Beschreibung
We consider on the real line, a differential-difference operator called Jacobi-Dunkl operator. This operator, like the others of Dunkl type, plays an important role in the description, in quantum mechanics, of the exactly resolvable models of Calogero-Morse-Sutherland. Its eigenfunction admits a Laplace integral representation whose kernel allows us to define the Jacobi-Dunkl transmutation operators that are shown to be positive. Then, for the Jacobi-Dunkl transform, we formulate inversion formulas and a Paley-Wiener theorem. Using the properties of the transmutation operators and the estimates of the heat kernel, we obtain a version of the Cowling-Price and Hardy theorems for the Jacobi-Dunkl transform. In the case of a bounded interval, we show that the eigenfunction of the Jacobi-Dunkl operator, equal to 1 at zero, is a trigonometric polynomial, related to the Jacobi polynomials. Then, we give a Laplace integral representation of this function called Jacobi-Dunkl polynomial. Finally, we study the harmonic analysis associated with this operator.
Autorenporträt
Frej CHOUCHENE is a past aggregate professor. He taught at the universities of Monastir and Sousse, mathematics at all levels of the licence and the preparatory cycle. He is also the author of several scientific works related to harmonic analysis and published in specialized journals.
Herstellerkennzeichnung:
OmniScriptum SRL
Str. Armeneasca 28/1, office 1
2012 Chisinau
MD
E-Mail: info@omniscriptum.com
