Reconstruction and best approximate inversion formulas for the modified Whittaker–Stockwell transform

We define and study the Stockwell transform \(\mathscr {S}_g\) ${\mathcal{S}}_g$ associated with the Whittaker operator \(L_{\alpha }:=-\frac{1}{4}\left[ x^2\frac{\text{ d}^2}{\text{ d }x^2}+(x^{-1}+(3-4\alpha )x)\frac{\text{ d }}{\text{ d }x}\right] \) $L_{\alpha }:=-\frac{1}{4}\left[x^2,\frac{{\text{d}}^2}{\text{d}{x}^2},+,(x^{-1}+(3-4\alpha )x),\frac{\text{d}}{\text{d}x}\right]$ , and prove a Plancherel theorem and an inversion formula. We define a reconstruction function \(f_{a,b}\) $f_{a,b}$ , and we prove Calderón’s reproducing inversion formula for the modified Whittaker–Stockwell transform \(\mathscr {S}_g\) ${\mathcal{S}}_g$ . We introduce and study the extremal function \(f^{*}_{\eta ,k}:=(\eta I+\mathscr {S}^{*}_g\mathscr {S}_g)^{-1}\mathscr {S}^{*}_g(k)\) $f_{\eta ,k}^{\ast }:=(\eta I+\mathcal{S}_g^{\ast }{\mathcal{S}}_g)_{}^{-1}\mathcal{S}_g^{\ast }(k)$ , and we deduce best approximate inversion formulas for the modified Whittaker–Stockwell transform \(\mathscr {S}_g\) ${\mathcal{S}}_g$ .