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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A class of Riesz-Fischer sequences

Author: Russell M. Reid
Journal: Proc. Amer. Math. Soc. 123 (1995), 827-829
MSC: Primary 42A70; Secondary 30D99, 42C15
MathSciNet review: 1223519
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Abstract: It is proved that if $ \{ {\lambda _n}\} $ is a sequence of real numbers whose differences are nondecreasing and satisfy $ \sum {{{({\lambda _{k + 1}} - {\lambda _k})}^{ - 2}} < \infty }$, then the set of complex exponentials $ \{ {e^{i{\lambda _n}x}}\}$ is a Riesz-Fischer sequence in $ {L_2}[ - A,A]$ for every $ A > 0$, which is to say that for any positive A, the equations $ \smallint _{ - A}^Af(x){e^{i{\lambda _n}x}}dx = {c_n}$ admit a solution f in $ {L_2}[ - A,A]$ for every sequence $ \{ {c_n}\} $ in $ {\ell _2}$. In particular, if $ {\lambda _n} = {n^p}$, then $ \{ {e^{i{\lambda _n}x}}\} $ is a Riesz-Fischer sequence when $ p > \frac{1}{2}$.

References [Enhancements On Off] (What's this?)

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Additional Information

PII: S 0002-9939(1995)1223519-0
Keywords: Riesz-Fischer sequence, moment problem, nonharmonic Fourier series
Article copyright: © Copyright 1995 American Mathematical Society