Pointwise characterization of smooth functions within the dunkl-type segal-bargmann space

Abstract

The Dunkl-type Segal-Bargmann space is the Hilbert space of holomorphic functions on ℂ of which even parts and odd parts are square-integrable with respect to certain measures related to the modified Bessel functions of the second kind. The Dunkl-type Segal-Bargmann space is then the (internal) direct sum of the even and odd subspaces. In this work, we characterize smooth functions in the Dunkl-type Segal-Bargmann space associated with the Dunkl operator.