Images of function and distribution spaces under the Bargmann transform

We consider a broad family of test function spaces and their dual (distribution) spaces. The family includes Gelfand–Shilov spaces, and a family of test function spaces introduced by Pilipović. 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.