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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality 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 problem
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by Robert M. Young PDF
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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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 56 (1976), 239-242
  • MSC: Primary 30A80
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0409832-3
  • MathSciNet review: 0409832