We consider a singular differential-difference operator Λ on the real line which includes, as particular case, the Dunkl operator associated with the reflection group Z2 on R. We exhibit a Laplace integral representation for the eigenfunctions of the operator Λ. From this representation, we construct a pair of integral transforms which turn out to be transmutation operators of Λ 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 Λ. In particular, we establish a Paley–Wiener theorem and a Plancherel theorem for the Fourier transform associated to Λ.
Miyachi's theorems associated with a differential- difference operator on the real line