On sharp constants in Marcinkiewicz-Zygmund and Plancherel-Polya inequalities
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Abstract:
The Plancherel-Polya inequalities assert that for $1<p<\infty$, and entire functions $f$ of exponential type at most $\pi$, \begin{equation*} A_{p}\sum _{j=-\infty }^{\infty }\left \vert f\left ( j\right ) \right \vert ^{p}\leq \int _{-\infty }^{\infty }\left \vert f\right \vert ^{p}\leq B_{p}\sum _{j=-\infty }^{\infty }\left \vert f\left ( j\right ) \right \vert ^{p}. \end{equation*} The Marcinkiewicz-Zygmund inequalities assert that for $n\geq 1$ and polynomials $P$ of degree $\leq n-1$, \begin{equation*} \frac {A_{p}^{\prime }}{n}\sum _{j=1}^{n}\left \vert P\left ( e^{2\pi ij/n}\right ) \right \vert ^{p}\leq \int _{0}^{1}\left \vert P\left ( e^{2\pi it}\right ) \right \vert ^{p}dt\leq \frac {B_{p}^{\prime }}{n} \sum _{j=1}^{n}\left \vert P\left ( e^{2\pi ij/n}\right ) \right \vert ^{p}. \end{equation*} 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
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Additional Information
- D. S. Lubinsky
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
- MR Author ID: 116460
- ORCID: 0000-0002-0473-4242
- Email: lubinsky@math.gatech.edu
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 3575-3584
- MSC (2010): Primary 30D15, 30D99, 41A17, 41A55; Secondary 26D15, 26D05
- DOI: https://doi.org/10.1090/S0002-9939-2014-12270-2
- MathSciNet review: 3238433