On harmonic analysis related with the generalized Dunkl operator

Bouzeffour, Fethi; Nemri, Akram; Fitouhi, Ahmed; Ghazouani, Sami

doi:10.1080/10652469.2011.618807

Cited by 5 publications

(3 citation statements)

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“…The study of harmonic analysis operators in the Dunkl setting has been carried out by many authors in recent years, see [6,7,[9][10][11]20]. In particular, the Dunkl transform pair, for f in a suitable function class takes the form [21]…”

Section: Non-symmetric Bessel Functionsmentioning

confidence: 99%

“…[6][7][8][9][10] Besides its mathematical interest, the operator T 0,ν has quantum-mechanical applications; it is naturally involved in the study of one-dimensional harmonic oscillators governed by Wigner commutation rules. [11] The present paper is devoted to study the differential-difference operator T α,ν defined in (1.1) and to show that it plays a central role in the theory of special functions and integral transforms.…”

Section: Introductionmentioning

confidence: 99%

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The non-symmetric Lommel function transform

Bouzeffour¹,

Chorfi²

2015

Integral Transforms and Special Functions

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In this work, we introduce a differential-difference operator T α,ν , which includes as particular cases the derivative operator and the Dunkl operator associated with root system of type A 1 . We show that the symmetrization of the eigenfunction of this operator is a solution of an inhomogeneous Bessel differential equation. Also we construct an intertwining operator between T α,ν and the Dunkl operator. As application, we study a polynomials set of Boas and Buck type by using the quasi-monomiality method related to this operator.

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Section: Non-symmetric Bessel Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The non-symmetric Lommel function transform

Bouzeffour¹,

Chorfi²

2015

Integral Transforms and Special Functions

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“…Dunkl operators [8] are commuting differential-difference operators associated with finite reflection groups. As generalizations of partial derivatives, these operators appear in a wide range of mathematical applications, such as Fourier analysis related to root systems [2], intertwining operator and angular momentum algebra [10,11], integrable Calogero-Moser-Sutherland models [9,23], harmonic analysis [4], the generation of orthogonal polynomials [16], and nonlinear wave equations [17]. Another field of application arises in quantum physics when conventional derivatives are replaced by Dunkl operators, hence leading to deformed versions of quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%