Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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$.

References [Enhancements On Off] (What's this?)

  • [1] Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. MR 0068627 (16,914f)
  • [2] D. P. Dryanov, M. A. Qazi, and Q. I. Rahman, Entire functions of exponential type in approximation theory, Constructive theory of functions, DARBA, Sofia, 2003, pp. 86-135. MR 2092333 (2006a:41008)
  • [3] Carolyn Eoff, The discrete nature of the Paley-Wiener spaces, Proc. Amer. Math. Soc. 123 (1995), no. 2, 505-512. MR 1219724 (95c:42011),
  • [4] F. Filbir and H. N. Mhaskar, Marcinkiewicz-Zygmund measures on manifolds, J. Complexity 27 (2011), no. 6, 568-596. MR 2846706,
  • [5] B. Ya. Levin, Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko; Translated from the Russian manuscript by Tkachenko. MR 1400006 (97j:30001)
  • [6] Eli Levin, D. S. Lubinsky, $ L_{p}$ Christoffel functions, $ L_{p}$ universality, and Paley-Wiener spaces, to appear in J. d'Analyse de Mathematique.
  • [7] D. S. Lubinsky, Marcinkiewicz-Zygmund inequalities: methods and results, Recent progress in inequalities (Niš, 1996) Math. Appl., vol. 430, Kluwer Acad. Publ., Dordrecht, 1998, pp. 213-240. MR 1609947 (99a:41045)
  • [8] D. S. Lubinsky, A survey of weighted polynomial approximation with exponential weights, Surv. Approx. Theory 3 (2007), 1-105. MR 2276420 (2007k:41001)
  • [9] M. Plancherel and G. Pólya, Fonctions entières et intégrales de fourier multiples, Comment. Math. Helv. 10 (1937), no. 1, 110-163 (French). MR 1509570,
  • [10] A. F. Timan, Theory of approximation of functions of a real variable, Dover Publications Inc., New York, 1994. Translated from the Russian by J. Berry; Translation edited and with a preface by J. Cossar; Reprint of the 1963 English translation. MR 1262128 (94j:41001)
  • [11] A. Zygmund, Trigonometric series. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1979 edition. MR 933759 (89c:42001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30D15, 30D99, 41A17, 41A55, 26D15, 26D05

Retrieve articles in all journals with MSC (2010): 30D15, 30D99, 41A17, 41A55, 26D15, 26D05

Additional Information

D. S. Lubinsky
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

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.

American Mathematical Society