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

Abstract

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 \(\mathscr{F}_{\alpha_d,*}(\mathbb{C}^d)\).

2020 Mathematics Subject Classification:

30H20, 32A15, 46C05

Keywords

Weinstein type Fock space, reproducing kernels, extremal functions, inversion formulas

Full text

Author Details

Fethi Soltani

Faculté des Sciences de Tunis
Laboratoire d'Analyse Mathématique et Applications
LR11ES11, Université de Tunis El Manar
Tunis 2092, Tunisia
and
Ecole Nationale d'Ingénieurs de Carthage
Université de Carthage
Tunis 2035, Tunisia
e-mail: fethi.soltani@fst.utm.tn

Hanen Saadi

Faculté des Sciences de Tunis
Laboratoire d'Analyse Mathématique et Applications
LR11ES11, Université de Tunis El Manar
Tunis 2092, Tunisia
e-mail: hanen.saadi@etudiant-fst.utm.tn


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