Plancherel-Polya inequality for entire functions of exponential type in $L^2(\mathbb{R}^n)$

E. V. Berestova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 3, pp. 27-33

Résumé

Let $\mathfrak{M}_{\sigma,n}^p$, $p>0$, be a set of entire functions $f$ of $n$ complex variables with exponential type $\sigma=(\sigma_1,\ldots,\sigma_n)$, $\sigma_k>0$, such that their restrictions to $\mathbb{R}^n$ belong to $L^p(\mathbb{R}^n)$. In 1937 Plancherel and Polya showed that $\sum_{k \in \mathbb{Z}^n}|f(k)|^p \le c_p(\sigma, n) \|f\|^p_{L^p(\mathbb{R}^n)}$ for $f\in \mathfrak{M}_{\sigma,n}^p$, where $c_p(\sigma, n)$ is a finite constant. We study the Plancherel-Polya inequality for $p=2$. If $0\sigma_k\le \pi$, then, by the Whittaker-Kotelnikov-Shannon theorem and its generalization to the multidimensional case established by Plancherel and Polya, we have $c_2(\sigma, n)=1$ and any function $f\in \mathfrak{M}_{\sigma,n}^2$ is extremal. In the general case, we prove that $c_2(\sigma, n)=\prod_{k = 1}^{n}\left\lceil~\sigma_k/\pi \right\rceil~$ and describe the class of extremal functions. We also write the dual problem $\big|\sum_{k \in \mathbb{Z}^n} (g\ast g)(k)\big| \le d_2(\sigma,n) \|g\|_2^2$, $g \in ~L^2\left(\Omega\right)$, prove that $c_2(\sigma,n)=d_2(\sigma,n)$, and describe the class of extremal functions.

Détail

Keywords: Plancherel-Polya inequality, Paley-Wiener space, entire function of exponential type
Mots-clés : Fourier transform.

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