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.