Within the conventional framework of a native space structure, a smooth kernel generates a
small native space, and radial basis functions stemming from the smooth kernel are intended to
approximate only functions from this small native space. In this paper, we embed the smooth
radial basis functions in a larger native space generated by a less smooth kernel and use them
to interpolate the samples. Our result shows that there exists a linear combination of spherical
radial basis functions that can both exactly interpolate samples generated by functions in the
larger native space and near best approximate the target function.

1. Introduction

Many scientific questions boil down to synthesizing an unknown but definite function from finitely many samples (xi,yi)i=1m. The purpose is to find a functional model that can effectively represent or approximate the underlying relation between the input xi and the output yi. If the unknown functions are defined on spherical domains, then data collected by satellites or ground stations are usually not restricted on any regular region and are scattered. Thus, any numerical method relying on the structure of a “grid” is doomed to fail.

The success of the radial basis function networks methodology in Euclidean space derives from its ability to generate estimators from data with essentially unstructured geometry. Therefore, it is natural to borrow this ideal to deal with spherical scattered data. This method, called the spherical radial basis function networks (SRBFNs), has been extensively used in gravitational phenomenon [1, 2], image processing [3, 4], and learning theory [5].

Mathematically, the SRBFN can be represented as
(1)Sn(x)∶=∑i=1nciϕ(ξi·x)∶=∑i=1nciϕξi(x),x∈Sd,
where ci∈R, ξi∈Sd, and ϕ:[-1,1]→R are called the connection weight, center, and activation function in the terminology of neural networks, respectively. Here and hereafter, Sd denotes the unit sphere embedded in the d+1-dimensional Euclidean space, Rd+1. Both the connection weights ci and the centers ξi are adjustable in the process of training. We denote by Φn the collection of functions formed as (1).

A basic and classical approach for training SRBFN is to construct exact interpolant based on the given samples (xi,yi)m, that is, to find a function Sn in Φn such that
(2)Sn(xi)=yi,i=1,…,m.

If the activation function ϕ is chosen to be positive definite [2], then the matrix A:=(ϕ(xi·xj))i,j=1m is nonsingular. Thus, the system of (2) can be easily solved by taking the scattered data as the centers of SRBFN. This method has been already named the spherical basis function (SBF) method. Under this circumstance, the connection weights are determined via
(3)c=A-1y,
where c=(c1,c2,…,cm)T, y=(y1,…,ym)T, UT denotes the transpose of the matrix (or vector) U, and A-1 denotes the inverse matrix of A.

For the SBF method, if the target function f belongs to the native space Nϕ associated with the activation function ϕ, then the best SBF approximant of f is characterized by the exact interpolation:
(4)∥f-ci∑i=1mϕxi∥Nϕ=infg∈Φm∥f-g∥Nϕ,
where {ci}i=1m are determined by (3). This property makes the SBF interpolation strategy popular in spherical scattered data fitting [6–14]. However, there are also two disadvantages for the SBF method. On one hand, since the centers of SRBFN interpolants are chosen as the scattered data, the interpolation capability depends heavily on their geometric distributions. This implies that we cannot obtain a satisfactory interpolation error estimate if the data are “badly” located on the sphere. On the other hand, the well known “native space barrier" [11, 12, 15] shows that (4) only holds for a small class of smooth functions if ϕ is smooth. Therefore, for functions outside Nϕ, the SBF interpolants are not guaranteed to be the best approximants.

Along the flavor of the previous papers [7, 11, 12], we use SRBFNs to interpolate functions in a large native space Nψ associated with the kernel ψ which is less smooth than ϕ and study its interpolation capability. Different from the previous work, the centers are chosen in advance to be quasi-uniform located on spheres, which makes the interpolation error depend on the number rather than the geometric distribution of centers. Our purpose is not to give the detailed error estimate of the SRBFN interpolation. Instead, we focus on investigating the relation between interpolation and approximation for SRBFN. Indeed, we find that there exists an SRBFN interpolant which is also the near best approximant of functions in Nψ, when the number of centers and geometric distribution of the scattered data satisfy a certain assumption. That is, for a suitable choice of n, there exists a function gn∈Φn such that

g exactly interpolates the samples (xi,yi)i=1m;

∥f-gn∥Nψ≤Cinfg∈Φn∥f-g∥Nψ,

where C is a constant depending only on d, α, and β.

2. Positive Radial Basis Function on the Sphere

It is easy to deduce that the volume of Sd, Ωd, satisfies
(5)Ωd∶=∫Sddω=2π(d+1)/2Γ((d+1)/2),
where dω denotes the surface area element on Sd. For integer k≥0, the restriction to Sd of a homogeneous harmonic polynomial of degree k on the unit sphere is called a spherical harmonic of degree k. The span of all spherical harmonics of degree k is denoted by Hkd, and the class of all spherical harmonics (or spherical polynomials) of degree k≤s is denoted by Πsd. It is obvious that Πsd=⊕k=0sHkd. The dimension of Hkd is given by
(6)Dkd∶=dimHkd={2k+d-1k+d-1(k+d-1k),k≥1;1,k=0,
and that of Πsd is ∑k=0sDkd=Dsd+1~sd.

Denote by {Yk,j}j=1Dkd an orthonormal basis of Hkd; then the following addition formula [16, 17] holds
(7)∑l=1DkdYk,l(x)Yk,l(y)=DkdΩdPkd+1(x·y),
where Pkd+1 is the Legendre polynomial with degree k and dimension d+1. The Legendre polynomial Pkd+1 can be normalized such that Pkd+1(1)=1 and satisfies the orthogonality relation
(8)∫-11Pkd+1(t)Pjd+1(t)(1-t2)(d-2)/2dt=ΩdΩd-1Dkdδk,j,
where δk,j is the usual Kronecker symbol.

Positive definite radial basis functions on spheres were introduced and characterized by Schoenberg [18]. Namely, a radial basis function φ is positive definite if and only if its expansion
(9)φ(x·y)=∑k=0∞ekDkdΩdPkd(x·y)
has all Fourier-Legendre coefficients ek≥0 and ∑k=0∞ek(Dkd/Ωd)<∞. We define the native space Nφ as
(10)Nφ∶={f(x)=∑k=0∞∑j=1Djdf^k,jYk,j(x):∑k=0∞ek-1∑j=1Djdf^k,j2<∞},
with its inner product
(11)〈f,g〉Nφ:=∑k=0∞ek-1∑j=1Djdf^k,jg^k,j,
where f^k,j:=∫Sdf(x)Yk,j(x)dω(x).

3. Interpolation and Near Best Approximation

Let Λ:={ηj}j=1n⊂Sd be a set of points and d(x,y)=arccosx·y be the spherical distance between xand y. We denote by hΛ:=maxx∈Sdminjd(x,ηj), qΛ:=(1/2)minj≠kd(ξj,ηk), and τΛ:=hΛ/qΛ the mesh norm, separation radius, and mesh ratio of Λ, respectively. It is easy to check that these three quantities describe the geometric distribution of points in Λ. The τ-uniform set Fτ:=Fτ(Sd) is defined by the family of all centers sets Ξ with τΛ≤τ.

Let ϕ and ψ satisfy
(12)ϕ(x·y)∶=∑k=0∞akDkdΩdPkd(x·y),(13)ψ(x·y)∶=∑k=0∞bkDkdΩdPkd(x·y)
with
(14)ak~(1+λk)-α,bk~(1+λk)-β.

The Sobolev embedding theorem [12] implies that if α,β>d/2, then Nϕ and Nψ are continuously embedded in C(Sd), and so there are reproducing kernel Hilbert spaces, with reproducing kernels being ϕ and ψ, respectively.

The aim of this section is to study the relation between the exact interpolation and best approximation for Φn with its centers set Ξn:={ξi}i=1n and activation function ϕ satisfying (12) and (14). It is obvious that such a Φn is a linear space. The following Theorem 1 shows that there exists an SRBFN interpolant which can near best approximate f∈Nψ in the metric of Nψ, where ψ satisfies (13) and (14).

Theorem 1.

Let X:={xi}i=1m be the set of scattered data with separation radius qX, and α>β>d/2. If Ξn∈Fτ and n≥CτqX-d, where Cτ>1 is a constant depending only on τ and d, then, for every f∈Nψ, there exists an SRBFN interpolant Sn∈Φn such that

Sn exactly interpolates the samples (xi,yi)i=1m,

∥f-Sn∥Nψ≤5infg∈Φn∥f-g∥Nψ.

Remark 2.

Similar results have been considered for spherical polynomials both in C(Sd) and Nψ. Narcowich et al. [11, 12] proved that there exists a spherical polynomial interpolant of degree at most L≥CqX which can also best approximate the target both in C(Sd) and Nψ.

To prove Theorem 1, we need the following three lemmas, which can be found in [12, Proposition 5.2], [12, Theorem 5.5], and [19, Example 2.10], respectively.

Lemma 3.

Let 𝒴 be a (possibly complex) Banach space, 𝒱 a subspace of 𝒴, and Z* a finite-dimensional subspace of 𝒴*, the dual of 𝒴. If for every z*∈Z* and some γ>1, γ independent of z*,
(15)∥z*∥𝒴*≤γ∥z*∣𝒱∥𝒱*,
then for y∈𝒴 there exists v∈𝒱 such that v interpolates y on Z*; that is, z*(y)=z*(v) for all z*∈Z*. In addition, v approximates y in the sense that ∥y-v∥𝒴≤(1+2γ)dist𝒴(y,𝒱).

Lemma 4.

Let β>d/2. If f∈Nϕ, then there is a u∈Φn such that
(16)∥f-u∥Nψ≤ChΞnα-β∥f∥Nϕ.

Lemma 5.

Let ψ be defined in (13) and (14) and β>d/2. Then for arbitrary set of real numbers {ci}i=1m, we have
(17)∥∑i=1mciψxi∥Nψ2≥CqX2β∑i=1mci2,
where C is a constant depending only on d and β.

Now we provide the proof of Theorem 1.

Proof of Theorem <xref ref-type="statement" rid="thm1">1</xref>.

We apply Lemma 3 to the case in which the underlying space is the native space 𝒴=Nψ. Let Z*=span{δxi}i=1m and 𝒱=Φn. So in order to prove Theorem 1, it suffices to prove that for arbitrary m real numbers {ci}i=1m such that
(18)∥∑i=1mciδxi∥Nψ*≤5∥∑i=1mciδxi|Φn∥Φn*.
Since Nψ is a reproducing kernel Hilbert space with ψ its reproducing kernel, we have
(19)∥∑i=1mciδxi∥Nψ*=supf∈Nψ,∥f∥Nψ=1|∑i=1mcif(xi)|∥∑i=1mciδxi∥Nψ*=supf∈Nψ,∥f∥Nψ=1|〈∑i=1mcif,ψxi〉Nψ|∥∑i=1mciδxi∥Nψ*=supf∈Nψ,∥f∥Nψ=1|〈∑i=1mciψxi,f〉Nψ|∥∑i=1mciδxi∥Nψ*=∥∑i=1mciψxi∥Nψ.
Similarly, we obtain
(20)∥∑i=1mciδxi|Φn∥Φn*=supg∈Φn,∥g∥Nψ=1|∑i=1mcig(xi)|∥∑i=1mciδxi|Φn∥Φn*=supg∈Φn,∥g∥Nψ=1|〈∑i=1mcig,ψxi〉Nψ|∥∑i=1mciδxi|Φn∥Φn*=supg∈Φn,∥g∥Nψ=1|〈∑i=1mciψxi,g〉Nψ|∥∑i=1mciδxi|Φn∥Φn*=∥s∥Nψ,
where s is the orthogonal projection of ∑i=1mciψxi to Φn in the metric of Nψ. Then the Pythagorean theorem yields that
(21)∥s∥Nψ2=∥∑i=1mciψxi∥Nψ2-∥∑i=1mciψxi-s∥Nψ2.
Thus, Lemma 3 with γ=2 yields that in order to prove (18), it suffices to prove
(22)∥∑i=1mciψxi-s∥Nψ∥∑i=1mciψxi∥Nψ≤32.
Let L∈ℕ, P∈ΠL be the best polynomial approximation of ∑i=1mciψxi in the metric of Nψ; then for arbitrary s′∈Φn, we have
(23)∥∑i=1mciψxi-s∥Nψ≤∥∑i=1mciψxi-P∥Nψ+∥P-s′∥Nψ.
It follows from [12, Page 382] that there exists a constant, κ, depending only on β such that for arbitrary L≥κqX-1, there holds
(24)∥∑i=1mciψxi-P∥Nψ≤34∥∑i=1mciψxi∥Nψ.

Let s′ be the best Φn approximation of P in the metric of Nψ. Then it follows from Lemma 4 and the well-known Bernstein inequality [17] that
(25)∥P-s′∥Nψ≤ChΞα-β∥P∥Nψ≤ChΞα-βLα∥P∥2.
Since P is the best polynomial approximation of ∑i=1mciψxi in the metric of Nψ, a simple computation yields
(26)∥P∥22=∑i=1mci2.
Furthermore, it follows from Lemma 5 that
(27)∥∑i=1mciψxi∥Nψ2≥CqX2β∑i=1mci2.
Then, Ξn∈Fτ together with L=max{[κqX-1]+1,[3β/2/(4C)βqX]} yields that
(28)∥P-s′∥Nψ≤ChΞα-βLαqX-β∥∑i=1mciψxi∥Nψ∥P-s′∥Nψ≤C(n-1/dqX)α-β∥∑i=1mciψxi∥Nψ,
where C is a constant depending only on β, d, and τ. Thus, there exists a constant Cτ depending only on α, β, d, and τ such that for arbitrary n≥CτqXd, there holds
(29)∥P-s′∥Nψ≤34∥∑i=1mciψxi∥Nψ.
Inserting (24) and (29) into (23), we finish the proof of (22) and then complete the proof of Theorem 1.

Acknowledgments

An anonymous referee has carefully read the paper and has provided to us numerous constructive suggestions. As a result, the overall quality of the paper has been noticeably enhanced, to which we feel much indebted and are grateful. The research was supported by the National 973 Programming (2013CB329404), the Key Program of National Natural Science Foundation of China (Grant no. 11131006), and the National Natural Science Foundations of China (Grant no. 61075054).

FreedenW.GervensT.SchreinerM.FreedenW.MichelV.Constructive approximation and numerical methods in geodetic research today—an attempt at a categorization based on an uncertainty principleTsaiY. T.ShihZ. C.All-frequency precomputed radiance transfer using spherical radial basis functions and clustered tensor approximationTsaiY. T.ChangC. C.JiangQ. Z.WengS. C.Importance sampling of products from illumination and BRDF using spherical radial basis functionsMinhH. Q.Some properties of Gaussian reproducing kernel Hilbert spaces and their implications for function approximation and learning theoryJetterK.StöcklerJ.WardJ. D.Error estimates for scattered data interpolation on spheresLevesleyJ.SunX.Approximation in rough native spaces by shifts of smooth kernels on spheresMhaskarH. N.NarcowichF. J.WardJ. D.Approximation properties of zonal function networks using scattered data on the sphereMhaskarH. N.NarcowichF. J.PrestinJ.WardJ. D.L^{p} Bernstein estimates and approximation by spherical basis functionsMortonT. M.NeamtuM.Error bounds for solving pseudodifferential equations on spheres by collocation with zonal kernelsNarcowichF. J.WardJ. D.Scattered data interpolation on spheres: error estimates and locally supported basis functionsNarcowichF. J.SunX.WardJ. D.WendlandH.Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functionsSunX.CheneyE. W.Fundamental sets of continuous functions on spheresSunX.ChenZ.Spherical basis functions and uniform distribution of points on spheresNarcowichF. J.SchabackR.WardJ. D.Approximations in Sobolev spaces by kernel expansionsMüllerC.WangK. Y.LiL. Q.SchoenbergI. J.Positive definite functions on spheresNarcowichF. J.SivakumarN.WardJ. D.Stability results for scattered-data interpolation on Euclidean spheres