Uncertainty Principles for the Continuous Dunkl Gabor Transform and the Dunkl Continuous Wavelet Transform

In this paper we consider the Dunkl operators T _j , j = 1, . . . , d, on and the harmonic analysis associated with these operators. We define a continuous Dunkl Gabor transform, involving the Dunkl translation operator, by proceeding as mentioned in 20] by C.Wojciech and G. Gigante. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. Then, we show that the portion of the continuous Dunkl Gabor transform lying outside some set of finite measure cannot be arbitrarily too small. Similarly, using the basic theory for the Dunkl continuous wavelet transform introduced by K. Trimèche in 18], an analogous of this result for the Dunkl continuous wavelet transform is given. Finally, an analogous of Heisenberg’s inequality for a continuous Dunkl Gabor transform (resp. Dunkl continuous wavelet transform) is proved.