Localization operators and inversion formulas for the Dunkl–Weinstein–Stockwell transform

Abstract

Abstract We define and study the Stockwell transform S g \mathscr{S}_{g} associated to the Dunkl–Weinstein operator Δ k , β \Delta_{k,\beta} and prove a Plancherel theorem and an inversion formula. Next, we define a reconstruction function f Δ f_{\Delta} and prove Calderón’s reproducing inversion formula for the Dunkl–Weinstein–Stockwell transform S g \mathscr{S}_{g} . Moreover, we define the localization operators L g ⁢ ( σ ) \mathcal{L}_{g}(\sigma) associated to this transform. We study the boundedness and compactness of these operators and establish a trace formula. Finally, we introduce and study the extremal function F η , k ∗ := ( η ⁢ I + S g ∗ ⁢ S g ) − 1 ⁢ S g ∗ ⁢ ( k ) F^{\ast}_{\eta,\smash{k}}:=(\eta I+\mathscr{S}^{\ast}_{g}\mathscr{S}_{g})^{-1}\mathscr{S}^{\ast}_{g}(k) , and we deduce best approximate inversion formulas for the Dunkl–Weinstein–Stockwell transform S g \mathscr{S}_{g} on the Sobolev space H k , β s ⁢ ( R + d + 1 ) \mathscr{H}^{s}_{k,\beta}(\mathbb{R}_{+}^{d+1}) .