Analytical and Numerical Approximation Formulas on the Dunkl-Type Fock Spaces
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Analytical and Numerical Approximation Formulas on the Dunkl-Type Fock Spaces Fethi Soltani1 · Akram Nemri1
Received: 8 May 2016 / Revised: 25 May 2016 / Accepted: 5 June 2016 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016
Abstract In this work, we establish some versions of Heisenberg-type uncertainty principles for the Dunkl-type Fock space Fk (Cd ). Next, we give an application of the classical theory of reproducing kernels to the Tikhonov regularization problem for operator L : Fk (Cd ) → H , where H is a Hilbert space. Finally, we come up with some results regarding the Tikhonov regularization problem and the Heisenberg-type uncertainty principle for the Dunkl-type Segal-Bargmann transform Bk . Some numerical applications are given. Keywords Dunkl-type Fock spaces · Bounded operators · Extremal functions · Tikhonov regularization · Uncertainty principles Mathematics Subject Classification (2010) 32A15 · 32A36 · 46E22
1 Introduction The Dunkl operators [7] are parameterized differential-difference operators acting on some Euclidean space. These operators extend the usual partial derivatives by additional reflection terms and give rise to a generalization of many multi-variable analytic structures like the exponential function, the Fourier transform, and the standard convolution. They provide
Fethi Soltani
[email protected] Akram Nemri [email protected] 1
Department of Mathematics, Faculty of Science, Jazan University, P.O.Box 277, Jazan 45142, Saudi Arabia
F. Soltani, A. Nemri
an useful tool in the study of special functions associated with root systems. During the last years, these operators have gained considerable interest in various fields of mathematics [8, 9, 12, 17], physics and in certain parts of quantum mechanics [14]. They allow the development of the Heisenberg uncertainty principle and the Tikhonov regularization problem on the Dunkl-type Fock spaces. The classical Fock spaces [2] form a family of Hilbert spaces, whose elements are entire functions of n complex variables. These spaces are associated with Fock’s realization of the creation and annihilation operators of Bose particles in quantum field theory. The study of several generalizations of the classical Fock spaces has a long and rich history in many different settings (see for instance [4–6, 35, 36]). In this paper, by using the theory of operators on Hilbert spaces [10, 11], we establish some versions of Heisenberg-type uncertainty principles for the Dunkl-type Fock space Fk (Cd ) [3, 25, 26]. In the Dunkl setting, the Heisenberg-type uncertainty principle for the Dunkl transform Dk [9, 12] is studied by R¨osler in [18] and by Soltani in [30–32]. This is an important support for giving in this work, the Heisenberg-type uncertainty principle for the Dunkl-type Segal-Bargmann transform Bk [3, 25, 26]. Next, building on the ideas of Matsuura et al. [15, 16], Saitoh [22, 24], Soltani [27], and Yamada et al. [34], we give an application of th
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