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 L2-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.
Citation:
D. V. Gorbachev, V. I. Ivanov, “Turán, Fejér and Bohman extremal problems for the multivariate Fourier transform in terms of the eigenfunctions of a Sturm-Liouville problem”, Sb. Math., 210:6 (2019), 809–835
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\by D.~V.~Gorbachev, V.~I.~Ivanov
\paper Tur\'an, Fej\'er and Bohman extremal problems for the multivariate Fourier transform in terms of the eigenfunctions of a~Sturm-Liouville problem
\jour Sb. Math.
\yr 2019
\vol 210
\issue 6
\pages 809--835
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Linking options:
https://www.mathnet.ru/eng/sm9057
https://doi.org/10.1070/SM9057
https://www.mathnet.ru/eng/sm/v210/i6/p56
This publication is cited in the following 7 articles:
Andrés Chirre, Dimitar Dimitrov, Emily Quesada-Herrera, Mateus Sousa, “An extremal problem and inequalities for entire functions of exponential type”, Proc. Amer. Math. Soc., 152:8 (2024), 3299
D. V. Gorbachev, V. I. Ivanov, “Logan–Hermite Extremal Problems for Entire Functions of Exponential Type”, Math. Notes, 113:1 (2023), 143–148
D. Gorbachev, V. Ivanov, S. Tikhonov, “Logan's problem for Jacobi transforms”, Can. J. Math., 2023, 1–31
R. Sousa, M. Guerra, S. Yakubovich, “A unified construction of product formulas and convolutions for Sturm-Liouville operators”, Anal. Math. Phys., 11:2 (2021), 87
D. V. Gorbachev, “Tochnye neravenstva Bernshteina — Nikolskogo dlya polinomov i tselykh funktsii eksponentsialnogo tipa”, Chebyshevskii sb., 22:5 (2021), 58–110
D. Gorbachev, V. Ivanov, S. Tikhonov, “Uncertainty principles for eventually constant sign bandlimited functions”, SIAM J. Math. Anal., 52:5 (2020), 4751–4782
D. V. Gorbachev, N. N. Dobrovolskii, “Ob ekstremalnykh zadachakh tipa Nikolskogo–Bernshteina i Turana dlya preobrazovaniya Danklya”, Chebyshevskii sb., 20:3 (2019), 394–400
Citation in format AMSBIB
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