Abstract
We show the following version of the Schur's product theorem. If M=(Mj,k)j,k=1n∈Rn×n is a positive semidefinite matrix with all entries on the diagonal equal to one, then the matrix N=(Nj,k)j,k=1n with the entries Nj,k=Mj,k2−1n is positive semidefinite. As a corollary of this result, we prove the conjecture of E. Novak on intractability of numerical integration on the 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 χ2-variables, and mean absolute values of trigonometric polynomials.
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