Continous linear canonical Dunkl wavelet transform: properties and applications
Ahmed Saoudi, Imen Kallel
TL;DR
This work builds a generalized time–frequency analysis framework in the linear canonical Dunkl setting by introducing the linear canonical Dunkl translation and convolution, enabling the continuous linear canonical Dunkl wavelet transform (LC-Dunkl wavelet transform). It develops admissibility, defines a family of wavelets $\\psi_{t,x}^{M}$ tied to the transform, and establishes core properties including a Plancherel-type formula, reconstruction, and a reproducing kernel, ensuring a stable inversion. The paper also derives uncertainty inequalities for the LC-Dunkl wavelet transform, demonstrating localization and energy distribution results. Overall, the approach unifies and extends several wavelet transform families within Dunkl analysis, providing a versatile tool for signal analysis in linear canonical and Dunkl contexts with potential applications to localization operators and further theoretical developments.
Abstract
The aim of this paper is to establish and study the linear canonical Dunkl wavelet transform. We begin by introducing the generalized translation operator and generalized convolution product for the linear canonical Dunkl transform and we establish their basic properties. Next, we introduce the new proposed wavelet transform and we investigate its fundamentals properties. In the end, we derive some uncertainty inequalities for the desired wavelet transform as applications.