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 Mat. Zametki, 2012, Volume 92, Issue 2, Pages 302–312 (Mi mz8874)

Rapid Decrease of Entire Functions along the Real Axis and the Uniqueness of the Fourier-Transform Pair

A. M. Sedletskii

M. V. Lomonosov Moscow State University

Abstract: We study integral classes of entire functions rapidly decreasing along the real axis, i.e., classes of entire functions whose rapid decrease along the real axis is brought about by the existence of definite weighted integrals. We establish a relationship between these classes and some classes known earlier, in which the decrease of functions along the real axis is given by uniform estimates. As an application, we obtain a concrete realization (also in the integral sense) of the well-known Wiener thesis that the “pair $f$ and $\widehat{f}$ cannot be very small at infinity”.

Keywords: integral class of entire functions, rapidly decreasing entire function, Fourier transform, Fourier-transform pair, definite weighted integral, generalized indicator of a function, Young duality of functions

DOI: https://doi.org/10.4213/mzm8874

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English version:
Mathematical Notes, 2012, 92:2, 270–279

Bibliographic databases:

Document Type: Article
UDC: 517.547.22+517.443

Citation: A. M. Sedletskii, “Rapid Decrease of Entire Functions along the Real Axis and the Uniqueness of the Fourier-Transform Pair”, Mat. Zametki, 92:2 (2012), 302–312; Math. Notes, 92:2 (2012), 270–279

Citation in format AMSBIB
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