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Some extremal functions in Fourier analysis. II

Authors: Emanuel Carneiro and Jeffrey D. Vaaler
Journal: Trans. Amer. Math. Soc. 362 (2010), 5803-5843
MSC (2000): Primary 41A30, 41A52, 42A05; Secondary 41A05, 41A44, 42A10
Published electronically: June 9, 2010
MathSciNet review: 2661497
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Abstract | References | Similar Articles | Additional Information

Abstract: We obtain extremal majorants and minorants of exponential type for a class of even functions on $ \mathbb{R}$ which includes $ \log \vert x\vert$ and $ \vert x\vert^\alpha$, where $ -1 < \alpha < 1$. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms, which include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev inequalities. A further application provides an Erdös-Turán-type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials.

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Additional Information

Emanuel Carneiro
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1082
Address at time of publication: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540

Jeffrey D. Vaaler
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78712-1082

Keywords: Exponential type, extremal functions
Received by editor(s): November 14, 2007
Received by editor(s) in revised form: June 23, 2008
Published electronically: June 9, 2010
Additional Notes: The first author’s research was supported by CAPES/FULBRIGHT grant BEX 1710-04-4.
The second author’s research was supported by the National Science Foundation, DMS-06-03282.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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