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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Gaussian subordination for the Beurling-Selberg extremal problem


Authors: Emanuel Carneiro, Friedrich Littmann and Jeffrey D. Vaaler
Journal: Trans. Amer. Math. Soc. 365 (2013), 3493-3534
MSC (2010): Primary 41A30, 41A52; Secondary 41A05, 41A44, 42A82
Published electronically: February 21, 2013
MathSciNet review: 3042593
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Abstract | References | Similar Articles | Additional Information

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 $ \vert x\vert^{\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.


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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
Email: vaaler@math.utexas.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05716-9
Keywords: Gaussian, exponential type, extremal functions, majorization, tempered distributions.
Received by editor(s): February 1, 2010
Received by editor(s) in revised form: July 12, 2011
Published electronically: February 21, 2013
Article copyright: © Copyright 2013 American Mathematical Society