A class of Riesz-Fischer sequences
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- by Russell M. Reid PDF
- Proc. Amer. Math. Soc. 123 (1995), 827-829 Request permission
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
- R. P. Boas Jr., A general moment problem, Amer. J. Math. 63 (1941), 361–370. MR 3848, DOI 10.2307/2371530 N. Levinson, Gap and density theorems, Academic Press, New York, 1980.
- 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, DOI 10.1090/coll/019
- 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 15553, DOI 10.5802/afst.424 R. M. Young, An introduction to nonharmonic fourier series, Amer. Math. Soc. Colloq. Publ., vol. 26, Amer. Math. Soc., Providence, RI, 1939.
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 827-829
- MSC: Primary 42A70; Secondary 30D99, 42C15
- DOI: https://doi.org/10.1090/S0002-9939-1995-1223519-0
- MathSciNet review: 1223519