This article is cited in 7 papers

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

D. V. Gorbachev, V. I. Ivanov

Tula State University, Tula, Russia

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 $L^2$-analogue of the Kotel'nikov-Nyquist-Whittaker-Shannon sampling theorem is put forward.
Bibliography: 42 titles.

Keywords: Sturm-Liouville problem, Fourier transform, Turán, Fejér and Bohman extremal problems, Gauss and Markov quadrature formulae.

UDC: 517.518.86

MSC: Primary 42B10; Secondary 41A55, 34B24

Received: 30.12.2017 and 18.11.2018

DOI: 10.4213/sm9057

 English version: Sbornik: Mathematics, 2019, 210:6, 809–835

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