A variant of Schur's product theorem and its applications

TL;DR

This paper presents a new version of Schur's product theorem, proves a conjecture on the intractability of certain numerical integrations, and explores implications for Bochner's theorem, covariance matrices, and trigonometric polynomials.

Contribution

It introduces a novel positive semidefinite matrix transformation and applies it to resolve a conjecture on numerical integration complexity.

Findings

New positive semidefinite matrix property established

Proved E. Novak's conjecture on numerical integration in specific spaces

Discussed implications for Bochner's theorem and covariance matrices

Abstract

We show the following version of the Schur's product theorem. If $M=(M_{j,k})_{j,k=1}^n\in{\mathbb R}^{n\times n}$ is a positive semidefinite matrix with all entries on the diagonal equal to one, then the matrix $N=(N_{j,k})_{j,k=1}^n$ with the entries $N_{j,k}=M_{j,k}^2-\frac{1}{n}$ is positive semidefinite. As a corollary of this result, we prove the conjecture of E. Novak on intractability of numerical integration on a space of trigonometric polynomials of degree at most one in each variable. Finally, we discuss also some consequences for Bochner's theorem, covariance matrices of $\chi^2$-variables, and mean absolute values of trigonometric polynomials. -- Please, have a look into page 6 of the preprint "Lower Bounds for the Error of Quadrature Formulas for Hilbert Spaces" for a discussion of the relation of Theorem 1 and Corollary 2 to Gegenbauer polynomials (pointed out by Dmitriy Bilyk (University of Minnesota)). --