Uniform approximation on the sphere by least squares polynomials

TL;DR

This paper studies uniform polynomial approximation on the sphere using least squares polynomials, proving optimal growth of Lebesgue constants and demonstrating near-best uniform convergence with numerical validation.

Contribution

It establishes the optimal growth rate of Lebesgue constants for least squares approximation on the sphere and introduces delayed arithmetic means that improve uniform convergence rates.

Findings

Lebesgue constants grow at the optimal rate

Delayed arithmetic means achieve near-best uniform approximation

Numerical experiments confirm theoretical results

Abstract

The paper concerns the uniform polynomial approximation of a function $f$, continuous on the unit Euclidean sphere of ${\mathbb R}^3$ and known only at a finite number of points that are somehow uniformly distributed on the sphere. First we focus on least squares polynomial approximation and prove that the related Lebesgue constants w.r.t. the uniform norm grow at the optimal rate. Then, we consider delayed arithmetic means of least squares polynomials whose degrees vary from $n-m$ up to $n+m$, being $m=\lfloor \theta n\rfloor$ for any fixed parameter $0<\theta<1$. As $n$ tends to infinity, we prove that these polynomials uniformly converge to $f$ at the near-best polynomial approximation rate. Moreover, for fixed $n$, by using the same data points we can further improve the approximation by suitably modulating the action ray $m$ determined by the parameter $\theta$. Some numerical experiments are given to illustrate the theoretical results.