05907641

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05907641 Hinrichs, Aicke Vyb{\'\i}ral, Jan On positive positive-definite functions and Bochner's theorem. J. Complexity 27, No. 3-4, 264-272 (2011). 2011 Elsevier Science (Academic Press), San Diego, CA EN Approximate quadratures

doi:10.1016/j.jco.2011.01.002

Authors' abstract: We recall an open problem on the error of quadrature formulas for the integration of functions from some finite dimensional spaces of trigonometric functions posed by {\it E. Novak} [J. Complexity 15, No. 3, 299--316 (1999; Zbl 0941.41016)] ten years ago and summarised recently in [{\it E. Novak} and {\it H. Wo\'zniakowski}, Tractability of multivariate problems. Volume I: Linear information. EMS Tracts in Mathematics 6. Z\"urich: European Mathematical Society (EMS) (2008; Zbl 1156.65001)]. It is relatively easy to prove an error formula for the best quadrature rules with positive weights which shows intractability of the tensor product problem for such rules. In contrast to that, the conjecture that also quadrature formulas with arbitrary weights cannot decrease the error is still open. We generalise Novak's conjecture to a statement about positive positive-definite functions and provide several equivalent reformulations, which show the connections to Bochner's Theorem and Toeplitz matrices. Dumitru Acu (Sibiu)