L^p Bernstein estimates and approximation by spherical basis functions

TL;DR

This paper develops L^p error estimates, Bernstein inequalities, and inverse theorems for approximation using spherical basis functions on the unit sphere, advancing the theoretical understanding of spherical approximation spaces.

Contribution

It introduces new Bernstein inequalities and inverse theorems for SBF approximation on the sphere, along with a novel characterization of Besov spaces in this context.

Findings

Established L^p error estimates for spherical basis function approximation.

Derived Bernstein inequalities relating Sobolev norms to coefficients.

Provided a new characterization of Besov spaces on the sphere.

Abstract

The purpose of this paper is to establish L^p error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates L^p Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the L^p norm of the function itself. An important step in its proof involves measuring the L^p stability of functions in the approximating space in terms of the l^p norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the L^P norm. Finally, we give a new characterization of Besov spaces on the n-sphere in terms of spaces of SBFs.