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

Abstract

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...

Intertwining Operators Associated with Dihedral Groups

“…The Poisson kernel associated to I k can be written in terms of the kernel associated to I 2 and the latter has an integral expression [10,Theorem 7.6.11]. 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.…”

On a Neumann-type series for modified Bessel functions of the first kind

“…, n, which correspond from a geometrical point of view to the n-fold covering of the unit circle z → z n . Recently, this suggestion was confirmed in [3] where (1) is expressed by means of Horn's hypergeometric function Φ 2 (see the definition below) in the variables cos(θ s ), s = 1, . .…”

“…formula we recall below. We consider, for fixed R > 0 and ξ ∈ [0, π], the following function If we set a s := R cos θ s , where θ s is defined in (5), then it is proved in [3] that its Laplace transform in the variable z is given by…”

An Introduction to Dunkl Theory and Its Analytic Aspects

“…For the root system B 2 , a complicated formula was obtained in [26] and a simpler one recently in [5]. The case of dihedral root systems I 2 pmq is currently investigated psee [22], [19] and the references thereinq.…”

“…See [73] for a similar result about generalized Bessel functions. ‚ The expressions (49) are substitutes for the integral representations (10) and (19). A different integral representation of F λ is established in [83].…”