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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Complete multipliers


Author: J. S. Byrnes
Journal: Trans. Amer. Math. Soc. 172 (1972), 399-403
MSC: Primary 42A64
DOI: https://doi.org/10.1090/S0002-9947-1972-0308676-1
MathSciNet review: 0308676
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Abstract: We investigate whether the completeness of a complete orthonormal sequence for $ {L^2}( - \pi ,\pi )$ is preserved if the sequence is perturbed by multiplying a portion of it by a fixed function. For the particular sequence $ \{ {(2\pi )^{ - 1/2}}{e^{inx}}\} $ we show that given any $ \psi \in {L^\infty }( - \pi ,\pi )$, except $ \psi = 0$ a.e., there is a nontrivial portion of $ \{ {(2\pi )^{ - 1/2}}{e^{inx}}\} $ which will maintain completeness under this perturbation.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0308676-1
Keywords: Completeness in $ {L^2}$, perturbation, complete orthonormal sequence, Fourier series, perturbation of Fourier series
Article copyright: © Copyright 1972 American Mathematical Society