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A positive radial product formula for the Dunkl kernel
TLDR
In this article, a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel, was shown to be positivity-preserving.Abstract:
It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for non-negative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial hereby means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.read more
Citations
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Book ChapterDOI
An Introduction to Dunkl Theory and Its Analytic Aspects
TL;DR: In this paper, the authors introduce rational and trigonometric Dunkl theory, starting with the historical examples of special functions associated with radial Fourier analysis on rank one symmetric spaces.
Journal ArticleDOI
Clifford algebras, Fourier transforms and quantum mechanics
TL;DR: In this article, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2).
Journal ArticleDOI
A transference theorem for the Dunkl transform and its applications
Feng Dai,Heping Wang +1 more
TL;DR: For a family of weight functions invariant under a finite reflection group, this article showed how weighted Lp multiplier theorems for the Dunkl transform on the Euclidean space Rd can be transferred from the corresponding results for h-harmonic expansions on the unit sphere Sd of Rd+1.
Journal ArticleDOI
Uncertainty principles for the Dunkl transform
Takeshi Kawazoe,Hatem Mejjaoli +1 more
TL;DR: In this article, a generalization and a variant of Cowling-Price's theorem, Beurling's theorem and Donoho-Stark's uncertainty principle are obtained for the Dunkl transform.
Posted Content
The class of Clifford-Fourier transforms
TL;DR: In this article, Brackx et al. studied the Fourier transform integral kernel of hypercomplex signals and their Fourier transforms from general principles, using four different yet equivalent definitions of the classical Fourier Transform.
References
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Book
A treatise on the theory of Bessel functions
TL;DR: The tabulation of Bessel functions can be found in this paper, where the authors present a comprehensive survey of the Bessel coefficients before and after 1826, as well as their extensions.
Book
Groups and geometric analysis
TL;DR: Geometric Fourier analysis on spaces of constant curvature Integral geometry and Radon transforms Invariant differential operators Invariants and harmonic polynomials Spherical functions and spherical transforms Analysis on compact symmetric spaces Appendix Some details Bibliography Symbols frequently used Index Errata.
Journal ArticleDOI
Differential-difference operators associated to reflection groups
TL;DR: In this article, a theory of spherical harmonics for measures invariant under a finite reflection group is presented, where the measures are products of powers of linear functions, whose zero-sets are the mirrors of the reflections in the group, times the rotation-invariant measure on the unit sphere in Rn.
Book
Orthogonal Polynomials of Several Variables
Charles F. Dunkl,Yuan Xu +1 more
TL;DR: In this article, the authors considered the properties of orthogonal polynomials on the unit sphere, root systems and Coxeter groups, and the Summability of Orthogonal expansions.