Bargmann transform, Zak transform, and coherent states

Abstract

It is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the Bargmann representation or by using the kq representation which was introduced by J. Zak. In this paper both methods are considered, in particular, in connection with expansions of generalized functions in what are called Gabor series. The setting consists of two spaces of generalized functions (tempered distributions and elements of the class S*) which appear in a natural way in the context of the Bargmann transform. Also, a thorough mathematical investigation of the Zak transform is given. This paper contains many comments and complements on existing literature; in particular, connections with the theory of interpolation of entire functions over the Gaussian integers are given.

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