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


Author: Robert M. Young
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
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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

DOI: https://doi.org/10.1090/S0002-9939-1976-0409832-3
Keywords: Interpolating sequence, entire functions of exponential type
Article copyright: © Copyright 1976 American Mathematical Society

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