(k,a)-generalized wavelet transform and applications - Mejjaoli, Hatem (Journal of pseudo-differential operators and applications, 2020)
We introduce the notion of the ( k , a )-generalized wavelet transform. Particular cases of such generalized wavelet transform are the classical and the Dunkl wavelet transforms. The restriction of the ( k , a )-generalized wavelet transform to radial functions is given by the generalized Hankel wavelet transform. We prove for this new transform Plancherel’s formula, inversion theorem and a Calderón reproducing formula. As applications on the ( k , a )-generalized wavelet transform, we give some applications of the theory of reproducing kernels to the Tikhonov regularization on the generalized Sobolev spaces. Next, we study the generalized wavelet localization operators.