Some extremal functions in Fourier analysis. II
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- by Emanuel Carneiro and Jeffrey D. Vaaler
- Trans. Amer. Math. Soc. 362 (2010), 5803-5843
- DOI: https://doi.org/10.1090/S0002-9947-2010-04886-X
- Published electronically: June 9, 2010
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Abstract:
We obtain extremal majorants and minorants of exponential type for a class of even functions on $\mathbb {R}$ which includes $\log |x|$ and $|x|^\alpha$, 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.References
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Bibliographic 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
- Email: ecarneiro@math.utexas.edu, ecarneiro@math.ias.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): 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. - © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 5803-5843
- MSC (2000): Primary 41A30, 41A52, 42A05; Secondary 41A05, 41A44, 42A10
- DOI: https://doi.org/10.1090/S0002-9947-2010-04886-X
- MathSciNet review: 2661497