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Proceedings of the American Mathematical Society

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

 
 

 

On perturbing bases of complex exponentials in $ L\sp{2}$ $ (-\pi ,\,\pi )$


Author: Robert M. Young
Journal: Proc. Amer. Math. Soc. 53 (1975), 137-140
MSC: Primary 30A98; Secondary 30A18, 46E15
DOI: https://doi.org/10.1090/S0002-9939-1975-0377075-7
MathSciNet review: 0377075
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Abstract: A sequence of complex exponentials $ \{ {e^{i{\lambda _n}t}}\} $ is said to be a Riesz basis for $ {L^2}( - \pi ,\;\pi )$ if each function in the space has a unique representation $ f = \Sigma {c_n}{e^{i{\lambda _n}t}}$, with $ A\Sigma \vert{c_n}{\vert^2} \leqslant \vert\vert f\vert{\vert^2} \leqslant B\Sigma \vert{c_n}{\vert^2}$. It is known, for example, that if $ \vert{\lambda _n} - n\vert \leqslant L < 1/4( - \infty < n < \infty )$, then $ \{ {e^{i{\lambda _n}t}}\} $ is a Riesz basis. In this note we show that not only the orthonormal basis $ \{ {e^{int}}\} $, but any Riesz basis of complex exponentials can be suitably perturbed.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0377075-7
Keywords: Riesz basis, exact frame, entire functions of exponential type, nonharmonic Fourier series
Article copyright: © Copyright 1975 American Mathematical Society

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