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- W2977012507 abstract "We show the following version of the Schur's product theorem. If $M=(M_{j,k})_{j,k=1}^nin{mathbb R}^{ntimes n}$ 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)). --" @default.
- W2977012507 created "2019-10-03" @default.
- W2977012507 creator A5080397395 @default.
- W2977012507 date "2019-09-25" @default.
- W2977012507 modified "2023-09-27" @default.
- W2977012507 title "A variant of Schur's product theorem and its applications." @default.
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