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New lower bounds for the integration of periodic functions

David Krieg, Jan Vybiral

TL;DR

This paper surveys a modern Schur-product–based approach to lower bounds for numerical integration in RKHS of periodic functions, showing its superiority over the classical bump-function method in both barely smooth and analytic regimes. It provides general lower bounds via sums of squares and norm-decreasing weight sequences, derives sharp asymptotics for isotropic and mixed small smoothness, and extends these insights to L2-approximation gaps and tractability. The results reveal a consistent logarithmic gap between sampling and approximation in small-smoothness spaces and demonstrate a curse of dimensionality for analytic classes, contrasting with certain tractability results for related univariate settings. Overall, the Schur technique yields tighter, more universal lower bounds and clarifies how smoothness and dimension jointly constrain numerical integration on periodic RKHS.

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.

New lower bounds for the integration of periodic functions

TL;DR

This paper surveys a modern Schur-product–based approach to lower bounds for numerical integration in RKHS of periodic functions, showing its superiority over the classical bump-function method in both barely smooth and analytic regimes. It provides general lower bounds via sums of squares and norm-decreasing weight sequences, derives sharp asymptotics for isotropic and mixed small smoothness, and extends these insights to L2-approximation gaps and tractability. The results reveal a consistent logarithmic gap between sampling and approximation in small-smoothness spaces and demonstrate a curse of dimensionality for analytic classes, contrasting with certain tractability results for related univariate settings. Overall, the Schur technique yields tighter, more universal lower bounds and clarifies how smoothness and dimension jointly constrain numerical integration on periodic RKHS.

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.
Paper Structure (15 sections, 17 theorems, 100 equations)

This paper contains 15 sections, 17 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Preliminaries
  4. Results
  5. Proof techniques for lower bounds
  6. Limits of the bump-function technique
  7. The Schur technique
  8. Some general lower bounds for periodic functions
  9. Sums of squares
  10. Norm-decreasing sequences
  11. Detour: The power of function values for $L_2$ approximation
  12. Lower bounds for small smoothness
  13. Isotropic smoothness
  14. Mixed smoothness
  15. Lower bounds for large smoothness

Key Result

Theorem 1

Let $\lambda \in \ell_1(\mathbb{Z})$ be given by eq:border-lambda. There exists an absolute constant $C>0$ such that for every even $n\ge 2$ and every $\varphi\in C([0,1])$ with ${\rm supp}\,\varphi\subset [0,1/(2n)]$ there exists $\{z_1,\dots,z_n\}\subset \{j/(2n):j=0,\dots,2n-1\}$ such that the fu satisfies

Theorems & Definitions (33)

  • Definition 1
  • Theorem 1
  • Proposition 1
  • proof
  • proof : Proof of Theorem \ref{['thm:no-bumps']}
  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • Theorem 2
  • ...and 23 more