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On the excess of sets of complex exponentials

Author(s): Nobuhiko Fujii; Akihiro Nakamura; Ray Redheffer
Journal: Proc. Amer. Math. Soc. 127 (1999), 1815-1818.
MSC (1991): Primary 30B60
Posted: February 17, 1999
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Abstract: For $-\infty<n<\infty$ let $\mu _n$ be complex numbers such that $\mu _n-n$ is bounded. For $n>0$ define $\lambda _n=\mu _n+a$, $\lambda _{-n}=\mu _{-n}-b$ where $a,b\ge 0$. Then the excesses $E$ in the sense of Paley and Wiener satisfy $E(\{\lambda _n\})\le E(\{\mu _n\})$.


References:

[1]
Levinson, Norman, Gap and Density Theorems, AMS Colloquium Publication XXVI (1940), Chapters I, III and IV. MR 2:180d

[2]
Paley, Raymond E. A. C and Norbert Wiener, Fourier Transforms in the Complex Domain, AMS Colloquium Publication XIX (1934), Chapter VI. CMP 97:13

[3]
Redheffer, Raymond M., Completeness of Sets of Complex Exponentials, Advances in Mathematics 24 (1977) 1-62. MR 56:5852

[4]
Schwartz, Laurent, Approximation d'une fonction quelconque par des sommes d'exponentielles imaginaires, Ann. Fac. Sci. Toulouse (1943), 111-176. Reprint (Paris 1959) with some additions.

[5]
Young, Robert M., An Introduction to Nonharmonic Fourier Series, Academic Press 1980, Chapter 3. MR 81m:42027


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Additional Information:

Nobuhiko Fujii
Affiliation: Department of Mathematics, Tokai University, 3-20-1 Orido, Shimizu, Shizuoka 424-8610, Japan
Email: nfujii@scc.u-tokai.ac.jp

Akihiro Nakamura
Affiliation: Department of Mathematics, Tokai University, 3-20-1 Orido, Shimizu, Shizuoka 424-8610, Japan

Ray Redheffer
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555

DOI: 10.1090/S0002-9939-99-04664-X
PII: S 0002-9939(99)04664-X
Received by editor(s): January 31, 1997
Received by editor(s) in revised form: September 20, 1997
Posted: February 17, 1999
Communicated by: J. Marshall Ash
Copyright of article: Copyright 1999, American Mathematical Society


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Akihiro Nakamura,Nobuhiko Fujii,Ray Redheffer, On the excess of sets of complex exponentials, Proceedings of the American Mathematical Society, 1999.


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