500 research outputs found
A positive radial product formula for the Dunkl kernel
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.Comment: 25 page
Real Paley-Wiener theorems and local spectral radius formulas
We systematically develop real Paley-Wiener theory for the Fourier transform
on R^d for Schwartz functions, L^p-functions and distributions, in an
elementary treatment based on the inversion theorem. As an application, we show
how versions of classical Paley-Wiener theorems can be derived from the real
ones via an approach which does not involve domain shifting and which may be
put to good use for other transforms of Fourier type as well. An explanation is
also given why the easily applied classical Paley-Wiener theorems are unlikely
to be able to yield information about the support of a function or distribution
which is more precise than giving its convex hull, whereas real Paley-Wiener
theorems can be used to reconstruct the support precisely, albeit at the cost
of combinatorial complexity. We indicate a possible application of real
Paley-Wiener theory to partial differential equations in this vein and
furthermore we give evidence that a number of real Paley-Wiener results can be
expected to have an interpretation as local spectral radius formulas. A
comprehensive overview of the literature on real Paley-Wiener theory is
included.Comment: 27 pages, no figures. Reference updated. Final version, to appear in
Trans. Amer. Math. So
Sonine Transform Associated to the Dunkl Kernel on the Real Line
We consider the Dunkl intertwining operator Vα and its dual tVα, we define and study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of the Dunkl Laplacian Δα and establish inversion formulas for the Dunkl Sonine operator Sα,β and its dual tSα,β. Also, we give a Plancherel formula for the operator tSα,β
Sonine Transform Associated to the Dunkl Kernel on the Real Line
We consider the Dunkl intertwining operator Vα and its dual tVα, we define and study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of the Dunkl Laplacian Δα and establish inversion formulas for the Dunkl Sonine operator Sα,β and its dual tSα,β. Also, we give a Plancherel formula for the operator tSα,β
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