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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?)

  • [1] R. P. Boas Jr., A general moment problem, Amer. J. Math. 63 (1941), 361–370. MR 0003848 (2,281d)
  • [2] N. Levinson, Gap and density theorems, Academic Press, New York, 1980.
  • [3] Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142 (98a:01023)
  • [4] Laurent Schwartz, Approximation d’un fonction quelconque par des sommes d’exponentielles imaginaires, Ann. Fac. Sci. Univ. Toulouse (4) 6 (1943), 111–176 (French). MR 0015553 (7,437d)
  • [5] R. M. Young, An introduction to nonharmonic fourier series, Amer. Math. Soc. Colloq. Publ., vol. 26, Amer. Math. Soc., Providence, RI, 1939.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1223519-0
PII: S 0002-9939(1995)1223519-0
Keywords: Riesz-Fischer sequence, moment problem, nonharmonic Fourier series
Article copyright: © Copyright 1995 American Mathematical Society