On Fourier transformation of a class of entire functions

I. Kh. Musin^a, M. I. Musin<sup>b

a</sup> Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa, Russia
^b Bashkir State University, Ufa, Russia

Abstract: We consider a space of entire functions of several complex variables decaying fast on $\mathbb R^n$ and such that their growth along $i\mathbb R^n$ is majorized by means of a family of weight functions. Under certain assumptions for the weight functions we obtain an equivalent description of this space in terms of estimates for partial derivatives of the functions in $\mathbb R^n$ and prove a Paley–Wiener type theorem.

Keywords: Gelfand–Shilov spaces, Fourier transform, entire functions, convex functions.

UDC: 517.982.3

MSC: 32A15, 42B10, 46E10, 46F05, 42A38

Received: 05.11.2014

 English version: Ufa Mathematical Journal, 2014, 6:4, 108–121

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