Images of function and distribution spaces under the Bargmann transform

Abstract

We consider a broad family of test function spaces and their dual (distribution) space. The family includes Gelfand-Shilov spaces, a family of test function spaces introduced by S. Pilipovic. We deduce different characterizations of such spaces, especially under the Bargmann transform and the Short-time Fourier transform. The family also include a test function space, whose dual space is mapped by the Bargmann transform bijectively to the set of entire functions.

arXiv comment53 pages. This version dates from June 2016. The main different between this version with the previous January 2016 version is that Proposition 1.2 for classifications of Gelfand-Shilov distributions with STFT is extended in full range. This leads to that Section 5 could be shorten down significantly

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