From Fourier to Gegenbauer: Dimension walks on spheres

TL;DR

This paper establishes a mathematical connection between Gegenbauer and Fourier/Legendre coefficients for isotropic positive definite functions on spheres, providing explicit formulas that facilitate analysis across different dimensions.

Contribution

It introduces a novel representation of Gegenbauer coefficients as linear combinations of Fourier and Legendre coefficients, with explicit formulas for these expansions.

Findings

Derived closed-form expressions for expansion coefficients.

Connected Gegenbauer coefficients to Fourier and Legendre coefficients.

Enhanced understanding of positive definite functions on spheres.

Abstract

We show that the even- resp. odd-dimensional Schoenberg coefficients in Gegenbauer expansions of isotropic positive definite functions on the d-sphere can be expressed as linear combinations of Fourier resp. Legendre coefficients, and we give closed form expressions for the coefficients involved in these expansions.