TRANSMUTATION OPERATORS AND PALEY-WIENER THEOREM ASSOCIATED WITH A SINGULAR DIFFERENTIAL-DIFFERENCE OPERATOR ON THE REAL LINE

We consider a singular differential-difference operator Lambda on the real line which includes, as particular case, the Dunkl operator associated with the reflection group Z(2) on R. We exhibit a Laplace integral representation for the eigenfunctions of the operator Lambda. From this representation, we construct a pair of integral transforms which turn out to be transmutation operators of Lambda into the first derivative operator d/dx. We exploit these transmutation operators to develop a new commutative harmonic analysis on the real line corresponding to the operator Lambda. In particular, we establish a Paley-Wiener theorem and a Plancherel theorem for the Fourier transform associated to Lambda.