# ${L}_{p}$ Error estimate for minimal norm SBF interpolation

- Jianjun Wang
^{1}Email author, - Chan-Yun Yang
^{2}and - Zhigang Gu
^{1}

**2013**:510

https://doi.org/10.1186/1029-242X-2013-510

© Wang et al.; licensee Springer. 2013

**Received: **25 February 2013

**Accepted: **27 August 2013

**Published: **8 November 2013

## Abstract

By the method of spherical splitting, the interpolation capability of the spherical basis function (SBF) is investigated. As the main result, we deduce the error estimate for the minimal norm SBF interpolation in the metric of the *p* th Lebesgue integral function space on the sphere. The result shows that the interpolation capability of SBF depends not only on the smoothness of the target function, but also on the geometric distributions of the interpolation knots.

**MSC:**41A36, 41A25.

## Keywords

## 1 Introductions and main results

Over the past decades, research in the field of function approximation of scattered data points gradually drifted from the polynomials to the radial basis functions (RBFs), and later to the spherical basis functions (SBFs) in a spherical coordinate. Recently, people, such as Wang and Li [1], Freeden *et al.* [2], and Muller [3], have moved their interests further to the topics of spherical approximation [1–7]. Based on the developments, spherical harmonic analysis was established and had some considerable progress. Meanwhile, ${L}_{p}$ error estimations for the SBF approximation were studied hereafter and stepped forward in the studies of Le Gia *et al.* [4], Hubbert and Morton [5], and Sloan and Wendland [7] after 2004. In 2007, Chen [8] established error bound for the minimal norm interpolation on a sphere. With this plentiful foundation, Lin and Cao [9] embedded firstly the smooth SBFs in a native space and specified the error bound between the best approximation and the target function via ${L}_{p}$ metric. As a consecutive study, the paper thus aims to derive a minimal norm interpolation with the ${L}_{p}$ measure.

*q*($q\ge 1$), ${S}^{q}$ is a unit sphere in ${R}^{q+1}$,

*i.e.*, ${S}^{q}=\{x=({x}_{1},{x}_{2},\dots ,{x}_{q+1})\in {R}^{q+1},{x}_{1}^{2}+{x}_{2}^{2}+\cdots +{x}_{q+1}^{2}=1\}$, and

*dω*represents a sufficient small elemental area on the spherical surface ${S}^{q}$. The total surface area of ${S}^{q}$ can hence be expressed as

*dω*, the ${L}^{p}({S}^{q})$ inner product of functions

*f*and

*g*is given as

where ${L}^{p}({S}^{q})$ represents a function space constructed by a complex function, $f:{S}^{q}\to C$, fulfilling ${\parallel f\parallel}_{p}<+\mathrm{\infty}$. An identical function in ${L}^{p}({S}^{q})$ is characterized as a function identical to the functions which have the same output values everywhere with the same inputs.

*k*th order homogeneous harmonic polynomials $p(x)$ restricted by ${S}^{q}$, and ${\mathrm{\Pi}}_{n}^{q}$ denotes the function constructed by all the

*k*th order, $k\le n$, spherical harmonic polynomials. The relationship between ${\mathrm{\Pi}}_{n}^{q}$ and ${H}_{k}^{q}$ can primarily be given as

*ξ*and

*η*. ${P}_{k}(q+1:x)$ denotes the

*k*th order Legendre polynomial satisfying ${P}_{k}(q+1:1)=1$ and complies with

*i.e.*, when $s>\frac{q}{2}$, ${H}^{s}({S}^{q})$ has a reproducing kernel

if ${A}_{n}$ satisfies ${A}_{n}\sim 1+{n}^{2}$ (see the reference [7]).

where ${P}_{n}(q+1,x)$ is the *n* th order Legendre polynomial. Here $K(\xi ,\eta )$ is obviously a spherical radial basis function.

*N*points taken from the unit sphere ${S}^{q}$ and has

*N*output values ${y}_{1},{y}_{2},\dots ,{y}_{N}$ corresponding to ${x}_{1},{x}_{2},\dots ,{x}_{N}$ through a function

*F*:

*F*is a problem subject to a minimal norm. By denoting ${S}_{N}(\xi )$ as an interpolation with the minimal norm, there is a kernel basis expression for the interpolation ${S}_{N}(\xi )$ (see the references [2, 7, 8]):

*i.e.*, $f({x}_{i})={y}_{i}$, $i=1,\dots ,N$. The interpolation problem becomes intrinsically a function approximation problem to estimate the error between $f(x)$ and ${S}_{N}(x)$. The approximation order of the interpolation can then be determined by the grid norm,

*h*, and the input ${x}_{i}\in X$. Here,

*h*is defined as

where $dist(\xi ,\eta )$ is the spherical distance between *ξ* and *η*, *i.e.*, $dist(\xi ,\eta )=arccos(\xi ,\eta )$, $\xi ,\eta \in {S}^{q}$.

*i.e.*,

for an arbitrary $f\in {H}^{s}({S}^{q})$, if ${S}_{N}$ is interpolated via the minimal norm.

for a given arbitrary $v\in {V}_{x}$. It implies that ${S}_{N}$ is an orthogonal projection of *f* to the span ${V}_{x}$ in the space ${H}^{s}({S}^{q})$.

*m*such that the grid norm $h\le \frac{1}{2m}$ for $X=\{{x}_{1},{x}_{2},\dots ,{x}_{N}\}$ is taken from ${S}^{q}$, we deduce that the ${L}^{p}$ ($p=+\mathrm{\infty}$) approximation via the minimal norm must satisfy

for an arbitrary $f\in {H}^{s}({S}^{q})$ (refer to [8]). Coming with these contributions, this study is sought to extend the significant results to a more general case of ${L}^{p}$ ($1\le p\le +\mathrm{\infty}$).

**Theorem 1.1**

*Assume that*$X=\{{x}_{1},{x}_{2},\dots ,{x}_{N}\}$

*is a set of*

*N*

*points taken from the unit sphere*${S}^{q}$,

*h*

*is the corresponding grid norm*,

*and there exists a positive integer*

*m*

*such that*$h\le \frac{1}{2m}$.

*For an arbitrary function*$f\in {H}^{s}({S}^{q})$, $s>\frac{q}{2}$,

*and its corresponding interpolation via the minimal norm*${S}_{N}$,

*there must exist a positive constant*

*C*,

*which is independent of*

*f*

*and*

*h*,

*such that the following estimations are satisfied*:

*and*

**Theorem 1.2**

*Assume that*$X=\{{x}_{1},{x}_{2},\dots ,{x}_{N}\}$

*is a set of*

*N*

*points taken from the unit sphere*${S}^{q}$,

*and the grid norm satisfies the condition*$h\le \frac{1}{2m}$.

*For an arbitrary function*$f\in {H}^{s}({S}^{q})$, $s>\frac{q}{2}$,

*and its corresponding interpolation via the minimal norm*${S}_{N}$,

*we have*

*and*

*where* *C* *denotes a positive constant independent of* *f* *and* *h*.

*γ*, the spherical cap $G({x}_{0},\gamma )$ is defined as

## 2 Lemmas

To completely prove Theorems 1.1 and 1.2, five lemmas are given as follows.

**Lemma 2.1** (refer to [5, 6, 9])

*Assume an integer*$q\ge 1$,

*constants*$M=2\sqrt{q}$

*and*${\delta}_{q}=\frac{1}{{(4q+1)}^{\frac{3}{2}}}$,

*and*${h}_{0}:=\frac{\theta}{M+{M}_{1}+{\delta}_{q}}$

*with an arbitrary positive number*${M}_{1}$,

*and*$\theta \in (0,\frac{\pi}{3})$,

*then there exists a point set*${Z}_{h}\subset {S}^{q}$

*with an arbitrary*$h\in (0,{h}_{0})$

*satisfying*

*If*${F}_{A}$

*represents the characteristic function of a given set*$A\in {S}^{q}$ ,

*there exists a positive integer*

*Q*

*independent of*

*h*

*satisfying*

*where* $\overline{M}=M+{M}_{1}$. *Furthermore*, *there exists a constant* ${C}_{Q}$ *independent of* *h* *such that* $|{Z}_{h}|\le {C}_{Q}{h}^{-q}$.

**Lemma 2.2** (refer to [5, 6, 9])

*By giving constants*$S\ge 0$, ${M}_{1}$,

*C*, $\overline{M}=M+{M}_{1}$, ${h}_{0}=\frac{C}{3\overline{M}}$

*and*$h\in (0,{h}_{0})$,

*there must exist an arbitrary*$f\in {H}^{s}({S}^{q})$

*such that*

**Lemma 2.3** *Assume that* ${\eta}_{1}^{\ast}$ *is an evaluation functional corresponding to* ${\eta}_{1}$ *in* ${H}^{s}({S}^{q})$, ${\eta}_{1}^{\ast}$ *in* ${H}^{s}({S}^{q})$ *can alternatively be expressed as* $K({\eta}_{1},\xi )$.

**Lemma 2.4**

*Assume that*${x}^{\ast},{x}_{1}^{\ast},{x}_{2}^{\ast},\dots ,{x}_{N}^{\ast}$

*are evaluation functionals in*${H}^{s}({S}^{q})$

*and their expressions in*${H}^{s}({S}^{q})$

*are*$u,{u}_{1},\dots ,{u}_{N}$.

*Together with*${S}_{N}$,

*which is interpolated via the minimal norm along*${x}_{1},{x}_{2},\dots ,{x}_{N}$,

*and*

*z*,

*which is the optimized approximation of*

*u*

*in the*$span\{{u}_{1},{u}_{2},\dots ,{u}_{N}\}$,

*we have*

*It should be noted that Lemmas* 2.3 *and* 2.4 *can be directly obtained from* [9].

**Lemma 2.5** (refer to [4, 5, 7])

*There exist two constants*$C>0$

*and*${h}_{1}>0$,

*dependent only on*

*s*

*and*

*d*,

*such that a function*

*g*($g\in {H}^{s}({S}^{q})$, $g{|}_{X}=0$

*for arbitrary*$X\in {S}^{d}$, $h\le {h}_{1}$)

*satisfies*

*when* $s>\frac{d}{2}$.

## 3 Proof of theorems

*Proof of Theorem 1.1*From Lemma 2.1, we have

where ${C}_{q}$ is a constant dependent only on *q* and satisfying $G(Mh)={C}_{q}{h}^{q}$.

With the fact that ${S}_{N}$ is the orthogonal projection of *f* in ${H}^{s}({S}^{q})$, the proof of Theorem 1.1 is thus completed. □

*Proof of Theorem 1.2*From the property of orthogonal projection of ${S}_{N}$, the Cauchy-Schwarz inequality, and Lemma 2.5, we can easily obtain

Together with the results of Theorem 1.1, Theorem 1.2 is thus established. □

## Declarations

### Acknowledgements

The first author and the third author would like to thank the Natural Science Foundation of China (Nos. 61273020, 11001227), the Fundamental Research Funds for the Central Universities (No. XDJK2010B005).

## Authors’ Affiliations

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