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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

On sharp constants in Marcinkiewicz-Zygmund and Plancherel-Polya inequalities


Author: D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 142 (2014), 3575-3584
MSC (2010): Primary 30D15, 30D99, 41A17, 41A55; Secondary 26D15, 26D05
Published electronically: July 1, 2014
MathSciNet review: 3238433
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Abstract: The Plancherel-Polya inequalities assert that for $ 1<p<\infty $, and entire functions $ f$ of exponential type at most $ \pi $,

$\displaystyle A_{p}\sum _{j=-\infty }^{\infty }\left \vert f\left ( j\right ) \... ...p}\sum _{j=-\infty }^{\infty }\left \vert f\left ( j\right ) \right \vert ^{p}.$    

The Marcinkiewicz-Zygmund inequalities assert that for $ n\geq 1$ and polynomials $ P$ of degree $ \leq n-1$,

$\displaystyle \frac {A_{p}^{\prime }}{n}\sum _{j=1}^{n}\left \vert P\left ( e^{... ...n} \sum _{j=1}^{n}\left \vert P\left ( e^{2\pi ij/n}\right ) \right \vert ^{p}.$    

We show that the sharp constants in both inequalities are the same; that is, $ A_{p}=A_{p}^{\prime }$ and $ B_{p}=B_{p}^{\prime }$. Moreover, the two inequalities are equivalent. We also discuss the case $ p\leq 1$.

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Additional Information

D. S. Lubinsky
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: lubinsky@math.gatech.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-12270-2
Keywords: Plancherel-Polya inequalities, Marcinkiewicz-Zygmund inequalities, Entire functions, quadrature sums
Received by editor(s): November 9, 2012
Published electronically: July 1, 2014
Additional Notes: Research supported by NSF grant DMS1001182 and US-Israel BSF grant 2008399
Communicated by: Walter Van Assche
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.