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Malaya Journal of Matematik

Uncertainty principles for the continuous wavelet transform associated with a Bessel type operator on the half line

Cyrine Baccar, Aicha Kabache

Published Jul 1, 2024• 1 citation
Publisher

Abstract

This paper presents uncertainty principles pertaining to generalized wavelet transforms associated with a second-order differential operator on the half line, extending the concept of the Bessel operator. Specifically, we derive a Heisenberg-Pauli-Weyl type uncertainty principle, as well as other uncertainty relations involving sets of finite measure

DOI
10.26637/mjm1203/007

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This paper presents uncertainty principles pertaining to generalized wavelet transforms associated with a second-order differential operator on the half line, extending the concept of the Bessel operator. Specifically, we derive a Heisenberg-Pauli-Weyl type uncertainty principle, as well as other uncertainty relations involving sets of finite measure
The deformed wavelet transform is a new addition to the class of wavelet transforms that heavily rely on a pair of generalized translation and dilation operators governed by the well-known Dunkl transform. In this study, we adapt the symmetrical properties of the Dunkl Laplacian operator to prove a class of quantitative uncertainty principles associated with the deformed wavelet transform, including Heisenberg’s uncertainty principle, the Benedick–Amrein–Berthier uncertainty principle, and the logarithmic uncertainty inequalities. Moreover, using the symmetry between a square integrable function and its Dunkl transform, we establish certain local-type uncertainty principles involving the mean dispersion theorems for the deformed wavelet transform.