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

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 $p\leq 1$.