Positivity of the Generalized Translation Associated with the q-Hankel Transform

Fitouhi, Ahmed; Dhaouadi, Lazhar

doi:10.1007/s00365-011-9132-0

Cited by 19 publications

(7 citation statements)

References 8 publications

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“…This is a continuous function on [0, 1) which satisfies Taking x = ν = 0, we observe that (see [6,Section 4.2])…”

Section: Casementioning

confidence: 98%

“…The proof of the positivity of the generalized translation operator in the q-case requires new techniques and manipulations using a new formulation of the q-analogue of Graf's addition formula. [6,9] We want to make clear how the positivity of the q-Bessel translation operator plays a role in q-Bessel Fourier analysis. In particular, we will prove a q-version of Young's inequality for the associated convolution.…”

Section: Introductionmentioning

confidence: 99%

“…In the first part, we describe the harmonic analysis around the q-Bessel Fourier transform studied by Fitouhi and co-workers. [1][2][3][4][5][6][7] The main motivation of this survey is the recent proof of the positivity of the generalized q-translation operator which plays a crucial role in this theory and which interacts with other fields.…”

Section: Introductionmentioning

confidence: 99%

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Positivity of the generalized translation associated with theq-Hankel transform and applications

Dhaouadi¹

2014

Integral Transforms and Special Functions

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A new formulation of the Graf-type addition formula related to the third Jackson q-Bessel function gives a solution for the problem of the positivity of the generalized q-translation operator associated with the q-Hankel transform. Next some applications in q-theory are treated, for instance the relationship between the q-Bessel-positive and negative-definite functions. We also show how the positivity of the q-Bessel translation operator plays a central role in q-Fourier analysis, namely in the study of Markov operators in the q-context. The paper concludes with the nonnegative product linearization of the q −2 -Lommel polynomials.

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“…This is a continuous function on [0, 1) which satisfies Taking x = ν = 0, we observe that (see [6,Section 4.2])…”

Section: Casementioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Positivity of the generalized translation associated with theq-Hankel transform and applications

Dhaouadi¹

2014

Integral Transforms and Special Functions

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“…These q-Bessel functions have been studied intensively, see e.g. [15], [16], [45] as well as other references. Notes.…”

Section: 2mentioning

confidence: 99%

q-special functions, basic hypergeometric series and operators

Koelink¹

2018

Preprint

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In the lecture notes we start off with an introduction to the q-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the q-hypergeometric difference equation is studied, and in particular we study solutions given in terms of power series at 0 and at ∞. Factorisations of the corresponding operator are considered in terms of a lowering operator, which is the q-derivative, and the related raising operator. Next we consider the q-hypergeometric operator in a special case, and we show that there is a natural Hilbert space -a weighted sequence space-on which this operator is symmetric. Then the corresponding eigenfunctions are polynomials, which are the little q-Jacobi polynomials. These polynomials form a family in the q-Askey scheme, and so many important properties are well known. In particular, we show how the orthogonality relations and the three-term recurrence for the little q-Jacobi polynomials can be obtained using only the factorisation of the corresponding operator. As a next step we consider the q-hypergeometric operator in general, which leads to the little q-Jacobi functions. We sketch the derivation of the corresponding orthogonality using the connection between various eigenfunctions. The link between the q-hypergeometric operators with different parameters is studied in general using q-analogues of fractional derivatives, and this gives transmutation properties for this operator. In the final parts of these notes we consider partial extensions of this approach to orthogonal polynomials and special functions. The first extension is a brief introduction to the Askey-Wilson functions and the corresponding integral transform. The second extension is concerned with a matrix-valued extension of the q-hypergeometric difference equation and its solutions.

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“…Specially, we need the positivity of the q -even translation operator [9] for proving the following inequality for f ∈ L 1 ( ℝ q , + ) ,…”

Section: Q-fourier Multiplier Operatorsmentioning

confidence: 99%

Analytical approximation formulas in quantum calculus

Nemri

Soltani

2016

Mathematics and Mechanics of Solids

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We study the class of q-Fourier multiplier operators T m : = F q ( F q ) , which are acted on the q-Sobolev space H * , q s ( ℝ q ) , and we obtain the exact expression and some properties for the extremal functions of the best approximation problem in quantum calculus inf f ∈ H * , q s ( ℝ q ) { η ∥ f ∥ H * , q s ( ℝ q ) 2 + ∥ g − T m f ∥ L 2 ( ℝ q , + ) 2 } , where η > 0 and g ∈ L 2 ( ℝ q , + ) . As an application, we provide numerical approximate formulas for a limit case η ↑ 0 ; using q-calculus, which generalizes the Gauss-Kronrod method studied given in [14] in one-dimensional space.

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Positivity of the Generalized Translation Associated with the q-Hankel Transform

Abstract: In this paper we study the positivity of the generalized q-translation associated with the q-Bessel Hahn Exton function which is deduced by a new formulation of the Graf's addition formula related to this function.

Cited by 19 publications

References 8 publications

Positivity of the generalized translation associated with theq-Hankel transform and applications

Positivity of the generalized translation associated with theq-Hankel transform and applications

q-special functions, basic hypergeometric series and operators

Analytical approximation formulas in quantum calculus

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