TRANSMUTATION OPERATORS AND PALEY-WIENER THEOREM ASSOCIATED WITH A CHEREDNIK TYPE OPERATOR ON THE REAL LINE

We consider a singular differential-difference operator Lambda on the real line which generalizes the one-dimensional Cherednik operator. We construct transmutation operators between Lambda and first-order regular differential-difference operators on R. We exploit these transmutation operators, firstly to establish a Paley-Wiener theorem for the Fourier transform associated with Lambda, and secondly to introduce a generalized convolution on R tied to Lambda.