On hyperinterpolation on the unit ball

TL;DR

This paper establishes improved estimates for the Lebesgue constants of hyperinterpolation operators on the unit ball, leveraging the connection between orthogonal polynomials on the sphere and ball.

Contribution

It provides new bounds on Lebesgue constants for hyperinterpolation on the unit ball using Gegenbauer weights, enhancing previous results especially in two dimensions.

Findings

Improved bounds on Lebesgue constants for hyperinterpolation on the unit ball.

Utilization of the relationship between sphere and ball orthogonal polynomials.

Enhanced understanding of hyperinterpolation stability on weighted domains.

Abstract

We prove estimates on the Lebesgue constants of the hyperinterpolation operator for functions on the unit ball $B^d \subset \RR^d$, with respect to Gegenbauer weight functions, $(1-|\xb|^2)^{\mu-1/2}$. The relationship between orthogonal polynomials on the sphere and ball is exploited to achieve this result, which provides an improvement on known estimates of the Lebesgue constant for hyperinterpolation operators on $B^2$.