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

Abstract

We investigate the Plancherel–Polya inequality $$\sum {_{k \in \mathbb{Z}}} |f(k){|^2} \leqslant {c_2}\left( \sigma \right)||f||_{{L^2}\left( \mathbb{R} \right)}^2$$ 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.