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We define and study the Stockwell transformSg\mathscr{S}_{g}associated to the Dunkl–Weinstein operatorΔk,β\Delta_{k,\beta}and prove a Plancherel theorem and an inversion formula. Next, we define a reconstruction functionfΔf_{\Delta}and prove Calderón’s reproducing inversion formula for the Dunkl–Weinstein–Stockwell transformSg\mathscr{S}_{g}. Moreover, we define the localization operatorsLg⁢(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 functionFη,k∗:=(η⁢I+Sg∗⁢Sg)−1⁢Sg∗⁢(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 transformSg\mathscr{S}_{g}on the Sobolev spaceHk,βs⁢(R+d+1)\mathscr{H}^{s}_{k,\beta}(\mathbb{R}_{+}^{d+1}).

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