Polynomial Expansions for Solutions of Higher-Order Bessel Heat Equations

Fitouhi, Ahmed; Mahmoud, N.H.; Mahmoud, Sid Ahmed Ould Ahmed

doi:10.1006/jmaa.1997.5203

Cited by 8 publications

(5 citation statements)

References 7 publications

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“…See [16,22] and [21, Chap 3&4] for more details. For λ ∈ C, define j γ (λ•) : C → C to be the function given by…”

Section: Preliminariesmentioning

confidence: 99%

“…In [16] the authors studied the polynomial expansion for the solutions of the heat equation associated with the operator B r , and very recently, in [12] the operator B r found its interpretation in the theory of special functions associated with complex reflection groups. A more general version of the hyper-Bessel operator is intensively studied by I.…”

Section: Introductionmentioning

confidence: 99%

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On Harmonic analysis associated with the hyper-Bessel operator on the complex plane

Hammouda,

Bennasr,

Fitouhi

2014

Preprint

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“…See [16,22] and [21, Chap 3&4] for more details. For λ ∈ C, define j γ (λ•) : C → C to be the function given by…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Harmonic analysis associated with the hyper-Bessel operator on the complex plane

Hammouda,

Bennasr,

Fitouhi

2014

Preprint

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“…Recently, in a nice and long paper, Cholewinski and Reneke [11] studied and extended, for the operator L 3,ν , the well known theory related to some singular differential operator of second order for which the literature is extensive. Next, Fitouhi et al [12,13] established a harmonic analysis related to this operator (for examples the eigenfunctions, the generalized translation, the Fourier-Airy transform, the heat equation, the heat polynomials, the transmutation operators, … ). Recently the Airy operator has gained considerable interest in various field of mathematics [9,14] and in certain parts of quantum mechanics [15].…”

Section: Introductionmentioning

confidence: 99%

Generalized Bessel Operator and Reproducing Kernel Theory

Soltani¹

2023

Operator Theory - Recent Advances, New Perspectives and Applications

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In 1961, Bargmann introduced the classical Fock space FC and in 1984, Cholewinsky introduced the generalized Fock space F2,νC. 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 F3,νC associated to the generalized Bessel operator L3,ν. The space F3,νC is a reproducing kernel Hilbert space (RKHS). This is the reason for defining the orthogonal projection operator, the Toeplitz operators and the Hankel operators associated to this space. Furthermore, we give an application of the theory of extremal function and reproducing kernel of Hilbert space, to establish the extremal function associated to a bounded linear operator T:F3,νC→H, where H be a Hilbert space. Finally, we come up with some results regarding the extremal functions, when T is a difference operator and an integral operator, respectively. Finally, we remark that it is now natural to raise the problem of studying the Bessel-type Segal-Bargmann transform associated to the space F3,νC. This problem is difficult and will be an open topic. This topic requires more details for the harmonic analysis associated to the operator L3,ν. We have the idea to continue this research in a future paper.

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“…Next, Fitouhi et al [7,8] established a harmonic analysis related to this operator for examples the eigenfunctions, the generalized translation, the Fourier-Airy transform, the heat equation, the heat polynomials, the transmutation operators, … During the last years, the Airy operator have gained considerable interest in various field of mathematics [3][4][5][6], [13][14][15] and in certain parts of quantum mechanics [1]. The results of this work will be useful when discussing the Fock space associated to this operator.…”

Section: Introductionmentioning

confidence: 99%

Generalized Fock space for the Airy operator

Alla¹,

Nemri²,

Soltani³

2016

Fifth International Conference on Advances in Applied Science and Environmental Engineering - ASEE 2016

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In this work, we introduce the Fock space ) (C F  associated to the Airy operator , and we establish Heisenberg-type uncertainty principle for this space. Next, we give an application of the theory of extremal function and reproducing kernel of Hilbert space, to establish the extremal function associated to a bounded linear operator, where H be a Hilbert space. Finally, we come up with some results regarding the extremal functions, when T is the difference operator and the Dunkl-difference operator, respectively.

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Polynomial Expansions for Solutions of Higher-Order Bessel Heat Equations

Abstract: In this paper we give for generalized Bessel operators studied by M. I. Klyuchantsev a representation theory for solutions of the related heat equations. A

Cited by 8 publications

References 7 publications

On Harmonic analysis associated with the hyper-Bessel operator on the complex plane

On Harmonic analysis associated with the hyper-Bessel operator on the complex plane

Generalized Bessel Operator and Reproducing Kernel Theory

Generalized Fock space for the Airy operator

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