖f − pm,M‖∞,[−M,M ].(5.11)

Choose a sequence Mmm such that, when m→∞,

Mm →∞, ‖f − pm,Mm‖∞,[−Mm,Mm] → 0. (5.12)

Then, for every m and every n ≥ nMm ,

f(An) = pm,Mm(An) +Rn,m,Mm +Nn,m,Mm ,

rank(Rn,m,Mm) ≤ r(Mm)N(n), ‖Nn,m,Mm‖ ≤ 2‖f − pm,Mm‖∞,[−Mm,Mm],

which implies that pm,Mm(An)nm is an a.c.s. for f(An)n. Moreover, pm,Mm(An)n ∼GLT pm,Mm(κ) byTheorem 5.5. Finally, pm,Mm(κ)→ f(κ) a.e. in [0, 1]d × [−π, π]d, due to (5.12). In conclusion, all the hypothesesof Theorem 5.1 are satisfied and so f(An)n ∼GLT f(κ).

The last issue we are interested in is to know if A−1n n ∼GLT κ

−1 in the case where Ann ∼GLT κ, each Anis invertible, and κ 6= 0 a.e. (so that κ−1 is a well-defined measurable function). More in general, we may ask ifA†nn ∼GLT κ−1 when An ∼GLT κ and κ 6= 0 a.e., being A†n the (Moore–Penrose) pseudoinverse of Ann.The answer to both the previous questions is affirmative, but some work is needed to bring out the related proofs.Note that these results cannot be inferred from Theorem 5.6, because the matrices An may fail to be Hermitianand, moreover, f(x) = x−1 is not a continuous function on R. We begin by introducing the concept of sparselyvanishing matrix-sequences.

Definition 5.2 (sparsely vanishing matrix-sequence). Let Ann be a matrix-sequence. We say that Ann issparsely vanishing (s.v.) if for every M > 0 there exists nM such that, for n ≥ nM ,

#i ∈ 1, . . . , N(n) : σi(An) < 1/MN(n)

≤ r(M),

where limM→∞ r(M) = 0.

It is clear from Definition 5.2 that, if Ann is s.v., then A†nn is s.u.; it suffices to recall that the singularvalues of A† are 1/σ1(A), . . . , 1/σr(A), 0, . . . , 0, where σ1(A) . . . σr(A) are the nonzero singular values of A(r = rank(A)).

66

Remark 5.3. Let Ann be a matrix-sequence. Then, Ann is s.v. if and only if

limM→∞

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) < 1/MN(n)

= 0. (5.13)

The proof of this equivalence is easy and follows the same line as the proof of the equivalence (1⇔ 2) in Proposi-tion 3.2; the details are left to the reader. Note that (5.13) can be rewritten as

limM→∞

lim supn→∞

1

N(n)

N(n)∑i=1

χ[0,1/M)(σi(An)) = 0.

Proposition 5.7. Let Ann be a matrix-sequence such that Ann ∼σ f for some measurable f : D ⊂ Rk → C.Then Ann is s.v. if and only if f 6= 0 a.e.

Proof. Fix M > 0 and take FM ∈ Cc(R) such that FM = 1 over [0, 1/M ], FM = 0 outside [−1/M, 2/M ] and0 ≤ FM ≤ 1 over R. Then,

#i ∈ 1, . . . , N(n) : σi(An) < 1/MN(n)

=1

N(n)

N(n)∑i=1

χ[0,1/M)(σi(An))

≤ 1

N(n)

N(n)∑i=1

FM(σi(An))n→∞−→ 1

µk(D)

∫D

FM(|f(x)|)dx

and

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) < 1/MN(n)

≤ 1

µk(D)

∫D

FM(|f(x)|)dx.

Since FM(|f(x)|)→ χf=0(x) a.e. and |F (|f(x)|)| ≤ 1, by the dominated convergence theorem we get

limM→∞

∫D

FM(|f(x)|)dx =µkf 6= 0µk(D)

.

Thus,

limM→∞

lim supn→∞

#i ∈ 1, . . . , N(n) : σi(An) < 1/MN(n)

= 0

if and only if f = 0 a.e. By Remark 5.3, this means that Ann is s.v. if and only if f = 0 a.e.

Theorem 5.7. Let Ann ∼GLT κ with κ 6= 0 a.e., then A†nn ∼GLT κ−1.

Proof. Take a sequence of matrix-sequences Bn,mnm such that Bn,mm ∼GLT ξm for eachm, and ξm → κ−1

in measure. This can be done because, by Lemma 2.2, there exists ξmm, with ξm of the form

ξm =Nm∑

j=−Nm

a(m)j ⊗ eij·θ, a

(m)j ∈ C([0, 1]d), Nm ∈ Nd,

such that ξm → κ−1 in measure; hence, it suffices to take Bn,m =∑Nm

j=−NmDn(a

(m)j )Tn(eij·θ), which is a GLT

sequence with symbol ξm (see Theorem 4.8, Corollary 5.1 and Theorem 5.5).For every m, by Theorem 5.5 we have Bn,mAn − IN(n)n ∼GLT ξmκ − 1, and ξmκ − 1 → 0 in measure.

Therefore, by Proposition 3.7, for every m there exists nm such that, for n ≥ nm,

Bn,mAn = IN(n) +Rn,m +Nn,m, (5.14)

67

rank(Rn,m) ≤ c(m)N(n), ‖Nn,m‖ ≤ ω(m),

where limm→∞ c(m) limm→∞ ω(m) = 0. Multiplying (5.14) byA†n, we obtain that, for everym and every n ≥ nm,

Bn,mAnA†n = A†n + (Rn,m +Nn,m)A†n. (5.15)

Since Ann is s.v. (by Theorem 5.3 and Proposition 5.7), A†nn is s.u. Hence, by Proposition 3.2, for all M > 0there is nM such that, for n ≥ nM ,

A†n = A†n,M + A†n,M

rank(A†n,M) ≤ r(M)N(n), ‖A†n,M‖ ≤M,

where limM→∞ r(M) = 0. Choosing Mm = [ω(m)]−1/2, from (5.15) we see that, for every m and every n ≥max(nm, nMm),

Cn,mAnA†n = A†n +R′n,m +N ′n,m, (5.16)

rank(R′n,m) ≤(c(m) + r(Mm)

)N(n), ‖N ′n,m‖ ≤ [ω(m)]1/2.

If the matrices An were invertible, then A†n = A−1n and (5.16) would imply that Bn,mnm is an a.c.s. for

A−1n n; this, in combination with the approximation result for GLT sequences (Theorem 5.1), would conclude the

proof. In the general case where the matrices An are not invertible will follow again from (5.16) and Theorem 5.1as soon as we have proved the following: for every m there is nm such that, for n ≥ nm,

AnA†n = IN(n) + Sn, rank(Sn) ≤ θ(m)N(n),

where limm→∞ θ(m) = 0. This is easy, because rank(Sn) = #i ∈ 1, . . . , N(n) : σi(An) = 0. Hence, theprevious claim follows directly from Definition 5.2 and from the fact that Ann is s.v.

5.4.1 The algebra generated by Toeplitz sequences

Fix a sequence of d-indices n = n(n)n such thatn→∞ as n→∞. In this section, we briefly discuss about thealgebra T over the complex field C generated by the Toeplitz sequences of the form Tn(g)n, g ∈ L1([−π, π]d).It is not difficult to see that

T =

r∑i=1

qi∏j=1

Tn(gij)n

: r, q1, . . . , qr ∈ N, gij ∈ L1([−π, π]d) for all i = 1, . . . , r and j = 1, . . . , qi

.

(5.17)It is clear from Theorem 4.6 and Corollary 5.1 that T is a sub-algebra of the GLT algebra A defined in (5.6).

Indeed, according to Corollary 5.3, r∑i=1

qi∏j=1

Tn(gij)n∼GLT

r∑i=1

qi∏j=1

1⊗ gij = 1⊗r∑i=1

qi∏j=1

gij.

Since∫

[0,1]d×[−π,π]d(1⊗ g) =

∫[−π,π]d

g, Theorem 5.3 and Definition 2.2 immediately give

r∑i=1

qi∏j=1

Tn(gij)n∼σ

r∑i=1

qi∏j=1

gij.

Similarly, if the matrices∑r

i=1

∏qij=1 Tn(gij) are Hermitian, Theorem 5.4 gives

r∑i=1

qi∏j=1

Tn(gij)n∼λ

r∑i=1

qi∏j=1

gij. (5.18)

68

The extension of the spectral distribution relation (5.18) to the case where the matrices∑r

i=1

∏qij=1 Tn(gij)

are not Hermitian has been the subject of a recent research; see [17, Theorem 9]. Note that, if we remove thehypothesis of ‘Hermitianity’, then we necessarily have to add some additional assumption. Indeed, (5.18) does nothold in general; a counterexample is provided, e.g., by the sequence of (1-level) Toeplitz matrices Tn(eijθ)n. Thehypothesis added in [17] is a topological assumption on the range of the functions gij . A completely analogoushypothesis was already used in [16, 19] and, especially, in the pioneering work by Tilli [55], in order to extend thespectral distribution relation expressed in Theorem 2.5 to the case where the generating function f is not real (andhence the related Toeplitz matrices Tn(f) are not Hermitian).

6 Conclusions: GLT as a Generalized Fourier AnalysisIn this revue we have reported unitarly the main features of the GLT sequences, by improving somehow thetechnical construction, by making the tools more easily usable in applications, and by proving a new key resultconcerning the stability of Hermitian GLT sequence under the action of a continuous function.

Furthermore we have clearly stated and proved the properties GLT1-GLT5 reportd in the Introduction. Morespecifically:

• item GLT1 is contained in Theorem 5.3 and Theorem 5.4;

• item GLT2 is contained in Theorem 5.5 and Theorem 5.7;

• item GLT3 is obtained by putting together Lemma 4.2, Theorem 4.6, and Corollary 5.1;

• item GLT4 is obtained by combining Lemma 4.2, Theorem 4.5, and Corollary 5.1;

• item GLT5 is proved in Theorem 4.4, taking into account the inclusion LT⊂GLT given Corollary 5.1.

In is worth observing, as already stated in [46], that the GLT theory and in particular items GLT1-GLT5represent a powerful tool for generalizing the local Fourier Analysis [5, 10] to much more general contexts inapproximating PDEs and IEs; see items GLT6, GLT7.

6.1 The algebra generated by diagonal sampling matrix-sequences, by zero distributedsequences, and by Toeplitz sequences: a tool for computing the symbol of PDE ap-proximations

As a direct consequence, of items GLT1-GLT5, the GLT algebra is a super-algebra of sequences containingthe algebra generated by diagonal sampling matrix-sequences, by zero distributed sequences, and by Toeplitzsequences. As a consequence (see Subsection 1.1, [45, 46, 6, 13, 14, 15, 23] and references therein), taking ageneral variable coefficients PDE on a [0, 1]d, we deduce that any reasonable approximation of the consideredPDE by local methods (Finite Differences, Finite Elements, Collocation/Galerkin IgA, Finite Volumes etc) leadsto matrix-sequences that can be approximated by linear combinations of products involving diagonal sampling andToeplitz matrices.

The case of a general domain can be recovered, especially in the IgA setting, via the use of a geometric map[6, 13, 14, 15], or by using the notion of reduced GLT sequence as stated in [46][Section 3.1.4], [45][Remark 2.1]:see also [45][pp 398-399, formula (59)] for a specific example of use, where it turns out that the domain where thePDE is defined has to be simply Peano-Jordan measurable, i.e., its characteristic function is Riemann integrable.

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6.2 Future work: reduced GLT, block GLT, and an automatic (symbolic) calculus of thesymbol

A future work should concern revisiting the work in [46][Section 3.1.4, Section 3.3] for giving a more applicableGLT theory with general domains and matrix-valued symbols: the latter case is not academic because of vectorPDEs (one of the simplest is the linear elasticity in saddle-point form) and because of quadrilateral Finite Elementsof degree p over a d dimensional domain, which leads to matrix-sequences having a symbol of size pd, even in thecase of a scalar equation.

In [26] we have treated such a case, but only for a constant coefficient equation, while a proper revisiting ofa block GLT theory would represent the ideal framework for dealing with general variable coefficients PDEs andFinite Elements of degree p, regularity k ∈ 0, . . . , p− 1, and dimensionality d.

Finally, a possible idea to be exploired is to design an automatic procedure for obtaining the symbol of anapproximated PDE, as a function of the principal operator, of the PDE coefficients, of the domain, and of the usedapproximation technique. Some hints are given in [45][Section 2] and [46][Question 3.1] in connection with theHörmander calculus [32].

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