Gaussian subordination for the Beurling-Selberg extremal problem
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- by Emanuel Carneiro, Friedrich Littmann and Jeffrey D. Vaaler PDF
- Trans. Amer. Math. Soc. 365 (2013), 3493-3534 Request permission
Abstract:
We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function $e^{-\pi \lambda x^2}$ by entire functions of exponential type. This leads to the solution of analogous extremal problems for a wide class of even functions that includes most of the previously known examples, plus a variety of new interesting functions such as $|x|^{\alpha }$ for $-1 < \alpha$; $\log \bigl ((x^2 + \alpha ^2)/(x^2 + \beta ^2)\bigr )$, for $0 \leq \alpha < \beta$; $\log \bigl (x^2 + \alpha ^2\bigr )$; and $x^{2n} \log x^2$ , for $n \in \mathbb {N}$. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomials and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one.References
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Additional Information
- Emanuel Carneiro
- Affiliation: IMPA–Instituto de Matematica Pura e Aplicada–Estrada Dona Castorina, 110, Rio de Janeiro, 22460-320, Brazil
- Email: carneiro@impa.br
- Friedrich Littmann
- Affiliation: Department of Mathematics, North Dakota State University, Fargo, North Dakota 58105-5075
- Email: friedrich.littmann@ndsu.edu
- Jeffrey D. Vaaler
- Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1082
- MR Author ID: 176405
- Email: vaaler@math.utexas.edu
- Received by editor(s): February 1, 2010
- Received by editor(s) in revised form: July 12, 2011
- Published electronically: February 21, 2013
- © Copyright 2013 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 3493-3534
- MSC (2010): Primary 41A30, 41A52; Secondary 41A05, 41A44, 42A82
- DOI: https://doi.org/10.1090/S0002-9947-2013-05716-9
- MathSciNet review: 3042593