Interpolation for entire functions of exponential type and a related trigonometric moment problem

Abstract:

A classical theorem of Hausdorff-Young shows that when $1 < p < 2$, the system of equations $\hat \varphi (n) = {c_n}( - \infty < n < \infty )$ admits a solution $\varphi$ in ${L^q}( - \pi ,\pi )$ whenever $\{ {c_n}\} \in {l^p}$. Here, as usual, $\hat \varphi$ denotes the complex Fourier transform of $\varphi$ and $q$ is the conjugate exponent given by ${p^{ - 1}} + {q^{ - 1}} = 1$. The purpose of this note is to show that if a set $\{ {\lambda _n}\}$ of real or complex numbers is “sufficiently close” to the inte gers, then the corresponding system $\hat \varphi ({\lambda _n}) = {c_n}$ is also solvable for $\varphi$ whenever $\{ {c_n}\} \in {l^p}$. The proof is accomplished by establishing a similar interpolation theorem for a related class of entire functions of exponential type.