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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 {\vert{\lambda _n} - {\mu _n}\vert < \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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0425499-2
Keywords: Nonharmonic Fourier series, Besselian basis, entire functions of exponential type
Article copyright: © Copyright 1976 American Mathematical Society

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