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On the zeros and Fourier transforms of entire functions in the Paley–Wiener space

Published online by Cambridge University Press:  24 October 2008

Konstantin M. Dyakonov
Konstantin M. Dyakonov
Affiliation:
Department of Mathematics II, St. Petersburg University of Electrical Engineering, Ul. Prof. Popova 5, St. Petersburg, 197376, Russia
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Abstract

Let f be an entire function of the form

where ø is a function in L2(ℝ) with compact support. If f|ℝ is real-valued then, for obvious reasons, (a) the supporting interval for ø is symmetric with respect to the origin, and

Assuming that f has no zeros in {Im z > 0}, we prove that the converse is also true: (a) and (b) together imply that f|ℝ takes values in αℝ, where α is a fixed complex number.

The proof relies on a certain formula involving the Dirichlet integral, which may be interesting on its own.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 2 , February 1996 , pp. 357 - 362
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

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