Polynomial Approximation in Sobolev Spaces on the Unit Sphere and the Unit Ball

TL;DR

This paper develops new Sobolev spaces on the sphere and ball, analyzing polynomial approximation properties including simultaneous approximation and the relationship between function and derivative approximations.

Contribution

It introduces novel Sobolev spaces on the sphere and ball and studies polynomial approximation within these spaces, extending previous work.

Findings

Defined new Sobolev spaces on the sphere and ball

Established results on simultaneous polynomial approximation

Explored the relation between best approximation of functions and derivatives

Abstract

This work is a continuation of the recent study by the authors on approximation theory over the sphere and the ball. The main results define new Sobolev spaces on these domains and study polynomial approximations for functions in these spaces, including simultaneous approximation by polynomials and relation between best approximation to a function and to its derivatives.