Heisenberg-Weyl Groups and Generalized Hermite Functions

We introduce a multi-parameter family of bases in the Hilbert space L-2(R) that are associated to a set of Hermite functions, which also serve as a basis for L-2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these generalized Hermite functions. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Heisenberg-Weyl group and some of their extensions.

Automated summary

We introduce a multi-parameter family of bases in the Hilbert space associated to Hermite functions, and show how these generalized Hermite functions are transformed under unitary representations of various groups.