Plancherel–Pólya Inequality for Entire Functions of Exponential Type in L2(ℝ)

We investigate the Plancherel–Pólya inequality ∑k∈ℤ|f(k)|2⩽c2(σ)||f||L2(ℝ)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum {_{k \in \mathbb{Z}}} |f(k){|^2} \leqslant {c_2}\left( \sigma \right)||f||_{{L^2}\left( \mathbb{R} \right)}^2$$\end{document} on the set of entire functions f of exponential type at most σ whose restrictions to the real line belong to the space L2(ℝ). We prove that c2(σ) = [σ/π] for σ > 0 and describe the extremal functions.