1. introduction. · tion analysis. the analysis in section 6, in which we establish that the...
TRANSCRIPT
MULTISCALE ANALYSIS IN SOBOLEV SPACES ON THE SPHERE∗
Q. T. LE GIA† , I. H. SLOAN‡ , AND H. WENDLAND§
Abstract. We consider a multiscale approximation scheme at scattered sites for functionsin Sobolev spaces on the unit sphere Sn. The approximation is constructed using a sequence ofscaled, compactly supported radial basis functions restricted to Sn. A convergence theorem forthe scheme is proved, and the condition number of the linear system is shown to stay boundedby a constant from level to level, thereby establishing for the first time a mathematical theory formultiscale approximation with scaled versions of a single compactly supported radial basis functionat scattered data points.
Key words. multiscale, approximation, interpolation, radial basis functions, unit sphere
AMS subject classifications. 65J10, 46E35
1. Introduction. Approximations based on spherical radial basis functions(SBFs) centered at scattered sites have many applications in the geosciences. Oftenthe ‘scale’ of the SBF is an important parameter. A common situation is that theSBF Φ is the restriction to the unit sphere Sn ⊆ Rn+1 of a radial basis function (RBF)Ψ(x) = ψ(|x|) for x ∈ Rn+1, where Ψ has support of radius 1 and |x| is the Euclideandistance in Rn+1. The scaled RBF Ψδ is then defined by Ψδ(x) = cδΨ(x
δ ) (which hassupport radius δ), and the scaled SBF Φδ is then the restriction of Ψδ to Sn.
In this paper we consider an approximation scheme based on a single underlyingRBF Ψ but using a sequence of scales δ1, δ2, . . . with limit zero. The motivatingidea is that geophysical data typically occurs at many length scales: for example,the topography of the Sahara Desert varies slowly, while that of the Himalayas variesrapidly. Therefore it seems natural to use SBFs of different scales to accommodatethe different length scales. But there is a significant theoretical difficulty in dealingwith more than one scale at the same time, namely that the associated reproducingkernel Hilbert space (the ‘native space’) has a different inner product for each scale.For this reason a multiresolution analysis within a single Hilbert space, of the kindfamiliar from wavelet analysis, does not seem possible for scaled SBFs on the sphere,or indeed for scaled RBFs on Euclidean regions. New tools seem to be needed.
The approximation scheme studied here follows a path inspired by papers ofSchaback (see [13]) and Floater & Iske [1], see also [18], which is intuitively plausiblebut until now unsubstantiated for either the sphere or Euclidean regions. The essentialnotion is that one begins with SBFs of large scale, and obtains a first approximationby interpolating at widely separated points; then forms the residual (or error) andfinds a correction by interpolating that residual with RBFs of a finer scale and withpoints closer together; and so on, finally obtaining an approximation which is a sum ofSBFs on a sequence with progressively finer scales. Floater and Iske [1] give numericalevidence to support the idea, but no analysis. We note that an analysis is available fora different multilevel scheme proposed by Narcowich, Schaback and Ward [9], in whichthe SBFs for the successive levels are obtained not by scaling, but rather by repeatedconvolution of the RBF for the finest scale. That scheme is intrinsically limited to
∗This work was supported by the Australian Research Council. The authors acknowledge a helpfulsuggestion by Yuesheng Xu concerning the presentation of the results
†University of New South Wales, Sydney, Australia ([email protected])‡University of New South Wales, Sydney, Australia ([email protected])§University of Oxford, Oxford, U.K. ([email protected])
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a finite number of levels, and seems to present difficulties in implementation. TheFloater and Iske [1] scheme is also limited to a finite number of levels, since it isformulated in terms of ’thinning’ a given data set. The present scheme, by contrast,can be carried to as many levels as desired, and is easy to implement if the underlyingRBF Ψ is available in closed form. We remark that, unlike the work of [1] and [9], wemake no assumption that the successive point sets at the different scales are nested.
Different approaches to the multiscale problem for the sphere have been exploredby Freeden and colleagues, [2, 3]. While the theory given, for example, in [3, Chapter11] contains an elegant multiresolution analysis, it differs from the present work in afundamental way, in that it does not include spatial discretization, except in the band-limited situation (in which case convolution integrals can be replaced by quadraturesums that are exact for appropriate spherical polynomials). In the present work,in contrast, spatial discretization is considered from the very beginning, and band-limited (i.e. polynomial) RBFs are excluded, since we require our RBFs to havecompact support.
Yet another multilevel approach, this time for Euclidean spaces, is that of Halesand Levesley [5]. In that approach the point sets used at each level differ only by achange of scale, making this approach inherently unsuitable for the case of the sphere.
After a section devoted to preliminaries, we consider in Section 3 interpolation forSBFs of a single scale. In that section and the next we define the scaled SBFs throughthe behavior of the Fourier coefficients instead of as above, but later, in Section 6,we show that the compactly supported RBFs of Wendland do indeed have Fouriercoefficients with the specified behavior.
The multiscale method itself is defined in Section 4. The main theorem, Theorem4.1, establishes that the L2 norm of the error, under appropriate assumptions, reduceslinearly from scale to scale as the approximation progresses.
In Section 5 the multiscale method is expressed in the framework of multiresolu-tion analysis.
The analysis in Section 6, in which we establish that the compactly supportedRBFs associated with particular Sobolev spaces have Fourier coefficients on the spherethat transform in the expected way, turns out to be non-trivial. Theorem 6.2 is per-haps of independent interest, since for the first time it establishes a clear mathematicalfoundation for the study of scaled SBFs on the sphere.
In Section 7 we discuss the cost and the conditioning of the approximation scheme.It turns out that the condition number of the linear system stays constant from levelto level, if we reduce the scale of the SBF and the separation radius of the points(see below) in proportion to each other. For the same reason the number of non-zeroentries per row also stays roughly constant, making for efficient implementation.
The cost of the method is essentially the same as that of the computation on thefinest scale.
A numerical example in Section 8 demonstrates the effectiveness of the scheme.In particular, Figures 8.3 and 8.4 compare the approximations on the one hand usingthe multiscale scheme, and on the other hand a “one-shot” approximation in whichwe proceed directly to an approximation with the finest scale and the smallest meshnorm. The multiscale results are unquestionably of better quality, yet the conditionnumber and the work involved essentially the same.
2. Interpolation using spherical basis functions. Let X = {x1, . . . ,xM} bea set of M distinct scattered points on the unit sphere Sn ⊂ Rn+1. The mesh norm
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hX of X is defined by
hX := supp∈Sn
minx∈X
θ(p,x),
where θ(p,x) := cos−1(p · x) is the geodesic distance of two points p and x on thesphere Sn. The separation radius is defined by
qX :=1
2minj 6=k
θ(xj ,xk),
and the Euclidean separation radius in Rn+1 by
qX :=1
2minj 6=k
|xj − xk| =1
2
√2(1 − cos qX). (2.1)
The mesh ratio
ρX := hX/qX ≥ 1
provides a measure of how uniformly points in X are distributed on Sn. In the caseof the circle (n = 1), ρX = 1 means that the points are equally spaced. In all othercases we have ρX > 1. If ρX is bounded above by a uniform constant, we say that Xis quasi-uniform.
Bizonal functions on Sn are functions that can be represented as φ(x · y) for allx,y ∈ Sn, where φ(t) is a continuous function on [−1, 1]. We shall be concernedexclusively with bizonal kernels of the type
Φ(x,y) = φ(x · y) =
∞∑
ℓ=0
aℓPℓ(n+ 1;x · y), aℓ > 0,
∞∑
ℓ=0
aℓ <∞, (2.2)
where {Pℓ(n+ 1; t)}∞ℓ=0 is the sequence of (n+ 1)-dimensional Legendre polynomialsnormalized to Pℓ(n+1; 1) = 1. Thanks to the seminal work of Schoenberg [14] and thelater work of [19], we know that such a Φ is positive definite on Sn, that is, the matrixA := [Φ(xi,xj)]
Mi,j=1 is positive definite for every set of distinct points {x1, . . . ,xM}
on Sn and every positive integer M .In particular, the following interpolation problem admits a unique solution: given
a continuous function f on Sn, a positive integer M , and a set of distinct pointsX = {x1, . . . ,xM} on Sn, there exists a sequence of numbers {bj}M
j=1 such that thefunction
IXf(x) =
M∑
j=1
bjΦ(x,xj) (2.3)
satisfies the interpolation condition
IXf(xk) = f(xk), 1 ≤ k ≤M.
The functions Φ(·,xj) = φ(· · xj), for 1 ≤ j ≤M , are called spherical basis functions(SBFs) in the literature.
For error analysis it is convenient to expand the kernel Φ(x,y) into a series ofspherical harmonics. A detailed discussion on spherical harmonics can be found in
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[8]. In brief, spherical harmonics are the restriction to Sn of homogeneous polyno-mials Y (x) in Rn+1 which satisfy ∆Y (x) = 0, where ∆ is the Laplacian operator inRn+1. The space of all spherical harmonics of degree ℓ on Sn, denoted by Hℓ, has anorthonormal basis
{Yℓk : k = 1, . . . , N(n, ℓ)},
where
N(n, 0) = 1 and N(n, ℓ) =(2ℓ+ n− 1)Γ(ℓ+ n− 1)
Γ(ℓ+ 1)Γ(n)for ℓ ≥ 1.
The space of spherical harmonics of degree ≤ L will be denoted by PL :=⊕L
ℓ=0Hℓ;it has dimension N(n+ 1, L). Every function f ∈ L2(S
n) can be expanded in termsof spherical harmonics,
f =
∞∑
ℓ=0
N(n,ℓ)∑
k=1
fℓkYℓk, fℓk =
∫
Sn
fYℓkdS,
where dS is the surface measure of the unit sphere.Using the addition theorem for spherical harmonics (see, for example, [8, page
18]), we can write
Φ(x,y) =
∞∑
ℓ=0
N(n,ℓ)∑
k=1
φ(ℓ)Yℓk(x)Yℓk(y), where φ(ℓ) =ωn
N(n, ℓ)aℓ, (2.4)
where ωn is the surface area of Sn. Throughout the paper, we make a further assump-tion that, φ(ℓ) ∼ (1 + ℓ)−2σ , i.e. there exist constants c1, c2 > 0 and σ > n/2 suchthat
c1(1 + ℓ)−2σ ≤ φ(ℓ) ≤ c2(1 + ℓ)−2σ, ℓ ≥ 0. (2.5)
The native space induced by Φ is defined to be
NΦ :=
f ∈ D′(Sn) : ‖f‖2Φ =
∞∑
ℓ=0
N(n,ℓ)∑
k=1
|fℓk|2
φ(ℓ)<∞
,
where D′(Sn) is the set of all tempered distributions defined on Sn. Alternatively, NΦ
is the completion of span{Φ(·,x) : x ∈ Sn} with respect to the norm ‖ · ‖Φ.The Sobolev space Hσ(Sn) with real parameter σ is defined by
Hσ(Sn) :=
f ∈ D′(Sn) : ‖f‖2
Hσ =∞∑
ℓ=0
N(n,ℓ)∑
k=1
(1 + ℓ)2σ|fℓk|2 <∞
.
For σ = 0 the space is just L2(Sn).
Under the condition (2.5), the norms ‖ · ‖Φ and ‖ · ‖Hσ are equivalent, since
c1/21 ‖f‖Φ ≤ ‖f‖Hσ ≤ c
1/22 ‖f‖Φ. (2.6)
Because σ > n/2 it follows from the embedding theorem that functions in Hσ(Sn)are continuous.
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3. Interpolation using scaled SBFs. For a given δ > 0, we define the scaledversion Φδ of the kernel Φ (with Φ1 = Φ) by
Φδ(x,y) =
∞∑
ℓ=0
N(n,ℓ)∑
k=1
φδ(ℓ)Yℓk(x)Yℓk(y) (3.1)
in which the Fourier coefficients satisfy the following asymptotic condition:
c1(1 + δℓ)−2σ ≤ φδ(ℓ) ≤ c2(1 + δℓ)−2σ, ℓ ≥ 0, (3.2)
with the coefficients c1 and c2 from (2.5) possibly relaxed so that (3.2) holds for allδ ≤ 1. In Section 6 we will show that the Fourier coefficients of the SBFs defined byrestricting scaled Wendland’s functions Ψδ(x) = δ−nΨ(x/δ) onto Sn behave as givenby (3.1) and (3.2).
For a given function f ∈ C(Sn), the interpolant IX,δf of f is defined by
IX,δf(x) :=
M∑
j=1
bjΦδ(x,xj) such that IX,δf(xk) = f(xk) for all xk ∈ X. (3.3)
For a function f ∈ Hσ(Sn), we define the norm corresponding to the scaled kernelΦδ(x,y) by
‖f‖Φδ=
∞∑
ℓ=0
N(n,ℓ)∑
k=1
|fℓk|2
φδ(ℓ)
1/2
. (3.4)
Clearly the norms ‖ · ‖Φδfor different δ are all equivalent.
Lemma 3.1. For δ ≤ 1 and all g ∈ Hσ(Sn) we have
‖g‖Φδ≤ c
−1/21 ‖g‖Hσ ≤ (c2/c1)
1/2‖g‖Φ, ‖g‖Φ ≤ (c2/c1)1/2δ−σ‖g‖Φδ
.
Proof. Since δ ≤ 1, on using (3.2) we have
‖g‖2Φδ
≤ 1
c1
∞∑
ℓ=0
N(n,ℓ)∑
k=1
|gℓk|2(1 + δℓ)2σ ≤ 1
c1
∞∑
ℓ=0
N(n,ℓ)∑
k=1
|gℓk|2(1 + ℓ)2σ
=:1
c1‖g‖2
Hσ ≤ c2c1‖g‖2
Φ.
We also have (1 + δℓ) = δ(1/δ+ ℓ) ≥ δ(1 + ℓ). Hence, (1 + ℓ)2σ ≤ δ−2σ(1 + δℓ)2σ, and
‖g‖2Φ ≤ 1
c1
∞∑
ℓ=0
N(n,ℓ)∑
k=1
|gℓk|2(1 + ℓ)2σ
≤ 1
c1δ−2σ
∞∑
ℓ=0
N(n,ℓ)∑
k=1
|gℓk|2(1 + δℓ)2σ ≤ δ−2σ c2c1‖g‖2
Φδ.
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Theorem 3.2. Let hX be the mesh norm of the finite set X ⊂ Sn, and letg ∈ Hσ(Sn) vanish on X. Then there exists c3 > 0 such that, for sufficiently smallhX and all δ ≤ 1,
‖g‖L2≤ c3
(hX
δ
)σ
‖g‖Φδ. (3.5)
If the interpolant of f ∈ Hσ(Sn) is defined as in (3.3) (with δ ≤ 1), then for sufficientlysmall hX
‖f − IX,δf‖L2≤ c3
(hX
δ
)σ
‖f‖Φδ. (3.6)
Proof. By [7, Theorem 3.3], because g|X = 0, there is a constant c such that
‖g‖L2≤ chσ
X‖g‖Hσ . (3.7)
Using Lemma 3.1 we have
‖g‖Hσ ≤ c1/22 ‖g‖Φ ≤ c
1/22 (c2/c1)
1/2δ−σ‖g‖Φδ. (3.8)
Inequality (3.5) then follows from (3.7) and (3.8) with the constant c3 = cc2/c1/21 . Now
take g = f − IX,δf . Since IX,δf is the orthogonal projection of f into span{Φδ(·,xj) :xj ∈ X} with respect to the ‖ · ‖Φδ
norm, it follows that
‖g‖Φδ= ‖f − IX,δf‖Φδ
≤ ‖f‖Φδ.
Hence inequality (3.6) follows from (3.5). 2
4. Multiscale interpolation. Suppose X1, X2, . . . is a sequence of point setson Sn with mesh norm h1, h2, . . . respectively. The mesh norms are assumed to satisfyhj+1 ≈ µhj for some fixed µ ∈ (0, 1).
We select an SBF Φ whose Fourier coefficients satisfy (2.5) for some σ > n/2. Forevery j = 1, 2, . . . we choose δj and from it derive the scaled SBF Φj = Φδj
, where δjis a scaling parameter proportional to hj , i.e. δj = νhj for some ν > 0. We denotethe interpolation operator for the j-th scaled SBF Φj by
IXj ,δjf =
∑
x∈Xj
bx(f)Φj(·,x), IXj ,δjf(x) = f(x) for x ∈ Xj .
We start with a widely spread set of points and use a basis function with scaleδ1 to recover the global behavior of the function f by computing f1 = s1 := IX1,δ1
f .The error, or residual, at the first step is e1 = f −f1. To reduce the error, at the nextstep we use a finer set of points X2 and a finer scale δ2, and compute a corrections2 = IX2,δ2
e1 and a new approximation f2 = f1 + s2, so that the new residual isf − f2 = e1 − IX2,δ2
e1; and so on.Stated as an algorithm, we set f0 = 0 and e0 = f and compute for j = 1, 2, . . . ,
sj = IXj ,δjej−1
fj = fj−1 + sj ,
ej = ej−1 − sj .
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Note that the algorithm makes fj + ej independent of j, from which it follows thatfj + ej = f0 + e0 = f , allowing us to identify ej = f − fj as the error (or the residual)at stage j, and fj as the approximation to f at the stage j. Since
fj =
j∑
i=1
si =
j∑
i=1
IXi,δiei−1,
we see that the approximation fj is a linear combination of spherical basis functionsat all scales up to level j. We can think of sj as adding additional “detail” to theapproximation fj−1 to produce fj . Also, since ej |Xj
= 0, it follows that fj = f − ej
interpolates f on Xj , or
fj |Xj= f |Xj
.
The next theorem is our main convergence result.
Theorem 4.1. Let X1, X2, . . . be a sequence of point sets on Sn with mesh normsh1, h2, . . . satisfying cµhj ≤ hj+1 ≤ µhj for j = 1, 2, . . . with fixed µ, c ∈ (0, 1) and h1
sufficiently small. Let Φδjbe a kernel satisfying (3.2) with scale factor δj = νhj and
ν satisfying 1/h1 ≥ ν ≥ β/µ with a fixed β > 0. Assume that the target function fbelongs to Hσ(Sn). Then there is a constant C = C(β) independent of j and f suchthat
‖ej‖Φδj+1≤ α‖ej−1‖Φδj
for j = 1, 2, . . . , (4.1)
with α = Cµσ, and hence there exists c4 > 0 such that
‖f − fk‖L2≤ c4α
k‖f‖Hσ for k = 1, 2, . . . .
Thus the multiscale approximation fk converges linearly to f in the L2 norm if µ <C−1/σ.
Proof. Noting that δk+1 ≤ δ1 = νh1 ≤ 1 and ek|Xk= 0, on using Theorem 3.2
with δ = δk+1 we obtain
‖f − fk‖L2= ‖ek‖L2
≤ c3hσkδ
−σk+1‖ek‖Φk+1
≤ c3(cβ)−σ‖ek‖Φk+1, (4.2)
since hk/δk+1 = hk/(νhk+1) ≤ 1/(cµν) ≤ 1/(cβ).
The remaining part of the proof is devoted to proving the recurrence (4.1). Fixingj ≥ 1 and writing Φδj
as Φj , we have, from (3.2) and (3.4),
‖ej‖2Φj+1
≤ 1
c1
∑
ℓ≤1/δj+1
N(n,ℓ)∑
k=1
|ejℓk|2(1 + δj+1ℓ)2σ +
1
c1
∑
ℓ>1/δj+1
N(n,ℓ)∑
k=1
|ejℓk|2(1 + δj+1ℓ)2σ
=:1
c1(S1 + S2).
For the sum S1, since δj+1ℓ ≤ 1,
S1 ≤∑
ℓ≤1/δj+1
N(n,ℓ)∑
k=1
|ejℓ,k|222σ ≤ 22σ‖ej‖2
L2. (4.3)
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Since ej = ej−1 − IXj ,δjej−1, by the last part of Theorem 3.2 we have
‖ej‖L2= ‖ej−1 − IXj ,δj
ej−1‖L2≤ c3
(hj
δj
)σ
‖ej−1‖Φj= c3ν
−σ‖ej−1‖Φj.
Therefore (4.3) gives
S1 ≤ c2322σν−2σ‖ej−1‖2
Φj.
For the sum S2, since δj+1ℓ > 1 we have, using δj+1/δj = hj+1/hj ≤ µ,
(1 + δj+1ℓ)2σ < (2δj+1ℓ)
2σ ≤ (2µδjℓ)2σ ≤ (2µ)2σ(1 + δjℓ)
2σ.
So with the aid of (3.2) and (3.4),
S2 ≤∑
ℓ>1/δj+1
N(n,ℓ)∑
k=1
(2µ)2σ|ejℓk|2(1 + δjℓ)2σ ≤ c2(2µ)2σ‖ej‖2
Φj≤ c2(2µ)2σ‖ej−1‖2
Φj,
where the key last step follows because ej = ej−1 − IXj ,δjej−1 is an orthogonal pro-
jection of ej−1 on a subspace of NΦj. (The subspace is the orthogonal complement of
span {Φj(·,x) : x ∈ Xj} in NΦj.) Hence, since ν ≥ β/µ,
‖ej‖2Φj+1
≤ 1
c1(c23ν
−2σ + c2µ2σ)22σ‖ej−1‖2
Φj≤ 1
c1(c23β
−2σ + c2)22σµ2σ‖ej−1‖2
Φj,
from which follows
‖ej‖Φj+1≤ Cµσ‖ej−1‖Φj
= α‖ej−1‖Φj
with C = 2σ(c23β−2σ + c2)
1/2/√c1, thus proving (4.1).
Using (4.2) and repeating (4.1) k times, we obtain, with c5 = c3(cβ)−σ,
‖f − fk‖L2≤ c5‖ek‖Φk+1
≤ c5α‖ek−1‖Φk≤ c5α
2‖ek−2‖Φk−1≤ . . . ≤ c5α
k‖f‖Φ1,
and with the help of Lemma 3.1 the theorem now follows, with c4 = c−1/21 c5 =
c−1/21 c3(cβ)−σ . 2
We remark that the condition ν ≥ β/µ is needed in the analysis because in (4.2)we chose to introduce the seemingly unnatural ‖ · ‖Φk+1
norm of ek. In turn, to prove(4.1) we bounded ‖ej‖Φj+1
(in the course of bounding S2) by (2µ)σ times the ‖ · ‖Φj
norm of ej . We needed to reach the latter norm because it is only for that norm thatwe know the crucial fact that the norm of ej is less than that of ej−1.
5. Multiresolution analysis. For a given RBF Ψ in Rn+1 and a sequence ofpoint sets X1, X2, . . . ⊂ Sn and a sequence of scales δ1, δ2, . . . converging to 0, themultiscale approximation scheme in Section 4 can be put into the following multires-olution framework.
For j ≥ 1, let Wj and Vj be the linear subspaces of Hσ(Sn) defined by
Wj := span{Φδj(·,x) : x ∈ Xj}
and
Vj := span{Φδi(·,x) : x ∈ Xi, i ≤ j},
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where Φδ is as in (3.1). Thus
V1 ⊂ V2 ⊂ . . . ⊂ Vj ⊂ Vj+1 ⊂ . . . ⊂ Hσ(Sn).
In the language of wavelets, Wj is the wavelet space and Vj the scale space. The spaceVj can be written as a direct sum of spaces Wi,
Vj =⊕
1≤i≤j
Wi, j ≥ 1
and hence if V0 = {0},
Vj = Vj−1
⊕Wj , j ≥ 1.
That the sum is direct follows under appropriate conditions from the following lemma.Lemma 5.1. Let Ψ be a compactly supported RBF as in [18] and let Φδi
fori = 1, 2, . . . be scaled SBFs constructed as in Section 4 where δ1, δ2, . . . are distinctscales with δi ≤ 1. Let Xi = {xi,1, . . . ,xi,Ni
} ⊂ Sn for i ≥ 1 be a set of Ni distinctpoints. Then for j ≥ 1 and ai,k ∈ R
j∑
i=1
Ni∑
k=1
ai,kΦδi(·,xi,k) = 0 implies ai,k = 0 for i = 1, . . . , j; k = 1, . . . , Ni. (5.1)
Proof. In (5.1), either all ai,k are zero, or one SBF, say Φδ1(·,x1,1), is a linear
combination of the others,
Φδ1(·,x1,1) =
j∑
i=1
Ni∑
k=1(i,k) 6=(1,1)
bi,kΦδi(·,xi,k). (5.2)
The SBF Φδi(xi,k, ·) is singular (in the sense of a discontinuity in an appropriate
derivative with respect to x) on the n− 1 dimensional sphere
Si,k := {x ∈ Sn : |x − xi,k| = δi}
and at xi,k, and is infinitely smooth elsewhere. For (i, k) 6= (1, 1) , Si,k ∩ S1,1 is ann− 2 dimensional sphere, hence
⋃
(i,k) 6=(1,1)
Si,k
⋃
{xi,k : 1 ≤ i ≤ j, 1 ≤ k ≤ Ni}
∩ S1,1
is of dimension n − 2 and so cannot contain S1,1. Therefore, there exists a pointx ∈ S1,1 at which the left hand side of (5.2) is singular while the right hand side isinfinitely smooth, which is a contradiction. Thus the sum in the lemma can vanishonly if ai,k = 0 for all i and k. 2
The sequence of spaces {Vj} is ultimately dense in L2(Sn), in the sense of the
following theorem.Theorem 5.2. Let X1, X2, . . . be a sequence of point sets on Sn with mesh norms
h1, h2, . . . satisfying cµhj ≤ hj+1 ≤ µhj for j = 1, 2, . . . with fixed µ, c ∈ (0, 1) and h1
10
sufficiently small. Let Φδjbe a kernel satisfying (3.2) with scale factor δj = νhj and
ν satisfying 1/h1 ≥ ν ≥ β/µ with a fixed β > 0. For all µ sufficiently small.
the closure of
∞⋃
j=1
Vj with respect to the norm ‖ · ‖L2is L2(S
n).
Proof. Theorem 4.1 shows that f ∈ Hσ(Sn) can be approximated arbitrarilyclosely in the L2 norm by linear combinations of functions in Vk, if k is sufficientlylarge. Since Hσ is dense in L2, the result extends to arbitrary functions in L2, provingthe theorem.
The algorithm presented in Section 4 can be generalized in the following way. Setf0 = 0 and e0 = f . For j = 1, 2, . . . compute
sj = Pj(ej−1) ∈Wj ,
where Pj : Hσ(Sn) → Wj is a linear projection. Then we update the approximationand the residual by
fj = fj−1 + sj ∈ Vj ,
ej = ej−1 − sj .
Theorem 4.1 gives a convergence proof for the special case Pj(g) = IXj ,δj(g). Note
that the proof does not extend to general projections Pj because we made essentialuse of the fact that interpolation is an orthogonal projection in the correspondingnative space.
6. Scaled SBFs defined from scaled RBFs. In this section we will show thatSBFs constructed by scaling compactly supported RBFs Ψ : Rn+1 → R have Fouriercoefficients φδ(ℓ) that satisfy the decay condition (3.2).
We start with a compactly supported radial basis functions Ψ(x) = ψ(|x|), typi-fied by Wendland’s radial basis functions [16]. We will assume that this radial basisfunction has Hτ (Rn+1) as its native space, which is equivalent to assuming that its
(n+ 1)-variate Fourier transform Ψ decays like
c1(1 + |ω|2)−τ ≤ Ψ(ω) ≤ c2(1 + |ω|2)−τ , ω ∈ Rn+1. (6.1)
Since Ψ(x) = ψ(|x|) is radial, the Fourier transform of Ψ can be computed via aHankel transform:
Ψ(ω) :=
∫
Rn+1
Ψ(x)e−ix·ωdx
= (2π)(n+1)/2r−(n−1)/2
∫ ∞
0
ψ(t)t(n+1)/2J(n−1)/2(rt)dt
= (2π)(n+1)/2r−(n+1)
∫ r
0
ψ(t/r)t(n+1)/2J(n−1)/2(t)dt
=: Fψ(r),
where r = |ω| and Jν denotes the Bessel function of the first kind of order ν.Then, we restrict the function (x,y) 7→ Ψ(x − y) to the unit sphere Sn ⊂ Rn+1.
Using |x − y|2 = |x|2 + |y|2 − 2x · y = 2 − 2x · y for x,y ∈ Sn we can conclude thatthe resulting function is indeed zonal:
Ψ(x− y) = ψ(|x − y|) = ψ(√
2 − 2x · y),
11
and hence we can define the kernel
Φ(x,y) := Ψ(x − y)|Sn , x,y ∈ Sn. (6.2)
The connection between the (n+ 1)-variate Fourier transform Ψ and the Fourier
coefficients φ(ℓ) is given by the following result.Lemma 6.1. [11, Theorem 4.1] Let the kernel Φ be defined by restricting an RBF
Ψ(x) = ψ(|x|) to the sphere Sn as in (6.2). Then
φ(ℓ) =
∫ ∞
0
rFψ(r)J2ν (r)dr, (6.3)
where ν = ℓ+ (n− 1)/2 and Jν(r) is the Bessel function of the first kind.A consequence of this lemma is that if Ψ generates the Sobolev space Hτ (Rn+1)
with τ > (n + 1)/2, i.e. its Fourier transform satisfies (6.1), then Φ generatesHτ−1/2(Sn), i.e. its Fourier coefficients satisfy (2.5) with σ = τ − 1/2, see [10, Propo-sition 4.1].
The next step is to introduce scaling. For a given δ > 0 and Ψ : Rn+1 → R we
define the scaled function Ψδ : Rn+1 → R
Ψδ(x) = δ−nΨ(x/δ). (6.4)
Note that this scaling preserves volumes in the n-dimensional space Rn, not in Rn+1,otherwise the scaling factor would be δ−n−1. Consequently, the Fourier transform ofΨδ scales as
Ψδ(ω) = δΨ(δω). (6.5)
This means that if Ψ generates Hτ (Rn+1) then so does Ψδ, and it follows from (6.1)that
c1δ(1 + δ2|ω|2)−τ ≤ Ψδ(ω) ≤ c2δ(1 + δ2|ω|2)−τ .
The scaled radial basis function Ψδ on Rn+1 induces the corresponding scaled SBF
Φδ on the unit sphere Sn,
Φδ(x,y) = Ψδ(x−y) = δ−nψ(|x−y|/δ) = δ−nψ(δ−1√
2 − 2x · y), x,y ∈ Sn, (6.6)
which has the Fourier coefficients
φδ(ℓ) =
∫ ∞
0
tFψδ(t)J2ν (t)dt =
∫ ∞
0
δtFψ(δt)J2ν (t)dt. (6.7)
We will spend the rest of this section in proving that these Fourier coefficients possessthe corresponding decay property (3.2), which is stated as the following theorem.
Theorem 6.2. If the radial function Ψ = ψ(| · |) has compact support in the unitball and a Fourier transform satisfying the condition (6.1) with τ > (n + 1)/2 thenthere are constants C1 and C2 depending only on τ and n such that the associatedscaled spherical basis function Φδ has Fourier coefficients satisfying
C1(1 + δℓ)−2τ+1 ≤ φδ(ℓ) ≤ C2(1 + δℓ)−2τ+1.
12
To achieve the goal, we first have to analyse the proof of [10, Proposition 4.1]carefully. To this end, we introduce the integral
A(ν, τ) =
∫ ∞
0
t−2τ+1J2ν (t)dt, ν > τ − 1, (6.8)
which can be explicitly computed (cf [15, Formula (2) in 13.41] and [10, Formula(4.9)]) as
A(ν, τ) =Γ(2τ − 1)Γ(ν − τ + 1)
22τ−1Γ2(τ)Γ(τ + ν)=
Γ(2τ − 1)
22τ−1Γ2(τ)
(ν − τ)Γ(ν − τ)
Γ(ν + τ). (6.9)
We will also make frequent use of Stirling’s formula for the Γ-function in the form√
2πxx−1/2e−x ≤ Γ(x) ≤√
2πxx−1/2e−xe1/(12x) (6.10)
and of the fact that
2
3|x| ≤ | log(1 + x)| ≤ 4
3|x| (6.11)
for |x| ≤ 1/4. We start with the upper bound.Proposition 6.3. If the radial function Ψ = ψ(| · |) has compact support in the
unit ball and a Fourier transform satisfying Ψ(ω) ≤ c2(1+ |ω|2)−τ with τ > (n+1)/2then there is a constant C2 > 0 depending only on τ and n such that the associatedscaled spherical basis function Φδ has Fourier coefficients satisfying
φδ(ℓ) ≤ C2(1 + δℓ)−2τ+1
for all ℓ ∈ N0 and all δ ∈ (0, 1].Proof. In a first step we will show that there is a constant c4 > 0 and an integerℓ0 ∈ N such that
φδ(ℓ) ≤ c4(δℓ)−2τ+1 (6.12)
for all ℓ ≥ ℓ0 and all δ > 0. The assumption on the decay of Ψ together with formula(6.7) lead to
φδ(ℓ) ≤ c2
∫ ∞
0
δtJ2ν (t)
(1 + (δt)2)τdt.
Since 1 + (δt)2 > (δt)2 we can conclude that
φδ(ℓ) ≤ c2δ−2τ+1
∫ ∞
0
t−2τ+1J2ν (t)dt = c2δ
−2τ+1A(ν, τ). (6.13)
Using the explicit formula (6.9) together with Stirling’s formula (6.10) and thefact that e1/(12(ν−τ)) ≤ e1/(12τ) provided that ν ≥ 2τ , we find for ν ≥ 2τ that
A(ν, τ) =Γ(2τ − 1)(ν − τ)Γ(ν − τ)
22τ−1Γ2(τ)Γ(ν + τ)
≤ Γ(2τ − 1)
22τ−1Γ2(τ)(ν − τ)
√2π(ν − τ)ν−τ−1/2e−(ν−τ)e1/(12(ν−τ))
√2π(ν + τ)ν+τ−1/2e−(ν+τ)
≤ Γ(2τ − 1)e2τe1/(12τ)
22τ−1Γ2(τ)
(ν − τ)ν−(τ−1/2)
(ν + τ)ν+τ−1/2
≤ Γ(2τ − 1)e2τe1/(12τ)
22τ−1Γ2(τ)
1
(ν2 − τ2)τ−1/2
(1 − τ
ν
1 + τν
)ν
.
13
For ν ≥ 2τ we have also ν2 − τ2 ≥ 34ν
2 and hence
1
(ν2 − τ2)τ−1/2≤ 22τ−1
3τ−1/2ν−2τ+1.
Assuming now ν ≥ 4τ , we see that (6.11) yields
log
(1 − τ
ν
1 + τν
)ν
= ν[log
(1 − τ
ν
)− log
(1 +
τ
ν
)]
= −ν[∣∣∣log
(1 − τ
ν
)∣∣∣ +∣∣∣log
(1 +
τ
ν
)∣∣∣]
≤ −ν · 2 · 2
3
τ
ν= −4
3τ.
Taking this all together gives for ν ≥ 4τ the upper bound
A(ν, τ) ≤ Γ(2τ − 1)e2τ/3e1/(12τ)
3τ−1/2Γ2(τ)ν−2τ+1,
which is consistent with the asymptotic result from [10, eq. (4.10)] for A(ν, τ). Thisupper bound together with (6.13) and ν = ℓ+ n−1
2 ≥ ℓ establishes our first step (6.12).In the second step, we show that there is a constant c5 > 0 such that
φδ(ℓ) ≤ c5 (6.14)
for all ℓ ∈ N0 and all δ > 0.Using formula (3.1), the addition theorem for spherical harmonics and the follow-
ing orthogonality∫ 1
−1
Pℓ(n+ 1; t)Pℓ′(n+ 1; t)(1 − t2)n−2
2 dt =ωn
ωn−1N(n, ℓ)δℓ,ℓ′ , (6.15)
we have
φδ(ℓ) = ωn−1
∫ 1
−1
φδ(t)Pℓ(n+ 1; t)(1 − t2)n−2
2 dt, (6.16)
in which φδ(x ·y) = Φδ(x,y). Since Ψ(x) = ψ(|x|) is a compactly supported functionwith support in the unit ball, we have ψ(r) = 0 for r ≥ 1. Hence, with the definition(6.6), the integral (6.16) is reduced to
φδ(ℓ) = δ−nωn−1
∫ 1
1−δ2/2
ψ
(√2 − 2t
δ
)Pℓ(n+ 1; t)(1 − t2)
n−2
2 dt. (6.17)
Using the fact that we integrate over the range 1 − δ2/2 ≤ t ≤ 1, we have
(1 − t2) = (1 + t)(1 − t) ≤ 2(1 − t) ≤ 2δ2
2= δ2.
Since the Legendre polynomials satisfy |Pℓ(n+1; t)| ≤ 1 for t ∈ [−1, 1], we can boundthe integral in (6.17) by
φδ(ℓ) ≤ ωn−1δ−2
∫ 1
1−δ2/2
|ψ(δ−1√
2 − 2t)|dt
= ωn−1
∫ 1
1/2
|ψ(√
2 − 2u)|du,
14
using the substitution (1 − t) = δ2(1 − u). Thus (6.14) holds for all ℓ ∈ N0.In the final step we use both of the above bounds. If δℓ ≥ 1 then we clearly have
(1 + δℓ)2τ−1 ≤ 22τ−1(δℓ)2τ−1,
which together with (6.12) gives, for ℓ ≥ max{ℓ0, 1/δ},
φδ(ℓ) ≤ 22τ−1c4(1 + δℓ)−2τ+1
For ℓ < max{ℓ0, 1/δ} we can use (6.14) and δ ≤ 1 to derive
φδ(ℓ) ≤ c5 = c5(1 + δℓ)2τ−1(1 + δℓ)−2τ+1
≤ c5 max1≤k≤max{ℓ0,1/δ}
(1 + δk)2τ−1(1 + δℓ)−2τ+1
≤ c5 max{(1 + δℓ0)2τ−1, 22τ−1}(1 + δℓ)−2τ+1
≤ c5 max{(1 + ℓ0)2τ−1, 22τ−1}(1 + δℓ)−2τ+1
This give the stated upper bound with
C2 = max{22τ−1c4, (1 + ℓ0)2τ−1c5}.
2
We will now address the lower bound. However since this is more technical webreak the actual proof up into several lemmas. We start by looking at the case whereδν is large. Here, we follow again [10] with special emphasis on the dependence on δ.
Lemma 6.4. Assume that the radial function Ψ = ψ(| · |) has a Fourier transform
satisfying Ψ(ω) ≥ c1(1 + |ω|2)−τ with τ > (n+ 1)/2. Then there is a constant c6 > 0depending only on τ such that, for ν = ℓ+ n−1
2 ≥ 1/δ and δ ≤ 1, we have
φδ(ℓ) ≥ c6(1 + δℓ)−2τ+1.
Proof. Let us introduce a constant c > 0, which we specify later on. The assumptionstogether with (6.7) allow us to bound the Fourier coefficients by
φδ(ℓ) ≥ c1
∫ ∞
0
δt
(1 + δ2t2)τJ2
ν (t)dt > c1
∫ ∞
c/δ
δt
(1 + δ2t2)τJ2
ν (t)dt,
where we used the constant c > 0 in the lower limit of the integral. In the latterintegral we now have t > c/δ, and hence, using 1 < δ2t2/c2,
1
(1 + δ2t2)τ> (δt)−2τ
(c2
1 + c2
)τ
,
and in turn
φδ(ℓ) > c1
(c2
1 + c2
)τ
δ−2τ+1
∫ ∞
c/δ
t−2τ+1J2ν (t)dt.
For ν > τ − 1, the latter integral can be decomposed, as in [10], as follows:
∫ ∞
c/δ
t−2τ+1J2ν (t)dt =
∫ ∞
0
t−2τ+1J2ν (t)dt−
∫ c/δ
0
t−2τ+1J2ν (t)dt
= A(ν, τ) −Bδ(ν, τ) = A(ν, τ)
[1 − Bδ(ν, τ)
A(ν, τ)
],
15
where A(ν, τ) is the integral from (6.8) and
Bδ(ν, τ) =
∫ c/δ
0
t−2τ+1J2ν (t)dt.
The rest of the proof contains two steps. We will show first that A(ν, τ) has thedesired dependence on ν, and then show that 1 −Bδ(ν, τ)/A(ν, τ) remains uniformlybounded away from zero under the given assumptions.
For the first task, the starting point is again the explicit formula (6.9). UsingStirling’s formula we find in a similar way to the proof of Proposition 6.3 that
A(ν, τ) =Γ(2τ − 1)
22τ−1Γ2(τ)
(ν − τ)Γ(ν − τ)
Γ(ν + τ)
≥ Γ(2τ − 1)
22τ−1Γ2(τ)(ν − τ)
√2π(ν − τ)ν−τ−1/2e−(ν−τ)
√2π(ν + τ)ν+τ−1/2e−(ν+τ)e1/(12(ν+τ))
≥ Γ(2τ − 1)e2τ
22τ−1Γ2(τ)e1/(24τ)
1
(ν2 − τ2)τ−1/2
(1 − τ
ν
1 + τν
)ν
,
using the fact that e1/(12(ν+τ)) < e1/(24τ) for ν ≥ τ . We obviously have
1
(ν2 − τ2)τ−1/2≥ ν−2τ+1.
Furthermore, the bound in (6.11) yields this time
log
(1 − τ
ν
1 + τν
)ν
= −ν[∣∣∣log
(1 − τ
ν
)∣∣∣ +∣∣∣log
(1 +
τ
ν
)∣∣∣]≥ −8
3τ,
provided that τ/ν ≤ 1/4. Thus we have for ν ≥ 4τ the bound
A(ν, τ) ≥ Γ(2τ − 1)e−2τ/3
22τ−1Γ2(τ)e1/(24τ)ν−2τ+1, (6.18)
which has the desired dependence on ν.It remains to look at the term 1 −Bδ(ν, τ)/A(ν, τ). As in [10] we use the bound
|Jν(x)| ≤ 2−νxν
Γ(ν + 1),
which holds for ν > −1/2 and x > 0, and Stirling’s formula to derive a bound forBδ(ν, τ): for ν > τ − 1 we have, with Γ(ν + 1) = νΓ(ν),
Bδ(ν, τ) =
∫ c/δ
0
t−2τ+1J2ν (t)dt ≤ 2−2ν
Γ2(ν + 1)
∫ c/δ
0
t2ν−2τ+1dt
=2−2ν−1
Γ2(ν + 1)(ν − τ + 1)
( cδ
)2ν−2τ+2
≤ 2−2ν−1
ν2(ν − τ + 1)
( cδ
)2ν−2τ+2 1
2πν2ν−1e−2ν
=2−2ν−2
π(ν − τ + 1)
( cδ
)2ν−2τ+2
ν−2ν−1e2ν .
16
This, together with the lower bound (6.18) on A(ν, τ) gives after some simplifications,for ν ≥ 4τ ,
Bδ(ν, τ)
A(ν, τ)≤ Γ2(τ)e1/(24τ)e
83
τ+2
2πΓ(2τ − 1)
1
ν − τ + 1
(2δν
ce
)2τ−2ν−2
.
If we now choose c = 2/e and use δν ≥ 1, then we see that
Bδ(ν, τ)
A(ν, τ)≤ Γ2(τ)e1/(24τ)e
83τ+2
2πΓ(2τ − 1)
1
ν − τ + 1=:
cτν − τ + 1
.
This expression becomes less than 1/2 for ν ≥ 2cτ + τ − 1. Thus, putting everythingtogether gives
φδ(ℓ) ≥ c1(δν)−2τ+1
for ν = ℓ+ n−12 ≥ max{kτ , 1/δ}, where kτ := max{4τ, 2cτ + τ − 1}. If necessary, we
can enlarge kτ so that without restriction kτ ≥ n− 1. Then, ν ≥ kτ ≥ n− 1 meansin particular ℓ ≥ (n− 1)/2 = ν − ℓ and hence ν ≤ 2ℓ, which gives
φδ(ℓ) ≥ c1(δν)−2τ+1 ≥ c12
−2τ+1(δℓ)−2τ+1 ≥ c12−2τ+1(1 + δℓ)−2τ+1,
for ν = ℓ+ n−12 ≥ max{kτ , 1/δ}, which settles our statement if kτ ≤ 1/δ. In the case
of kτ > 1/δ we have to make sure that the statement also holds for 1/δ ≤ ν ≤ kτ .This can be seen as follows. We have for all ν ≤ kτ , using 1 ≤ kτδ,
φδ(ℓ) ≥ c1
∫ kτ
0
δt
(1 + δ2t2)τJ2
ν (t)dt
≥ c1δ−2τ+12−τk−2τ
τ
∫ kτ
0
tJ2ν (t)dt
≥ c1δ−2τ+12−τk−2τ
τ infν≤kτ
∫ kτ
0
tJ2ν (t)dt
=: c6δ−2τ+1 ≥ c6(1 + ℓδ)−2τ+1,
where c6 > 0 because there are only a finite number of values of ν = ℓ+ n−12 less than
or equal to kτ . Hence, we have the stated lower bound for δν ≥ 1. 2
It remains to investigate the behaviour of φδ(ℓ) for small values of δν. For this,the following result, which is interesting on its own will be useful.
Lemma 6.5. Let ν ≥ 1/2. Let λ be any positive zero of Jν . Then,
λ[J ′ν(λ)]2 >
2
π
(λ2 − ν2)1/2
λ.
Proof. We will use the Weber functions, also known as Bessel functions of thesecond kind defined for non-integer index ν 6∈ Z by
Yν(x) =Jν(x) cos(νπ) − J−ν(x)
sin(νπ)
and by Yn := limν→n Yν for integer index, see [15, section 3.54, pages 63/64]. Theygive another fundamental solution to the Bessel differential equation. Furthermore,
17
according to [15, formula (1), section 3.63, page 76] the Wronskian of the system{Jν , Yν} is given by
Jν(x)Y ′ν (x) − J ′
ν(x)Yν(x) =2
πx(6.19)
for all ν and all x 6= 0. We will also employ the bounds
2
πx< J2
ν (x) + Y 2ν (x) <
2
π
1
(x2 − ν2)1/2, (6.20)
which hold for x ≥ ν ≥ 1/2, see [15, formula (1), section 13.74, page 447].According to [15, formula (1), section 15.3, page 485] the smallest positive root
λ1 of Jν satisfies λ1 > ν, thus we can set x = λ in (6.20) where λ is any positive rootof Jν to derive
λ[J ′ν(λ)]2 =
4
π2λ[Yν(λ)]2. (6.21)
From (6.19) we have
Y 2ν (λ) = J2
ν (λ) + Y 2ν (λ) <
2
π
1
(λ2 − ν2)1/2,
thus (6.21) gives
λ[J ′ν(λ)]2 >
2
π
(λ2 − ν2)1/2
λ.
2
Lemma 6.6. Let ν = ℓ+ n−12 ≥ 1/2. For every x ≥ 4ν there is at least one zero
of Jν in the interval [x, 2x].Proof. Let jν,k denote the kth positive zero of Jν . From [6, page 251] we have, forν ≥ 1/2,
jν,k+1 − jν,k ≤ π√1 − ν2− 1
4
j2ν,k
,
which gives, since the sequence of zeros is increasing in k,
jν,k+1 − jν,k
jν,k≤ π√
j2ν,k − ν2 + 1/4≤ π√
j2ν,1 − ν2 + 1/4≤ π√
2ν + 1/4, (6.22)
where the last inequality comes from the lower bound jν,1 ≥√ν(ν + 2), see [15,
formula (5), section 15.3, page 486]. From (6.22) it follows easily that
jν,k+1 − jν,k
jν,k≤ 1 : (6.23)
for ν ≥ 5 this follows from the last bound in (6.22), while for ν = 12 , 1,
32 , . . . ,
92 the
result follows from the penultimate expression in (6.22) by direct computation. Thebound is sharp for ν = 1/2 and is easily satisfied for 1 ≤ ν ≤ 9
2 . The inequality (6.23)is equivalent to
jν,k+1 ≤ 2jν,k. (6.24)
18
Consider first the case x > jν,1, and fix k ≥ 1 by letting jν,k be the biggest of thezeros of Jν which is smaller than x. Then we have, by (6.24),
jν,k < x ≤ jν,k+1 ≤ 2jν,k < 2x,
thus jν,k+1 ∈ [x, 2x]. To complete the argument, we consider the case 4ν ≤ x ≤ jν,1.
We know by [15, formula (5), section 15.3, page 486] that jν,1 <√
2(ν + 1)(ν + 3),and hence
ν < jν,1 ≤ ν√
2
√(1 +
1
ν
) (1 +
3
ν
)≤
√42ν < 8ν for ν ≥ 1
2.
For 4ν ≤ x ≤ jν,1 we have
4ν ≤ x ≤ jν,1 < 8ν ≤ 2x,
and hence jν,1 ∈ [x, 2x], completing the proof. 2
This now leads to a lower bound for the Fourier coefficients in the case of δν ≤ 1.Lemma 6.7. For δν ≤ 1 we have
φδ(ℓ) ≥ c7.
Proof. For any λ ≥ 1/δ we have the lower bound
φδ(ℓ) ≥ c1
∫ λ
0
δt
(1 + δ2t2)τJ2
ν (t)dt
≥ c12−τδ(δλ)−2τ
∫ λ
0
tJ2ν (t)dt
= c12−τδ(δλ)−2τ λ
2
2
[[J ′
ν(λ)]2 +
(1 − ν2
λ2
)J2
ν (λ)
],
where the explicit form of the integral comes from [4, formula (24), page 70]. If wepick λ as a zero of Jν , this reduces with the help of Lemma 6.5 to
φδ(ℓ) ≥ c12−τ−1(δλ)−2τ+1λ[J ′
ν(λ)]2 (6.25)
≥ c12−τ−1(δλ)−2τ+1 2
π
(λ2 − ν2)1/2
λ. (6.26)
Now choose λ = λν,4/δ, where for x ≥ ν
λν,x := min{λ : λ ≥ x and Jν(λ) = 0}.
Since
λν, 4δ≥ 4
δ≥ 4ν,
and since the function f(λ) := (λ2 − ν2)1/2/λ is strictly monotonically increasing on[ν,∞), we have
(λ2ν,4/δ − ν2)1/2
λν,4/δ= f(λν,4/δ) ≥ f(4ν) =
√15
4.
19
Since lemma 6.6 gives
x ≤ λν,x ≤ 2x for x ≥ 4ν,
we also have an upper bound
λν, 4δ≤ 2
4
δ=⇒ δλν, 4
δ≤ 8.
Substituting the two bounds into (6.26), we find
φδ(ℓ) ≥ c12−7τ+1
√15
π=: c7,
proving the lemma. 2
Proposition 6.3 establishes the upper bound while Lemmas 6.4 and 6.7 withC1 := min{c6, c7} give the lower bound stated in Theorem 6.2.
7. Condition number and cost. We will now analyse the condition numberand cost of the multiscale algorithm. For simplicity, we will drop the index indicatingthe current level. Thus in the present section X = Xj is the point set at the step j.
As in the previous section, we will assume that the spherical basis function Φδ isdefined by the restriction of a positive definite radial function Ψδ = δ−nΨ(·/δ), wherethe unscaled basis function Ψ generates Hτ (Rn+1), i.e. it satisfies (6.1).
In each step of the multiscale algorithm, we have to compute an interpolant ofthe form (3.3), where the coefficients bj for j = 1, . . . ,M are determined by a linearsystem
AX,δb = f . (7.1)
Here, AX,δ = (ai,j) is the interpolation matrix with entries ai,j = Φδ(xi,xj) =δ−nΨ((xi − xj)/δ) and f = (f(x1), . . . , f(xM ))T . The nature of the multiscale algo-rithm guarantees that we can assume that δ = νhX with ν > 1 a fixed number, i.e. δis proportional to hX and this number ν is independent of the level.
Since the matrix AX,δ is symmetric and positive definite, an iterative methodsuch as the conjugate gradient method can be used to solve (7.1) efficiently. Thecomplexity of the conjugate gradient method depends on the condition number of thematrix AX,δ and on the cost of a matrix-vector multiplication.
We start with a discussion of the condition number. It is well known, see forexample [18, Section 12.2] that the lower bound on the Fourier transform in (6.1)gives for the RBF with scale δ and the point set X a lower bound on the minimaleigenvalue of the form
λmin(AX,1) ≥ cq2τ−(n+1)X , (7.2)
with a constant c > 0 independent of X (but depending on δ1), where we are usingthe Euclidean separation radius qX in Rn+1 from (2.1). We now introduce the dataset X/δ = {x1/δ, . . . ,xM/δ}, which obviously has Euclidean separation radius
qX/δ = qX/δ,
and make the simple observation that
AX,δ = (Φδ(xi,xj)) =
(δ−nΨ
(xi − xj
δ
))= δ−nAX/δ,1.
20
From this and (7.2) we obtain
λmin(AX,δ) ≥ cδ−n
(qXδ
)2τ−(n+1)
.
Thus, we have the following result.Lemma 7.1. The minimum eigenvalue of the matrix AX,δ can be bounded by
λmin(AX,δ) ≥ cδ−n
(qXδ
)2τ−(n+1)
with a constant c > 0 independent of X and δ.We state an upper bound of the maximum eigenvalue of Aδ in the following
lemma.Lemma 7.2. Suppose Ψ : Rn+1 → R has support in the unit ball and is bounded
by 1. If the Euclidean separation radius qX is less than 1/2 then
λmax(AX,δ) ≤ Cδ−n
(qXδ
)−n−1
.
Proof. According to [17, Theorem 4.2] we have
λmax(AX,1) ≤(
4
qX
)n+1
and the same result together with the previous scaling argument leads to
λmax(AX,1) ≤ 4n+1δ−n
(qXδ
)−n−1
,
which is the desired estimate. 2
From the previous lemmas, we have the following theorem.Theorem 7.3. The condition number κ(AX,δ) of the interpolation matrix AX,δ
is bounded by
κ(AX,δ) ≤ C
(δ
qX
)2τ
,
where C > 0 is independent of X and the scaling parameter δ.Thus Theorem 7.3 leads to the conclusion that if δ/qX ≤ c then the condition
number should remain constant. However, since we already know that δ = νhX , thesetwo assumptions lead to
νqX ≤ νhX = δ ≤ cqX .
Bearing in mind that the Euclidean and geodesic norm are comparable, this meansthat the data set has to be quasi-uniform.
We finish with a result on the cost of the multilevel algorithm.Theorem 7.4. Suppose that the data sets used in each step of the multiscale al-
gorithm are quasi-uniform. Then, for a given precision the conjugate gradient methodcan solve the linear system (7.1) on each level in O(M logM) time, where M denotesthe number of points at that level.
21
Proof. The most expensive operation in the conjugate gradient method is the matrixvector product. However, in our case the interpolation matrix AX,δ is sparse and thenumber of nonzero entries in each row is bounded by a constant, which is independentof the level. Hence, each matrix-vector multiplication can be done in linear time.
According to Theorem 7.3 the condition number on each level is bounded by thesame constant, which means that the number of iterations of the conjugate gradientmethod is also bounded by a constant independent of the level.
Hence, if the nonzero entries of the matrix AX,δ are known, the conjugate gradientmethod can solve the system in linear time. However, with scattered data points thenonzero entries are at unknown locations, and have to be sought in a preprocessingstep. This boils down to finding, for every xi, all data sites xj within the supportof Ψδ(xi − ·). This is a classical problem in computational geometry and efficientmethods based on tree-structures are known to solve this in O(M logM) time, if Mis the number of points used for the finest level. 2
In general, we will keep the number of steps in the multiscale algorithm smallcompared to the number of data sites on the final level, such that this number can beseen as a constant. Hence, the overall complexity for solving the original interpolationproblem is again of the form O(M logM) , if M is the number of points at the finestlevel.
8. Numerical experiments. In the experiments, the sets X1, . . . , X5 are gen-erated using a modified version of Saff’s equal area algorithm [12] so that latitudesand longitudes (in degree) of the points are odd multiples of half a degree. The car-dinalities, mesh norms and separation radii of these sets are listed in Table 8.1. The
X1 X2 X3 X4 X5
M 32 125 500 2000 8000hX 0.4372 0.2288 0.1165 0.0613 0.0367qX 0.3020 0.1281 0.0637 0.0263 0.0091
Table 8.1
Cardinalities, mesh norms and separation radii of the sets of points
function we wish to interpolate is the topographic data from NOAA (US Departmentof Commerce), which is available in MATLAB as an array of size (180, 360) (whichcan be invoked by the command load topo) as shown in Figure 8.1. The functionvalues represent the average heights in meters above and below the mean sea level atcells of size 1 degree times 1 degree. The RBF used in the experiment is the Wendlandfunction
ψ(r) = ψ3,1(r) = (1 − r)4+(4r + 1)
and the scaled version is
ψδ(r) = δ−2(1 − r/δ)4+(4r/δ + 1)
where at stage j, we set δ = δj. We remark that the target function does not belongto the native space.
In Table 8.2 we show the errors and the condition numbers as reported by MAT-LAB at each stage of three independent multilevel experiments, using always thesame 5 point sets given in Table 8.1, but with different starting values for the scale ofthe SBF (the three starting values of δ1 being 2, 1, 1/2 respectively). The condition
22
Fig. 8.1. Exact function
numbers are in good agreement with Theorem 7.3, and by the usual standards areall small. In particular, Experiment 3 has condition numbers reflecting the fact thatAδ is essentially a constant matrix, but at the cost of a very large errors. The errors‖ej‖ in the table are approximations to the true errors in the mean-square sense: inprinciple we define
‖ej‖ :=
(1
4π
∫
S2
|f(x) − fj(x)|2dx)1/2
=
(1
4π
∫ π
0
∫ 2π
0
|f(θ, φ) − fj(θ, φ)|2 sin θdφdθ
)1/2
,
and in practice approximate this by the product midpoint rule at 1 degree intervals,
1
4π
2π2
|G|∑
x(θ,φ)∈G
|f(θ, φ) − fj(θ, φ)|2 sin θ
1/2
,
where G is the grid containing the centres of rectangles of size 1 degree times 1 degreeand |G| = 180 · 360 = 64800.
The approximations f3, f4, f5, the details s4, s5 and the signed error f5 − f forExperiment 2 are shown in Figure 8.2. In Table 8.3 we show the results obtainedwith different scales by the “one-shot” approximation. Finally, in Figures 8.3 and 8.4we show for comparison the final multiscale approximation obtained with Experiment2, and therefore finishing at scale 1/16, and the one-shot approximation at scale 16.The latter shows a marked graininess, reflecting the poor overlap of the scale 16 radialbasis functions on the 8, 000 point set.
REFERENCES
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radial basis functions, J. Comput. Appl. Math., 73 (1996), pp. 65–78.
23
Fig. 8.2. Approximations f3, f4, f5 and details s4, s5 and signed error f5 − f for Experiment 2.
[2] W. Freeden, Multiscale modelling of spaceborne geodata, B. G. Teubner, Stuttgart, 1999.[3] W. Freeden, T. Gervens, and M. Schreiner, Constructive Approximation on the Sphere
with Applications to Geomathematics, Oxford University Press, Oxford, 1998.[4] A. Gray and G.B. Mathews, A treatise on Bessel functions and their applications to physics,
McMillan, London, 1952.[5] S. J. Hales and J. Levesley, Error estimates for multilevel approximation using polyharmonic
splines, Numer. Algorithms, 30 (2002), pp. 1–10.[6] H. Hochstadt, The functions of mathematical physics, Dover, New York, 1986.[7] Q. T. Le Gia, F. J. Narcowich, J. D. Ward, and H. Wendland, Continuous and discrete
least-squares approximation by radial basis functions on spheres, J. Approx. Theory, 143(2006), pp. 124–133.
[8] C. Muller, Spherical Harmonics, Springer, Berlin, 1966.[9] F. J. Narcowich, R. Schaback, and J. D. Ward, Multilevel interpolation and approximation,
Appl. Comput. Harmon. Anal., 7 (1999), pp. 243–261.[10] F. J. Narcowich, X. Sun, and J. D. Ward, Approximaton power of RBFs and their associated
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(1997), pp. 5–11.[13] R. Schaback, Multivariate interpolation and approximation by translates of a basis function,
24
Level j 1 2 3 4 5δj 2 1 1/2 1/4 1/8
Experiment 1 κ(Aδj) 42.22 53.54 49.45 83.47 174.21
‖ej‖ 1955.40 1490.14 999.75 637.25 371.81δj 1 1/2 1/4 1/8 1/16
Experiment 2 κ(Aδj) 1.71 2.14 2.16 4.09 7.80
‖ej‖ 1961.31 1432.02 965.86 615.98 364.83δj 1/2 1/4 1/8 1/16 1/32
Experiment 3 κ(Aδj) 1.00 1.00 1.00 1.01 1.29
‖ej‖ 2831.27 2321.06 1852.14 1466.27 1169.33Table 8.2
Errors and condition numbers of multiscale approximation
δ 2 1 1/2 1/4 1/8 1/16κ(Aδ) 1.18e+8 3.70e+6 1.19e+5 4.14e+3 174.21 7.80‖e‖ 373.61 373.65 373.54 373.19 375.39 894.21
Table 8.3
One shot approximations with different scales
in Approximation Theory VIII, Vol. 1: Approximation and Interpolation, C. K. Chui andL. L. Schumaker, eds., Singapore, 1995, World Scientific Publishing, pp. 491–514.
[14] I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J., 9 (1942), pp. 96–108.[15] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press,
Cambridge, 1966.[16] H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions
of minimal degree, Adv. Comput. Math., 4 (1995), pp. 389–396.[17] , Error estimates for interpolation by compactly supported radial basis functions of min-
imal degree, J. Approx. Theory, 93 (1998), pp. 258–272.[18] , Scattered Data Approximation, Cambridge Monographs on Applied and Computational
Mathematics, Cambridge University Press, Cambridge, UK, 2005.[19] Y. Xu and E. W. Cheney, Strictly positive definite functions on spheres, Proc. Am. Math.
Soc., 116 (1992), pp. 977–981.
25
Fig. 8.3. Multiscale approximation f5 with finishing scale δ = 1/16.
Fig. 8.4. One shot approximation with δ = 1/16.