New lower bounds for the integration of periodic functions

TL;DR

This paper surveys and extends new techniques for establishing lower bounds on the error of numerical integration of multivariate periodic functions, demonstrating their superiority over traditional bump-function methods.

Contribution

It introduces a refined proof technique based on Hilbert space structure and the Schur product theorem, improving lower bounds for quadrature errors.

Findings

New lower bounds are sharper and more general.

The Hilbert space-based technique outperforms bump-function methods.

Extensions include broader classes of periodic functions.

Abstract

We study the integration problem on Hilbert spaces of (multivariate) periodic functions. The standard technique to prove lower bounds for the error of quadrature rules uses bump functions and the pigeon hole principle. Recently, several new lower bounds have been obtained using a different technique which exploits the Hilbert space structure and a variant of the Schur product theorem. The purpose of this paper is to (a) survey the new proof technique, (b) show that it is indeed superior to the bump-function technique, and (c) sharpen and extend the results from the previous papers.