Dunkl kernel associated with dihedral groups

Deleaval, Luc; Демни, Низар; Youssfi, Hassan

doi:10.1016/j.jmaa.2015.07.029

Cited by 11 publications

(18 citation statements)

References 19 publications

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“…This is used to derive a complicated integral formula for the kernel in [2]. It is worth mentioning that there has also been attempt on explicit expression for the kernel V [e i ·,y ](x) in the dihedral group setting [4], but the result is in series rather than in integral. Our partial closed form of the intertwining operator gives satisfactory formulas when y = y p,k in both cases.…”

Section: Poisson Kernels For H-harmonics and Sieved Gegenbauer Polynomentioning

confidence: 99%

Intertwining Operators Associated with Dihedral Groups

2019

Constr Approx

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The Dunkl operators associated to a dihedral group are a pair of differential-difference operators that generate a commutative algebra acting on differentiable functions in R 2 . The intertwining operator intertwines between this algebra and the algebra of differential operators. The main result of this paper is an integral representation of the intertwining operator on a class of functions. As an application, closed formulas for the Poisson kernels of h-harmonics and sieved Gegenbauer polynomials are deduced when one of the variables is at vertices of a regular polygon, and similar formulas are also derived for several other related families of orthogonal polynomials.

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Section: Poisson Kernels For H-harmonics and Sieved Gegenbauer Polynomentioning

confidence: 99%

Intertwining Operators Associated with Dihedral Groups

2019

Constr Approx

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“…However, this is not made concrete there. In [9], the authors use the shift principle of [24] and act with multiple combinations of the Dunkl operators on the Dunkl Bessel function to derive the Dunkl kernel in the dihedral setting. However, there the Dunkl Bessel function was only known in a few cases.…”

Section: Now We Have Our First Main Results In This Sectionmentioning

confidence: 99%

Explicit formulas for the Dunkl dihedral kernel and the (κ,a)-generalized Fourier kernel

Constales

Bie

Lian

2018

Journal of Mathematical Analysis and Applications

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In this paper, a new method is developed to obtain explicit and integral expressions for the kernel of the (κ, a)-generalized Fourier transform for κ = 0. In the case of dihedral groups, this method is also applied to the Dunkl kernel as well as the Dunkl Bessel function. The method uses the introduction of an auxiliary variable in the series expansion of the kernel, which is subsequently Laplace transformed. The kernel in the Laplace domain takes on a much simpler form, by making use of the Poisson kernel. The inverse Laplace transform can then be computed using the generalized Mittag-Leffler function to obtain integral expressions. In case the parameters involved are integers, explicit formulas are obtained using partial fraction decomposition.New bounds for the kernel of the (κ, a)-generalized Fourier transform are obtained as well.Recently, a lot of attention has been given to various generalizations of the Fourier transform. This paper focusses on two in particular, namely the Dunkl transform [14,7] and the (κ, a)-generalized Fourier transform [4]. Both transforms depend on a number of parameters, and as such it is an open problem, except for certain special cases, to find concrete formulas for their integral kernels. Our aim in this paper is to develop a new method for finding explicit expressions as well as integral expressions for these kernels. Explicit expressions can be obtained when some of the arising parameters take on rational or integer values. The integral expressions we will obtain are valid in full generality and are expressed in terms of the generalized Mittag-Leffler function (see [22] or the subsequent Definition 1).Essentially our method works as follows. Consider the following series expansion, for x, y ∈ R mwith λ = (m − 2)/2, z = |x||y|, ξ = x, y /z, J j+λ (z) the Bessel function and C λ j (ξ) the Gegenbauer polynomial. It is not so easy to recognize that this is the classical Fourier kernel e −i x,y .However, when we introduce an auxiliary variable t in the kernel as followswe can take the Laplace transform in t of K m (x, y, t). Simplifying the result by making use of the Poisson kernel (see subsequent Theorem 2) then yieldsof which we immediately compute the inverse Laplace transform as K m (x, y, t) = t m−2 2 e −it x,y and the classical Fourier kernel is recovered by putting t = 1. We develop this method in full generality for the Dunkl kernel related to dihedral groups, as well as for the (κ, a)-generalized Fourier transform when κ = 0. The restriction to dihedral groups is necessary, because only then the Poisson kernel for the Dunkl Laplace operator is known, see [15] or subsequent Theorem 11.Let us describe our main results. The Laplace transform of the (0, a)-generalized Fourier transform is obtained in Theorem 4. When a = 2/n and m is even, the result is a rational function and we can apply partial fraction decomposition to obtain an explicit expression, see Theorem 6. We prove that the kernel for a = 2/n is bounded by 1 in Theorem 9, for both even and odd dimensions. For arbit...

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“…Remark 3.9. ‚ Mind the possible formal confusion between (22) and the classical Bessel function of the first kind J ν . ‚ Conversely, the Dunkl kernel E λ pxq can be recovered by applying to the generalized Bessel function J λ pxq a linear differential operator in x whose coefficients are rational functions of λ psee [62, proposition 6.8.(4)]q.…”

Section: Dunkl Kernelmentioning

confidence: 99%

“…‚ Conversely, the Dunkl kernel E λ pxq can be recovered by applying to the generalized Bessel function J λ pxq a linear differential operator in x whose coefficients are rational functions of λ psee [62, proposition 6.8.(4)]q. ‚ In dimension 1, K reg is the complement of´N´1 2 in C. The generalized Bessel function (22) reduces to the modified Bessel function encountered in Subsection 2.2 :…”

Section: Dunkl Kernelmentioning

confidence: 99%

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An Introduction to Dunkl Theory and Its Analytic Aspects

Anker

2017

Trends in Mathematics

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Intertwining operator and (dual) Abel transform 22 3.6. Generalized translations, convolution and product formula 23 3.7. Comments, references and further results 25 4. Trigonometric Dunkl theory 27 4.1. Cherednik operators 27 4.2. Hypergeometric functions 28 4.3. Cherednik transform 30 4.4. Rational limit 30 4.5. Intertwining operator and (dual) Abel transform 31 4.6. Generalized translations, convolution and product formula 32 4.7. Comments, references and further results 33 Appendix A. Root systems 34 References 38

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Dunkl kernel associated with dihedral groups

Cited by 11 publications

References 19 publications

Intertwining Operators Associated with Dihedral Groups

Intertwining Operators Associated with Dihedral Groups

Explicit formulas for the Dunkl dihedral kernel and the (κ,a)-generalized Fourier kernel

An Introduction to Dunkl Theory and Its Analytic Aspects

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