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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on a trigonometric moment problem


Author: Robert M. Young
Journal: Proc. Amer. Math. Soc. 49 (1975), 411-415
MSC: Primary 42A80
MathSciNet review: 0367548
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Abstract: A sequence $ \{ {\lambda _n}\} _{n = - \infty }^\infty $ is said to be an interpolating sequence for $ {L^2}( - \pi ,\pi )$ if the system of equations

$\displaystyle {c_n} = \int_{ - \pi }^\pi {f(t)} {e^{i{\lambda _n}t}}dt\quad ( - \infty < n < \infty )$

admits a solution $ f$ in $ {L^2}( - \pi ,\pi )$ whenever $ \{ {c_n}\} \in {l^2}$. If the solution is unique then $ \{ {\lambda _n}\} $ is said to be a complete interpolating sequence. It is shown that if the imaginary part of $ {\lambda _n}$ is uniformly bounded and if $ \vert\operatorname{Re} ({\lambda _n}) - n\vert \leq L < 1/4( - \infty < n < \infty )$, then $ \{ {\lambda _n}\} $ is a complete interpolating sequence and $ \{ {e^{i{\lambda _n}t}}\} $ is a Schauder basis for $ {L^2}( - \pi ,\pi )$. It is also shown that this result is sharp in the sense that the condition $ \vert{\lambda _n} - n\vert < 1/4$ is not sufficient to guarantee that $ \{ {\lambda _n}\} $ is an interpolating sequence.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0367548-5
PII: S 0002-9939(1975)0367548-5
Keywords: Interpolating sequence, frames, Paley-Wiener space
Article copyright: © Copyright 1975 American Mathematical Society