A variant of Schur's product theorem and its applications

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)). --