Extremal functions and approximate inversion formulas for the Weinstein type Segal-Bargmann space

In 1961, Bargmann introduced the classical Fock space \(\mathscr{F}(\mathbb{C}^d)\) and in 1984, Cholewinski introduced the generalized Fock space \(\mathscr{F}^{\ast}_{\alpha}(\mathbb{C}^d)\), were \(\alpha=(\alpha_1,\dots,\alpha_d)\). 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 \(\mathscr{F}_{\alpha_d,*}(\mathbb{C}^d)\) associated to the Weinstein operator \(\Delta_W\) with \(\alpha_d>-1/2\). This space satisfies the inclusions \(\mathscr{F}^*_{\alpha}(\mathbb{C}^d)\subset\mathscr{F}_{\alpha_d,*}(\mathbb{C}^d) \subset\mathscr{F}(\mathbb{C}^d)\). We prove that the space \(\mathscr{F}_{\alpha,\ast}(\mathbb{C}^d)\) is a reproducing kernel Hilbert space (RKHS). Next, we examine the extremal functions associated to the difference operator \(\mathscr{D}\) and to the primitive operator \(\mathscr{P}\), respectively. Furthermore, we establish approximate inversion formulas for the operators \(\mathscr{D}\) and \(\mathscr{P}\), on \(\mathscr{F}_{\alpha_d,*}(\mathbb{C}^d)\).