New uncertainty principles for the $(k,a)$-generalized wavelet transform
DOI:
https://doi.org/10.33044/revuma.2051Abstract
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.
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