Orthogonal expansions for generalized Gegenbauer weight function on the unit ball

TL;DR

This paper develops orthogonal polynomial expansions for a specific weight function on the unit ball, deriving formulas for reproducing kernels and analyzing the summability of Fourier expansions.

Contribution

It introduces new orthogonal polynomial expansions for a generalized Gegenbauer weight on the unit ball, including explicit reproducing kernel formulas.

Findings

Derived concise formulas for reproducing kernels.

Analyzed summability properties of Fourier orthogonal expansions.

Extended results to related weight functions on the simplex.

Abstract

Orthogonal polynomials and expansions are studied for the weight function $h_\kappa^2(x) \|x\|^{2\nu} (1-\|x\|^2)^{\mu-1/2}$ on the unit ball of $\mathbb{R}^d$, where $h_\kappa$ is a reflection invariant function, and for related weight function on the simplex of $\mathbb{R}^d$. A concise formula for the reproducing kernels of orthogonal subspaces is derived and used to study summability of the Fourier orthogonal expansions.