Abstract
For s∈ℝ, denote by Pksf the “projection” of a function f in Dℝd into the eigenspaces of the Dunkl Laplacian Δk corresponding to the eigenvalue −s2. The parameter k comes from Dunkl’s theory of differential-difference operators. We shall characterize the range of Pks on the space of functions f∈Dℝd supported inside the closed ball BO,R¯. As an application, we provide a spectral version of the Paley-Wiener theorem for the Dunkl transform.
Highlights
Analysis of the Dunkl Laplacian operator Δk on Rd commenced in the early 90’s, inspired by numerous results in the Euclidean setting, as well as some extensions of this to flat symmetric spaces
The purpose of this paper is to study a family of eigenfunctions for the Dunkl Laplacian derived through the use of the inversion formula for the Dunkl transform
As an application of the main result, we prove a spectral version of the complex Paley-Wiener theorem for the Dunkl transform Fk given in [23]
Summary
Analysis of the Dunkl Laplacian operator Δk on Rd commenced in the early 90’s, inspired by numerous results in the Euclidean setting, as well as some extensions of this to flat symmetric spaces. There have been increasing interests in the study of problems involving the Dunkl Laplacian and have received a lot of attention, see for instance [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. The purpose of this paper is to study a family of eigenfunctions for the Dunkl Laplacian derived through the use of the inversion formula for the Dunkl transform. Our main result may be interpreted as a contribution to the spectral theory of the Dunkl Laplacian. As an application of the main result, we prove a spectral version of the complex Paley-Wiener theorem for the Dunkl transform Fk given in [23].
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