The Case of Phase Space

Abstract

The main point of the chapter is a development of the theory of transmutation operators on the phase space associated to the Heisenberg group. The basic constructions heavily depend on many properties of Laguerre functions. They arise as eigenfunctions of the special Hermite operator  $\mathfrak{L}$ . A number of important properties of Laguerre functions and their generalizations are proved at the beginning of the chapter. A holomorphic extension of the discrete Fourier–Laguerre transform leads to the transform ℱ ^(p,q) _l , which is an analog of the spherical transform in the case under consideration. The basic properties of this transform are studied. In particular, the Paley–Wiener theorem for ℱ ^(p,q) _l is proved using the Koornwinder integral representation of Jacobi functions for Laguerre functions. Then transmutation operators on ℂⁿ associated with the Laguerre polynomials expansion are investigated.