Dunkl two-wavelet theory and localization operators

Abstract

In this paper the notion of a Dunkl two-wavelet is introduced. The resolution of the identity formula for the Dunkl continuous wavelet transform is then formulated and proved. Calderon’s type reproducing formula in the context of the Dunkl two-wavelet theory is proved. The two-wavelet localization operators in the setting of the Dunkl theory are then defined. The Schatten–von Neumann properties of these localization operators are established, and for trace class localization operators, the traces and the trace class norm inequalities are presented. It is proved that under suitable conditions on the symbols and two Dunkl wavelets, the boundedness and compactness of these localization operators on \(L^{p}_{k}(\mathbb {R}^{d})\), \(1 \le p \le \infty \). Finally typical examples of localization operators are presented.