Jacobi-Dunkl transforms and applications

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.