Abstract
For the Dunkl Laplacian associated with a Coxeter group, the Von Neumann spectral decomposition is given. As a consequence, for the Dunkl kernel an integral representation with respect to the Lebesgue measure is given (DIR). Also, this integral representation (DIR) is used to write the Dunkl kernel as a radial integral representation with respect to a radial probability measure \(({\textbf {DIR}})_{rad}\). Moreover, we compute the Schwartz kernel associated with the spectral density of the Dunkl Laplacian and we prove that it is nothing but the kernel of the generalized spectral projector as given in Ben Said and Mejjaoli (J Funct Spaces, 2020). Furthermore, we use this Schwartz kernel to build a more general functional calculus for the Dunkl Laplacian. Finally, we apply this functional calculus to give an integral representation for the wave kernel of the Dunkl Laplacian.
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Acknowledgements
The authors are very grateful for the referee for this valuable suggestions. They are also very thank full to the professor Ahmed Touhami for having done several reading and for helping us a lot in the realization of the Tex file. Finally, they are would like to thank the audience of the seminar of “Functions and operators theory” especially the professors Omar EL-Fellah, E.H. Zeroili and A. Belhaj.
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Askour, N.E., El Mourni, A. & El Yazidi, I. Spectral decomposition of Dunkl Laplacian and application to a radial integral representation for the Dunkl kernel. J. Pseudo-Differ. Oper. Appl. 14, 28 (2023). https://doi.org/10.1007/s11868-023-00522-w
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DOI: https://doi.org/10.1007/s11868-023-00522-w