A variant of Schur's product theorem and its applications

09/25/2019

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by   Jan Vybiral, et al.

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We show the following version of the Schur's product theorem. If M=(M_j,k)_j,k=1^n∈R^n× 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^2-1/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 χ^2-variables, and mean absolute values of trigonometric polynomials.