Optimal Lebesgue constants for least squares polynomial approximation on the (hyper)sphere

TL;DR

This paper studies the growth of Lebesgue constants for least squares polynomial approximation on spheres, establishing conditions for optimal growth rates that ensure comparable uniform approximation to hyperinterpolation.

Contribution

It provides new sufficient conditions under which least squares polynomials on spheres achieve minimal Lebesgue constant growth, aligning their approximation quality with hyperinterpolation.

Findings

Identifies conditions for optimal Lebesgue constant growth on spheres.

Shows least squares and hyperinterpolation have comparable uniform approximation.

Provides theoretical bounds for polynomial approximation on spheres.

Abstract

We investigate the uniform approximation provided by least squares polynomials on the unit Euclidean sphere $\mathbb{S}^q$ in $\mathbb{R}^{q+1}$, with $q\ge 2$. Like any other polynomial projection, the study concerns the growth, as the degree $n$ tends to infinity, of the associated Lebesgue constant, i.e., of the uniform norm of the least squares operator. If the least squares polynomial of degree $n$ is based on a set of points, which are nodes of a positive weighted quadrature rule of degree of exactness $2n$, then we state two different sufficient conditions for having an optimal Lebesgue constant that increases with $n$ at the minimal projections order. Hence, under our assumptions least squares and hyperinterpolation polynomials provide a comparable approximation with respect to the uniform norm.