Time–Frequency Concentration, Heisenberg Type Uncertainty Principles and Localization Operators for the Continuous Dunkl Wavelet Transform on $$\mathbb {R}^{d}$$ R d

Abstract

We consider the continuous Dunkl wavelet transform $$\Phi ^{D}_{h} $$ associated with the Dunkl operators on $$\mathbb {R}^{d}$$ . We analyze the concentration of this transform on sets of finite measure. In particular, Donoho–Stark and Benedicks type uncertainty principles are given. Next, we prove many versions of Heisenberg type uncertainty principles for $$\Phi ^{D}_{h} $$ . Quantitative Shapiro’s dispersion uncertainty principle and umbrella theorem are proved for the Dunkl continuous wavelet transform. Finally, we investigate the localization operators for $$\Phi ^{D}_{h} $$ , in particular we prove that they are in the Schatten-von Neumann class.