The expansion in ultraspherical polynomials: a simple procedure for the fast computation of the ultraspherical coefficients

TL;DR

This paper introduces a fast algorithm leveraging Fourier transforms to efficiently compute ultraspherical polynomial coefficients, significantly reducing computational complexity for function expansions.

Contribution

The paper presents a novel method that uses an Abel-type transform and FFT to compute ultraspherical coefficients in O(N log N) time, enhancing efficiency over previous approaches.

Findings

Computes ultraspherical coefficients in O(N log N) time.

Uses Fourier transform techniques for efficient calculations.

Provides a simple procedure applicable to various functions.

Abstract

We present a simple and fast algorithm for the computation of the coefficients of the expansion of a function f(cos u) in ultraspherical (Gegenbauer) polynomials. We prove that these coefficients coincide with the Fourier coefficients of an Abel-type transform of the function f(cos u). This allows us to fully exploit the computational efficiency of the Fast Fourier Transform, computing the first N ultraspherical coefficients in just O (N log_2 N) operations.