Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 010, 22 pages      arXiv:1101.5551      https://doi.org/10.3842/SIGMA.2013.010

The Clifford Deformation of the Hermite Semigroup

Hendrik De Bie a, Bent Ørsted b, Petr Somberg c and Vladimir Souček c
a) Department of Mathematical Analysis, Ghent University, Galglaan 2, 9000 Gent, Belgium
b) Department of Mathematical Sciences, University of Aarhus, Building 530, Ny Munkegade, DK 8000, Aarhus C, Denmark
c) Mathematical Institute of Charles University, Sokolovská 83, 186 75 Praha, Czech Republic

Received September 21, 2012, in final form January 29, 2013; Published online February 05, 2013

Abstract
This paper is a continuation of the paper [De Bie H., Ørsted B., Somberg P., Souček V., Trans. Amer. Math. Soc. 364 (2012), 3875–3902], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many results obtained in [Ben Saïd S., Kobayashi T., Ørsted B., Compos. Math. 148 (2012), 1265–1336]. We establish the analogues of Bochner's formula and the Heisenberg uncertainty relation in the framework of the (holomorphic) Hermite semigroup, and also give a detailed analytic treatment of the series expansion of the associated integral transform.

Key words: Dunkl operators; Clifford analysis; generalized Fourier transform; Laguerre polynomials; Kelvin transform; holomorphic semigroup.

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