Sparse spectral approximations for computing polynomial functionals

TL;DR

This paper introduces a fast spectral approximation method for nonlinear polynomial functionals, demonstrating convergence for smooth functions under broad spectral basis conditions, including Fourier and Hermite.

Contribution

The paper presents a novel, efficient algorithm for spectral approximation of polynomial functionals with proven convergence for smooth functions.

Findings

The new method is computationally faster than existing approaches.

Convergence is guaranteed for functions with sufficient smoothness.

Applicable to Fourier, Hermite, and other spectral bases.

Abstract

We give a new fast method for evaluating sprectral approximations of nonlinear polynomial functionals. We prove that the new algorithm is convergent if the functions considered are smooth enough, under a general assumption on the spectral eigenfunctions that turns out to be satisfied in many cases, including the Fourier and Hermite basis.