# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Interpolation for entire functions of exponential type and a related trigonometric moment problemHTML articles powered by AMS MathViewer

by Robert M. Young
Proc. Amer. Math. Soc. 56 (1976), 239-242 Request permission

## 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.
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