Sampling theorem based Fourier-Legendre transform

TL;DR

This paper explores the use of sampling theorem to expand products of Legendre and Gegenbauer functions into Legendre polynomials, providing new insights into their mathematical properties and limitations.

Contribution

It introduces a sampling theorem-based method to expand products of Legendre and Gegenbauer functions, overcoming restrictions present in Jacobi functions.

Findings

Expansion of Legendre function products using sinc coefficients

Limitations for Jacobi functions beyond two factors

New mathematical framework for function expansions

Abstract

The product of any number of Legendre functions, under a restricted domain, can be expanded by the corresponding Legendre polynomials, with the coefficient being the sinc function. While an analogous expansion can be made for any number of Gengenbauer functions, it is not allowed for more than two Jacobi functions. To obtain such an expansion, the sampling theorem is of great availability.