Segal –Bargmann Transform for The Lévy Functionals space and Sobolev Spaces

Abstract

We study the basics of holomorphic function spaces , and their holomorphically equivalent Segal-Bargmann spaces . We also consider the canonical commutation relations to derive the Segal – Bargmann transform and invstigate the representations of Lie group and Lie algebra . We give analytic functions on the Hilbert space and estimates of bounds of the Segal – Bargmann transform on the Lebesgue space functions . We investigate the Segal-Bargmann transform and functional calculus on matrix spaces and the Theory of Semi-circular and circular systems . We study the Segal-Bargmann transform on a symmetric space of the compact type and for Lévy functional . We also show the pointwise bound for holomorphic functions which are square-integrable with respect to the density fuctions and hence the generalized Segal-Bargmann transform on holomorphic Sobolev spaces .