Efficient Legendre polynomials transforms: from recurrence relations to Schoenberg's theorem

TL;DR

This paper explores efficient methods for expanding functions into Legendre and Gegenbauer polynomials, comparing algebraic/symbolic techniques with an approach based on Schoenberg's theorem to improve computational efficiency.

Contribution

It introduces an alternative approach based on Schoenberg's theorem for efficient Legendre polynomial expansion, complementing existing algebraic methods.

Findings

The algebraic/symbolic approach is detailed and effective.

The Schoenberg-based method offers a computationally efficient alternative.

Results demonstrate improved efficiency in polynomial expansion computations.

Abstract

We report results on various techniques which allow to compute the expansion into Legendre (or in general Gegenbauer) polynomials in an efficient way. We describe in some detail the algebraic/symbolic approach already presented in Ref.1 and expand on an alternative approach based on a theorem of Schoenberg.