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Proceedings of the American Mathematical Society

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

 
 

 

Some stability theorems for nonharmonic Fourier series


Author: Robert M. Young
Journal: Proc. Amer. Math. Soc. 61 (1976), 315-319
MSC: Primary 42A64
DOI: https://doi.org/10.1090/S0002-9939-1976-0425499-2
MathSciNet review: 0425499
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Abstract: The theory of nonharmonic Fourier series in ${L^2}( - \pi ,\pi )$ is concerned with the completeness and expansion properties of sets of complex exponentials $\{ {e^{i{\lambda _n}t}}\}$. It is well known, for example, that the completeness of the set $\{ {e^{i{\lambda _n}t}}\}$ ensures that of $\{ {e^{i{\mu _n}t}}\}$ whenever $\sum {|{\lambda _n} - {\mu _n}| < \infty }$. In this note we establish two results which guarantees that if $\{ {e^{i{\lambda _n}t}}\}$ is a Schauder basis for ${L^2}( - \pi ,\pi )$, then $\{ {e^{i{\mu _n}t}}\}$ is also a Schauder basis whenever $\{ {\mu _n}\}$ is “sufficiently close” to $\{ {\lambda _n}\}$.


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Keywords: Nonharmonic Fourier series, Besselian basis, entire functions of exponential type
Article copyright: © Copyright 1976 American Mathematical Society