Concentration Uncertainty Principles for the Dunkl-Bessel Transform and Dunkl-Bessel Wavelet Transform

This paper delves into the Dunkl-Bessel operator on \({\mathbb {R}}^{d+1}_+\) ${\mathbb{R}}_{+}^{d+1}$ and its corresponding harmonic analysis. We prove an analogous of a time-frequency localization theorem for orthonormal sequences in \(L^2_{k,\beta }({\mathbb {R}}_+^{d+1})\) $L_{k,\beta }^2({\mathbb{R}}_{+}^{d+1})$ and a strong version of uncertainty inequality for orthonormal bases of \(L^2_{k,\beta }({\mathbb {R}}_+^{d+1})\) $L_{k,\beta }^2({\mathbb{R}}_{+}^{d+1})$ . As consequence we obtain a quantitative version of Shapiro’s uncertainty principle on the time-frequency concentration of orthonormal sequences. Then, we prove for the Dunkl-Bessel wavelet transform, a sharp quantitative form of Shapiro’s mean dispersion theorem with generalized dispersion. Finally, we show the Umbrella theorem for this transform.