A fast FFT-based discrete Legendre transform

TL;DR

This paper introduces a highly efficient FFT-based algorithm for computing the discrete Legendre transform and its inverse, significantly reducing computational complexity using advanced transform techniques and Taylor series expansions.

Contribution

The paper presents a novel algorithm that combines Legendre-Chebyshev coefficient conversion with FFT-based Taylor expansion, achieving near-optimal computational complexity for the discrete Legendre transform.

Findings

Algorithm achieves $ ext{O}(N( ext{log} N)^2/ ext{log} ext{log} N)$ complexity.

Numerical results demonstrate high performance and accuracy.

Intermediate step reduces to $ ext{O}(N ext{log} N)$ for Chebyshev evaluations.

Abstract

An $\mathcal{O}(N(\log N)^2/\log\!\log N)$ algorithm for computing the discrete Legendre transform and its inverse is described. The algorithm combines a recently developed fast transform for converting between Legendre and Chebyshev coefficients with a Taylor series expansion for Chebyshev polynomials about equally-spaced points in the frequency domain. Both components are based on the FFT, and as an intermediate step we obtain an $\mathcal{O}(N\log N)$ algorithm for evaluating a degree $N-1$ Chebyshev expansion at an $N$-point Legendre grid. Numerical results are given to demonstrate performance and accuracy.