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Proceedings of the American Mathematical Society

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

 

 

Inequalities for a perturbation theorem of Paley and Wiener


Author: Robert M. Young
Journal: Proc. Amer. Math. Soc. 43 (1974), 320-322
MSC: Primary 42A64
MathSciNet review: 0340948
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Abstract: A classical theorem of Paley and Wiener states that the set of functions $ \{ {e^{i{\lambda _n}t}}\} _{n = - \infty }^\infty $ forms a basis for $ {L^2}( - \pi ,\pi )$ whenever the following condition is satisfied:

$\displaystyle ( \ast )\quad \vert\vert\sum {{c_n}({e^{i{\lambda _n}t}} - {e^{in... ...^2} \leqq {\theta ^2}\,\sum {\vert{c_n}{\vert^2}} } \quad (0 \leqq \theta < 1).$

It is known that ($ \ast$) holds whenever $ {\lambda _n}$ is real and $ \vert{\lambda _n} - n\vert \leqq L < \frac{1}{4}( - \infty < n < \infty )$, and may fail to hold if $ \vert{\lambda _n} - n\vert = \frac{1}{4}$.

In this note we show, more generally, that the condition $ \vert{\lambda _n} - n\vert < \frac{1}{4}$ is also insufficient to ensure ($ \ast$).


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DOI: https://doi.org/10.1090/S0002-9939-1974-0340948-4
Keywords: Nonharmonic Fourier series, frames, functions of exponential type
Article copyright: © Copyright 1974 American Mathematical Society