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Noncommutative quantum decomposition of Gegenbauer white noise process

Department of Mathematics, College of Science Al-Zulfi, Majmaah University, P. O. Box 66, Al Majma’ah 11952, Saudi Arabia

Department of Mathematics, Nabeul Preparatory Institute for Engineering Studies, University of Carthage, Carthage, Tunisia

E-mail Address: anis.r@mu.edu.sa

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https://doi.org/10.1142/S0219025722500187Cited by:0 (Source: Crossref)

Abstract

The main purpose of this paper is to derive a general structure of Gegenbauer white noise analysis as a counterpart class of non-Lévy white noise. First, we start with a new detailed construction of the Gegenbauer Fock space $Γ_{β} (ℋ)$ which serves to obtain the quantum decomposition associated with the Gegenbauer white noise processes. More precisely, based on the notion of quantum decomposition and the orthogonalization of polynomials of noncommutative Gegenbauer white noise $ω (t) : = b_{t}^{*} + b_{t} + b_{t}^{*} b_{t} {\tilde{b}}_{t}$ , we study the chaos property of the noncommutative $L^{2}$ -space with respect to the vacuum expectation $τ$ . Next, we determine the distribution of the Gegenbauer operator $J (χ_{D}) = 〈 ω, χ_{D} 〉$ and as a consequence we give some useful properties of the Gegenbauer white noise process.

Communicated by Uwe Franz

Keywords:

AMSC: 60J65, 60J45, 60J51, 60H40

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