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On a space of entire functions rapidly decreasing on \(\mathbb R^n\) and its Fourier transform. (English) Zbl 1345.46015

Summary: A space of entire functions of several complex variables rapidly decreasing on \(\mathbb R^n\) and such that their growth along \(i\mathbb R^n\) is majorized with the help of a family of weight functions is considered in this paper. For such a space an equivalent description in terms of estimates on all of its partial derivatives as functions on \(\mathbb R^n\) and a Paley-Wiener type theorem are obtained.

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46F05 Topological linear spaces of test functions, distributions and ultradistributions
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