On Fourier transformation of a class of entire functions

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.