On the excess of sets of complex exponentials

Fujii, Nobuhiko; Nakamura, Akihiro; Redheffer, Ray

doi:10.1090/S0002-9939-99-04664-X

On the excess of sets of complex exponentials
HTML articles powered by AMS MathViewer

by Nobuhiko Fujii, Akihiro Nakamura and Ray Redheffer PDF

Proc. Amer. Math. Soc. 127 (1999), 1815-1818 Request permission

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

Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
Paley, Raymond E. A. C and Norbert Wiener, Fourier Transforms in the Complex Domain, AMS Colloquium Publication XIX (1934), Chapter VI.
Raymond M. Redheffer, Completeness of sets of complex exponentials, Advances in Math. 24 (1977), no. 1, 1–62. MR 447542, DOI 10.1016/S0001-8708(77)80002-9
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.
Robert M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 591684