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A note on band-limited minorants of an Euclidean ball

Author: Felipe Gonçalves
Journal: Proc. Amer. Math. Soc. 146 (2018), 2063-2068
MSC (2010): Primary 42B35; Secondary 33C10
Published electronically: December 12, 2017
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Abstract: We study the Beurling-Selberg problem of finding band-limited $ L^1$-functions that lie below the indicator function of an Euclidean ball. We compute the critical radius of the support of the Fourier transform for which such construction can have a positive integral.

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  • [1] Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011
  • [2] Emanuel Carneiro, Vorrapan Chandee, Friedrich Littmann, and Micah B. Milinovich, Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function, J. Reine Angew. Math. 725 (2017), 143-182. MR 3630120,
  • [3] J. Carruth, N. Elkies, F. Gonçalves, and M. Kelly,
    On Selberg's Box Minorant Problem,
    (preprint at arXiv:1702.04579 [math.CA]).
  • [4] Henry Cohn and Noam Elkies, New upper bounds on sphere packings. I, Ann. of Math. (2) 157 (2003), no. 2, 689-714. MR 1973059,
  • [5] Á. Elbert, Some recent results on the zeros of Bessel functions and orthogonal polynomials, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), 2001, pp. 65-83. MR 1858270,
  • [6] D. V. Gorbachev, An extremal problem for entire functions of exponential spherical type, which is connected with the Levenshteĭn bound for the density of a packing of $ {\mathbb{R}}^n$ by balls, Izv. Tul. Gos. Univ. Ser. Mat. Mekh. Inform. 6 (2000), no. 1, Matematika, 71-78 (Russian, with English and Russian summaries). MR 2018751
  • [7] Jeffrey J. Holt and Jeffrey D. Vaaler, The Beurling-Selberg extremal functions for a ball in Euclidean space, Duke Math. J. 83 (1996), no. 1, 202-248. MR 1388849,
  • [8] Friedrich Littmann, Quadrature and extremal bandlimited functions, SIAM J. Math. Anal. 45 (2013), no. 2, 732-747. MR 3038107,
  • [9] Jeffrey D. Vaaler, Some extremal functions in Fourier analysis, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 183-216. MR 776471,
  • [10] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746

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Additional Information

Felipe Gonçalves
Affiliation: Mathematical and Statistical Sciences, University of Alberta, CAB 632, Edmonton, Alberta, T6G 2G1 Canada

Received by editor(s): May 2, 2017
Received by editor(s) in revised form: July 3, 2017
Published electronically: December 12, 2017
Communicated by: Harold P. Boas
Article copyright: © Copyright 2017 American Mathematical Society

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