a matrix approach for the semiclassical and coherent ... · a matrix approach for the semiclassical...

A Matrix Approach for the Semiclassical andCoherent Orthogonal Polynomials

Lino Gustavo Garza GaonaJoint work with: Natalia C. Pinzón, Luis E. Garza & Francisco

Marcellán

Universidad Carlos III de Madrid

V Encuentro Iberoamericano de Polinomios Ortogonales ysus Aplicaciones

June 8 - 12, Instituto de Matemáticas, UNAM, México, D.F.

Lino Gustavo Garza Gaona (UC3M) EIBPOA 2015 UNAM, June 8-12 2015 1 / 35

Contents

1 IntroductionOrthogonal polynomials (OP)A new matrix interpretationCoherent pairs on the real line

2 Matrix characterization for the semiclassical orthogonal polynomialsSemiclassical Orthogonal PolynomialsMatrix Interpretation

3 Matrix characterization for the coherence of orthogonal polynomials(1,0)-Coherent pairs(M,0)-Coherence


Introduction

Contents





Introduction Orthogonal polynomials (OP)

Orthogonal polynomials (OP)

Let us consider a linear functionalU : P→ C defined on the linear space ofpolynomials with complex coefficients. A sequence of monic polynomials{Pn(x)}n≥0 such that

deg(Pn(x)) = n and 〈U, Pn(x)Pm(x)〉 = knδn,m with kn , 0, n,m ≥ 0,

is said to be the sequence of monic orthogonal polynomials (SMOP) associatedwithU.




The existence of a SMOP can be characterized in terms of the infinite Hankelmatrix H = [ui+ j]i,i>0, where un = 〈U, xn〉 , n ≥ 0, are called the moments ofU.Indeed, {Pn(x)}n≥0 exists if and only if the leading principal submatrices

Hn = [ui+ j]ni, j=0, n > 0

of H are nonsingular. In this situation,U is said to be a quasi-definite or regular.On the other hand, if we have det Hn > 0, n > 0,U is said to be positive definite,and it has the integral representation

〈U, p(x)〉 =∫

Ep(x)dµ(x),

where µ is a nontrivial probability measure supported on some infinite subsetE ⊂ R.




The most familiar sequences of orthogonal polynomials are the so-called classicalfamilies: Jacobi, Laguerre and Hermite.They correspond to the cases when:

E has bounded support (E = [−1, 1]),E is the positive real axis, and

E = R,

and the corresponding probability measures are the Beta, Gamma and normaldistributions, respectively.

There are several ways to characterize the classical orthogonal polynomials:

as polynomial solutions of a hypergeometric differential equation (Bochner),

as polynomials expressed by a Rodrigues formula, and

as the only sequences of orthogonal polynomials whose derivatives alsoconstitute an orthogonal family (Hahn’s property).



TTRR & Jacobi matricesOne of the most important properties of monic orthogonal polynomials is that theysatisfy the three term recurrence recurrence relation

xPn(x) = Pn+1(x) + bnPn(x) + anPn−1(x), n > 0 (1)

where P−1(x) := 0, bn ∈ R, n > 0, and an , 0, n > 1. IfU is positive definite, thenwe have an > 0. In a matrix form,

xP(x) = JP(x),

where P(x) = [P0(x), P1(x), . . .]T and J is the tridiagonal infinite matrix

J =

b0 1 0 . . .

a1 b1 1. . .

0 a2 b2. . .

.... . .

. . .. . .

, (2)

called the monic Jacobi matrix associated with {Pn(x)}n≥0.Lino Gustavo Garza Gaona (UC3M) EIBPOA 2015 UNAM, June 8-12 2015 7 / 35

Introduction A new matrix interpretation

Matrix interpretation

Recently, in a work by Verde Star1, a new characterization of classical orthogonalpolynomials was introduced by using matrix analysis.

Let write

Pn(x) =n∑

j=0

an, jx j, n ≥ 0,

and define the infinite matrix A = [an, j]06 j6n,n>0. Notice that A is a lower triangularmatrix whose n-th row contains the coefficients of the n-th degree orthogonalpolynomial. Furthermore, since Pn is monic, the diagonal entries are an,n = 1 and,therefore, A is nonsingular.

We say that A is the matrix associated with the sequence {Pn(x)}n≥0. If thepolynomials are classical, we will say that A is classical.

1L. Verde-Star: “Characterization and construction of classical orthogonal polynomials using amatrix approach”. Linear Algebra and its Applications 438 (2013) 3635-3648.



Matrix interpretationIf A is non zero, we say that A has index m if m is the minimum integer such that Ahas at least one nonzero entry in the m−th diagonal, and we say that A is(n,m)−banded if there exists a pair of integer numbers (n,m) such that n ≤ m andall the nonzero entries of A lie between the diagonals of indices n and m.

Let define the matrices

D =

0 0 0 0 . . .1 0 0 0 . . .0 2 0 0 . . .0 0 3 0 . . ....

......

. . .. . .

, D =

0 1 0 0 . . .0 0 1/2 0 . . .0 0 0 1/3 . . .0 0 0 0 . . ....

......

. . .. . .

,

X =

0 1 0 0 . . .0 0 1 0 . . .0 0 0 1 . . .

0 0 0 0. . .

......

.... . .

. . .

, X =

0 0 0 0 . . .1 0 0 0 . . .0 1 0 0 . . .0 0 1 0 . . ....

......

. . .. . .

.



Matrix interpretation

With these elements we get the following matrix characterization for theorthogonality of a sequence of polynomials.

Theorem

Let {Pn(x)}n≥0 be a monic polynomial sequence and let A be its associated matrix.Then, the sequence {Pn(x)}n≥0 is orthogonal with respect to some linear functionalif and only if the matrix L = AXA−1 is (-1,1)-banded with nonzero entries in thediagonals of index -1 and 1.


Introduction Coherent pairs on the real line

Coherent pairs on the real line

If we assume that a vector of measures (dµ0, dµ1) is coherent, then we cananalyze the asymptotics properties of the corresponding sequence of Sobolevorthogonal polynomials.

We say that two non-trivial probability measures measures, dµ0 and dµ1,constitute a coherent pair if there exists a fixed constant k ∈ N0 such that, for eachn ∈ N, the monic orthogonal polynomial Pn(·; dµ1) can be expressed as a linearcombination of the set P′n+1(·; dµ0), . . . , P′n−k(·, dµ0).

The coherence is classified in terms of k.




The simplest case was studied by A. Iserles, P.E. Koch, S. P. Norsett and J. M.Sanz-Serna2.

They considered a pair of non-trivial probability measures (dµ0, dµ1) supported onthe real line with finite moments of all orders and stated that a necessary andsufficient condition for these measures to be a (1,0)-coherent pair is that thereexist nonzero constants {an}n≥1 such that their corresponding sequences of monicorthogonal polynomials (SMOP), {Pn(x)}n≥0 and {Qn(x)}n≥0 respectively, satisfy

P′n+1(x)n + 1

+ anP′n(x)

n= Qn(x), an , 0, n ≥ 1. (3)

2A. Iserles, P.E. Koch, S. P. Norsett, J. M. Sanz-Serna, On Polynomials Orthogonal with Respect toCertain Sobolev Inner Products., J. Approx. Theory, 65 (1991) 151-175.



Coherent pairs on the real lineThis condition of coherence arises as a sufficient condition for the existence of arelation

Pn+1(x) +n + 1

nanPn(x) = S n+1(x; λ) + cn(λ)S n(x; λ), n ≥ 1, (4)

where {cn(λ)}n≥1 are rational functions in λ > 0 and {S n(x; λ)}n≥0 is the SMOPassociated with the Sobolev inner product

〈p(x), r(x)〉λ =∫ ∞

−∞

p(x)r(x)dµ0(x) + λ∫ ∞

−∞

p′(x)r′(x)dµ1(x), λ > 0, (5)

where p(x) and r(x) are polynomials with real coefficients. They studied the casewhen the first measure dµ0 is either the Gamma or the Beta function whosecorresponding sequences of orthogonal polynomials are the Laguerre and Jacobi,respectively.




In 1997, in 3, H. G. Meijer determined all (1,0)-coherent pairs (U,V) of regularlinear functionals. He proved that at least one of the linear functionals (U orV)must be classical (Laguerre or Jacobi). Moreover, he showed that there are onlytwo cases.

He also determined all symmetrically (1,0)-coherent pairs, providing similar resultsto those obtained in the non symmetrical case.

3H. G. Meijer. Determination of All Coherent Pairs. J. Approx. Theory 89, (1997), 321-343.Lino Gustavo Garza Gaona (UC3M) EIBPOA 2015 UNAM, June 8-12 2015 14 / 35

Matrix characterization for the semiclassical orthogonal polynomials

Contents





Matrix characterization for the semiclassical orthogonal polynomials Semiclassical Orthogonal Polynomials

Semiclassical OP

Let φ(x) = at xt + . . . + a0, ψ(x) = blxl + . . . + b0 be non zero polynomials such thatatbl , 0, t ≥ 0, l ≥ 1. (φ, ψ) is said to be an admissible pair if either t − 1 , l ort − 1 = l and nal+1 + bl , 0, n ≥ 0. A quasi-definite linear functionalU is said to besemiclassical if there exists an admissible pair (φ, ψ) such thatU satisfies

D(φU) = ψU, (6)

where D denotes the distributional derivative. The corresponding sequence oforthogonal polynomials is called semiclassical.

The class of a semiclassical linear functional is the non negative integer

s = min{max{deg(φ) − 2, deg(ψ) − 1} : (φ, ψ) is an admissible pair

}.



Structure RelationsThere are several characterizations of semiclassical orthogonal polynomials interms of the so called structure relations.

Theorem

LetU be a quasi-definite linear functional and let {Pn(x)}n≥0 be its correspondingSMOP. Then, the following statements are equivalent

There exist non zero polynomials φ, ψ of degrees t ≥ 0, l ≥ 1, respectively,such that (6) holds.a (First structure relation) There exist a polynomial φ of degree t andsequences {an,k} such that {Pn(x)} satisfies

φ(x)P[1]n (x) =

n+t∑k=n−s

an,kPk(x), n ≥ s, an,n−s , 0, n ≥ s + 1, (7)

where s is a positive integer such that t ≤ s + 2 and P[1]n (x) =

P′n+1(x)n + 1

.

a P. Maroni. Une Théorie Algébrique des Polynômes Orthogonaux. Application aux PolynômesOrthogonaux Semi-Classiques. IMACS Annals Comput. Appl. Math. 9 (1991), 95-130.



Structure Relations

Theorem (cont.)


b (Second structure relation) There exist non-negative integers t, s, andsequences {an,k}, {bn,k}, such that

n+s∑k=n−s

an,kPk(x) =n+s∑

k=n−t

bn,kP[1]k (x), n ≥ max{s, t},

holds, where an,n+s = bn,n+s = 1, n ≥ max{s, t + 1},

bF. Marcellán and R. Sfaxi. Second Structure Relation for Semiclassical OrthogonalPolynomials. J. Comput. Appl. Math. 200 (2007), 537-554.



Structure Relations

Theorem (cont.)


c There exist a non-negative integer s and sequences {bn, j}n≥0 and {cn, j}n≥0such that {Pn(x)} satisfies the structure relation

s∑j=0

bn,n− jPn− j(x) =s+2∑j=0

cn,n− jP[1]n− j(x), bn,n = cn,n = 1, n ≥ s + 1. (8)

cA. Branquinho and M. N. Rebocho. On the Semiclassical Character of OrthogonalPolynomials Satisfying Structure Relations. J. Difference Equ. Appl. 18 (2012), 111-138.


Matrix characterization for the semiclassical orthogonal polynomials Matrix Interpretation

Matrix Interpretation

Notice that (8) can be expressed in matrix form as

BA = CA, (9)

where B is a (0, s)−banded monic matrix, and C is a (0, s + 2)−banded monicmatrix. With this representation, we can state the following

Theorem

Let {Pn(x)}n>0 be a SMOP with respect to some linear functionalU. Then,U issemiclassical of class at most s if and only if there exists a semi-infinite(0, s)−banded monic matrix B such that BAA−1 is a (0, s + 2)−banded monicmatrix.




Now, let assume that the linear functionalU in (6) is positive definite and it isassociated with an absolutely continuous positive measure µ supported in[a, b] ⊂ R, which can be expressed as dµ(x) = ω(x)dx, with the weight ω satisfyinglimx→a+ xnφ(x)ω(x) = limx→b− xnφ(x)ω(x) = 0, n ≥ 0. The Pearson equation (6) canbe expressed in terms of the weight as

(φω)′ = ψω.

In such a case, there exists a sequence of orthonormal polynomials {pn(x)}n>0,and the corresponding Jacobi matrix is the symmetric matrix

J =

b0 a1 0 . . .

a1 b1 a2. . .

0 a2 b2. . .

.... . .

. . .. . .

, satisfying xp(x) = J p(x), with p(x) =

p0(x)p1(x)...

.




In this context, the first structure relation for semiclassical polynomials given in (7)also holds for the corresponding sequence of orthonormal polynomials {pn(x)}n>0associated with the semiclassical functionalU. It can be expressed in a matrixform as

φ(x)p′(x) = XT Hp(x), (10)

where H is a (−t, s)−banded matrix whose elements, starting from the row s, arethe coefficients appearing in (7) given in terms of {pn(x)}n>0, andp′(x) = [p′0(x), p′1(x), . . .]T .




The following result establishes a relation between H and J.

Theorem

Let {pn(x)}n>0 be a semiclassical sequence of orthonormal polynomials and let Hbe the (−t, s)−banded matrix associated with the first structure relation (10). Then,we have

(i) [J, XT H] = φ(J),(ii) H + HT = −ψ(J),

where [J, XT H] = JXT H − XT H J and φ, ψ are the polynomials appearing in thePearson equation.


Matrix characterization for the coherence of orthogonal polynomials

Contents





Matrix characterization for the coherence of orthogonal polynomials (1,0)-Coherent pairs

(1,0)-Coherent pairs

A pair of regular linear functionals (U,V) in the linear space of polynomials withcomplex coefficients is said to be a (1, 0)-coherent pair of order 1, or simply(1, 0)−coherent pair, if their corresponding SMOP {Pn(x)}n≥0 and {Qn(x)}n≥0 satisfythe structure relation

P[1]n (x) + cnP[1]

n−1(x) = Qn(x), n ≥ 0,

that we already defined in the introduction. The following lemma relates thematrices corresponding to sequences of orthogonal polynomials associated with acoherent pair of linear functionals.




Lemma

If {Pn(x)}n≥0 and {Qn(x)}n≥0 are SMOP with associated matrices A and Q,respectively, then

(AXA−1)2AQ−1 = AQ−1N2 (11)

holds, where A = DAD and N is the Jacobi matrix associated with Q.

Proof.

From Theorem 1 we can see that QX = NQ holds. Hence X = Q−1NQ and, as aconsequence, X2 = Q−1NQQ−1NQ = Q−1N2Q. Thus AX2Q−1 = AQ−1N2, orequivalently

(AXA−1)2AQ−1 = AQ−1N2.

�




Theorem

Let {Pn(x)}n≥0 and {Qn(x)}n≥0 be SMOP with associated matrices A and Q,respectively. Then {Pn(x)}n≥0 and {Qn(x)}n≥0 constitute a (1, 0)−Coherent pair ifand only if QA−1 is lower bidiagonal with ones in the main diagonal and nonzeroentries in the subdiagonal.

Proof.

Assume ({Pn(x)}n≥0, {Qn(x)}n≥0) is a (1, 0)−coherent pair, i.e., the relation

P[1]n (x) + cnP[1]

n−1(x) = Qn(x), cn , 0, n ≥ 0, (12)

holds. Since A is the matrix associated with {P[1]n (x)}n≥0, then (12) can be written

in matrix form asA +CXA = Q,




Cont.

where C = diag(c0, c1, . . . ). Another way to write the above equation is

(I +CX)A = Q, (13)

and, since A is nonsingular,(I +CX) = QA−1.

So QA−1 is clearly lower bidiagonal with ones in the diagonal and non zero entriesin the subdiagonal, since cn , 0, n ≥ 0.

For the converse, if QA−1 = T is bidiagonal with ones in the main diagonal andnon zero elements in the subdiagonal, then

T A = Q,

so {Pn(x)}n≥0 and {Qn(x)}n≥0 constitute a (1, 0)−Coherent pair of SMOP. �




Notice that, in the particular case when {Pn(x)}n≥0 is an orthogonal classical family,then {P[1]

n (x)}n≥0 is also orthogonal (and classical), and it has an associated Jacobimatrix M = AXA−1. In this situation (11) becomes

M2T − T N2 = 0,

which is a particular case of a Sylvester equation. As a consequence, thecoherence coefficients {cn}n≥0 can be obtained by using some numericalalgorithm.



(1,0)-Coherent pairs of order mThe notion of coherence can be generalized for higher order of derivatives asfollows. A pair of regular linear functionals (U,V) in the linear space ofpolynomials with complex coefficients is said to be a (1, 0)-coherent pair of orderm, if their corresponding sequences of monic orthogonal polynomials (SMOP){Pn(x)}n≥0 and {Qn(x)}n≥0 satisfy the structure relation

P[m]n (x) + cnP[m]

n−1(x) = Qn(x), n ≥ 0, (14)

where {cn}n≥0 is a sequence of complex numbers such that cn , 0 for n ≥ 1, c0 is afree parameter, P−1(x) = 0, and P[m]

n (x) denotes the monic polynomial of degree n

P[m]n (x) =

P(m)n+m(x)

(n + 1)m, n ≥ 0,

where (n + 1)m is the Pochhammer symbol defined by(α)n = α(α + 1) · · · (α + n − 1), n ≥ 1, and (α)0 = 1.



(1,0)-Coherent pairs of order m

Notice that, defining Dm and Dm to be diagonal matrices of index m and index −m,respectively, as follows

Dm =

0 0 0 0 . . .... 0 0 0 . . .

(n + 1)m... 0 0 . . .

0 (n)m... 0 . . .

......

. . ....

. . .

, Dm =

0 . . . 1/(n + 1)m 0 . . .

0 0 . . . 1/(n)m. . .

0 0 0 . . .. . .

0 0 0 0 . . ....

......

. . .. . .

,

and then, using the same argument as before, we obtain the following results.



(1,0)-Coherent pairs of order m

Lemma

If {Pn(x)}n≥0 and {Qn(x)}n≥0 are SMOP with associated matrices A and Q,respectively, then

(A[m]XA[m]−1)2A[m]Q−1 = A[m]Q−1N2 (15)

holds, where A[m] = DmADm is the matrix associated with the sequence{P[m]

n (x)}n≥0 and N is the Jacobi matrix associated with Q.

Theorem

Let {Pn(x)}n≥0 and {Qn(x)}n≥0 be SMOP with associated matrices A and Q,respectively. Then {Pn(x)}n≥0 and {Qn(x)}n≥0 constitute a (1, 0)−coherent pair oforder m if and only if QA[m]−1

is lower bidiagonal with ones on the main diagonaland nonzero entries in the subdiagonal.


Matrix characterization for the coherence of orthogonal polynomials (M,0)-Coherence

(M,0)-Coherent pairs

A pair of regular linear functionals (U,V) in the linear space of polynomials withcomplex coefficients is said to be a (M, 0)-coherent pair of order 1, or simply(M, 0)-coherent pair, if their corresponding sequences of monic orthogonalpolynomials (SMOP) {Pn(x)}n≥0 and {Qn(x)}n≥0 satisfy the structure relation

M∑i=0

ci,nP[1]n+1−i(x) = Qn(x), n ≥ 0, (16)

where {ci,n}n≥0, 0 ≤ i ≤ M, is a sequence of complex numbers such that cM,n , 0 ifn ≥ M and ci,n = 0 if i > n.



(M,0)-Coherent pairs

Theorem

Let {Pn(x)}n≥0 and {Qn(x)}n≥0 be the SMOP with associated matrices A and Q,respectively. Then {Pn(x)}n≥0 and {Qn(x)}n≥0 constitute a (M, 0)−Coherent pair if,and only if QA−1 is (0,M)−banded with ones on the main diagonal.

Proof.

Use the matrix form of the coherence relation equation

A + YA = Q,

where Y = [C1X +C2X2 + · · · +CM XM] and Ci is a diagonal matrix of index 0 withentries cn+i,n, n ≥ 0, 1 ≤ i ≤ M, and apply the previous arguments. �



Thank you


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