Rapidly Computing Sparse Legendre Expansions via Sparse Fourier Transforms

TL;DR

This paper introduces a fast, near-linear time algorithm for computing sparse Legendre polynomial expansions, significantly reducing computational complexity compared to traditional methods, with promising theoretical and numerical results.

Contribution

The paper presents a novel approach that leverages sparse Fourier transforms to efficiently compute sparse Legendre expansions in sublinear time.

Findings

Algorithms achieve near-optimal approximation with sublinear runtime.

The methods outperform traditional $ ext{O}(N ext{log} N)$ algorithms for sparse cases.

Numerical experiments confirm theoretical efficiency and accuracy.

Abstract

In this paper we propose a general strategy for rapidly computing sparse Legendre expansions. The resulting methods yield a new class of fast algorithms capable of approximating a given function $f:[-1,1] \rightarrow \mathbb{R}$ with a near-optimal linear combination of $s$ Legendre polynomials of degree $\leq N$ in just $(s \log N)^{\mathcal{O}(1)}$-time. When $s \ll N$ these algorithms exhibit sublinear runtime complexities in $N$, as opposed to traditional $\Omega(N \log N)$-time methods for computing all of the first $N$ Legendre coefficients of $f$. Theoretical as well as numerical results demonstrate the promise of the proposed approach.