DIFFERENCE AND PRIMITIVE OPERATORS ON THE DUNKL-TYPE FOCK SPACE $\mathscr {F}_{\alpha }(\mathbb {C}^{d})$

Abstract

In 1961, Bargmann introduced the classical Fock space $\mathscr {F}(\mathbb {C}^{d})$ and in 1984, Cholewinsky introduced the generalized Fock space $\mathscr {F}_{\alpha ,e}(\mathbb {C}^{d})$. These two spaces are the aim of many works, and have many applications in mathematics, in physics, and in quantum mechanics. In this work, we introduce and study the Fock space $\mathscr {F}_{\alpha }(\mathbb {C}^{d})$ associated to the Dunkl operators $T_{\alpha _{j}}$ with $\alpha _{j}>-1/2$ for all $j=1,\ldots ,d$. This space is an extension of the Dunkl-type Fock space $\mathscr {F}_{\alpha }(\mathbb {C})$ constructed by Sifi and Soltani in 2002. We prove that the space $\mathscr {F}_{\alpha }(\mathbb {C}^{d})$ is a Hilbert space with reproducing kernel. Next, we give an application of the classical theory of reproducing kernels to the Tikhonov regularization problem for the bounded linear operator $\mathscr {L}:\mathscr {F}_{\alpha }(\mathbb {C}^{d})\rightarrow \mathscr {H}$, where $\mathscr {H}$ is a Hilbert space. Finally, we come up with some results regarding the extremal functions, when $\mathscr {L}$ is the difference operator and the primitive operator, respectively.