Turán, Fejér and Bohman extremal problems for the multivariate Fourier transform in terms of the eigenfunctions of a Sturm-Liouville problem

Abstract

Abstract The Turán, Fejér and Bohman extremal problems for the multivariate Fourier transform in terms of the eigenfunctions of a Sturm-Liouville problem on the Cartesian product of half-lines are solved under natural conditions on a weight function defined as a product of one-dimensional weight functions. Extremal functions are constructed. A multivariate Markov quadrature formula is proved based on the zeros of eigenfunctions of the Sturm-Liouville problem. This quadrature formula is shown to be sharp on entire multivariate functions of exponential type. A Paley-Wiener type theorem is proved for the multivariate Fourier transform. A weighted -analogue of the Kotel’nikov-Nyquist-Whittaker-Shannon sampling theorem is put forward. Bibliography: 42 titles.