POSITIVITY OF THE INTERTWINING OPERATOR AND HARMONIC ANALYSIS ASSOCIATED WITH THE JACOBI-DUNKL OPERATOR ON R

We consider a differential-difference operator Lambda(alpha,beta), alpha > -1/2, beta is an element of R on R. The eigenfunction of this operator equal to 1 at zero is called the Jacobi-Dunkl kernel. We give a Laplace integral representation for this function and we prove that for alpha >= beta >= -1/2, alpha not equal -1/2, the kernel of this integral representation is positive. This result permits us to prove that the Jacobi-Dunkl intertwining operator and its dual are positive. Next we study the harmonic analysis associated with the operator Lambda(alpha,beta) (Jacobi-Dunkl transform, Jacobi-Dunkl translation operators, Jacobi-Dunkl convolution product, Paley Wiener and Plancherel theorems ...).