$$(\varvec{k},\varvec{a})$$-generalized wavelet transform and applications

Abstract

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 Calderon 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.