New uncertainty principles for the $(k,a)$-generalized wavelet transform

Abstract

We present the basic (k, a)-generalized wavelet theory and prove several Heisenberg-type inequalities for this transform. After reviewing Pitt's and Beckner's inequalities for the (k, a)-generalized Fourier transform, we connect both inequalities to show a generalization of uncertainty principles for the (k, a)-generalized wavelet transform. We also present two concentration uncertainty principles, namely the Benedicks-Amrein-Berthier's uncertainty principle and local uncertainty principles. Finally, we connect these inequalities to show a generalization of the uncertainty principle of Heisenberg type and we prove the Faris-Price uncertainty principle for the (k, a)-generalized wavelet transform.

Topics & Concepts