The Continuous Wavelet Transform Associated with a Differential-Difference Operator and Applications

Abstract

In this paper we consider a class of singular differential-difference operators on the real line. We define and study the continuous wavelet transform associated with these operators. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. As applications, we establish Calderón's formulas and give practical real inversion formula for the generalized continuous wavelet transform. At the end of the paper, analogous of Heisenberg's inequality for the generalized continuous wavelet transform on Chébli-Triméche hypergroups and a special case of Jacobi-Dunkl operator are proved.