New inversion formulas for the Hankel transformations (Q1365595)

The authors consider the Hankel transform written in the form: \[ h_\nu (f)(y)= \int^\infty_0 x^\nu C_\nu (xy)f(x)dx \tag{*} \] where \(C_\nu(z) =z^{-\nu/2} J_\nu (2\sqrt z)\), in which \(J_\nu\) denotes as usual the Bessel function of the first kind and order \(\nu\). Here \(C_\nu\) is the so-called Bessel-Clifford function extensively studied by the reviewer [see Collect. Math. 18 (1966-1967) Barcelona, 55-174; 67; Zbl 0153.40301)]. The Hankel transformation defined by (*) was introduced and classically investigated by the reviewer [Sobre la transformación de Hankel, Actas VIII Reunión Anual Mat. Espan. Santiago Compost. 47-60 (1969; Zbl 0211.14102)] and later studied by \textit{J. M. R. Méndez} [La transformación integral de Hankel-Clifford, Ph.D. Thesis, Secret. Publ. Univers. La Laguna, La Laguna (1971); Col. Monog. No. 8 (1981)], \textit{J. M. R. Méndez} and \textit{M. M. Socas} [J. Math. Anal. Appl. 154, No. 2, 543-557 (1991; Zbl 0746.46031)] and by \textit{J. J. Betancor} [Port. Math. 46, No. 3, 229-243 (1989; Zbl 0736.46035)] in certain distribution spaces. Making use of the operator \[ H^\nu_{k,t} [F]= \int^\infty_0 {y^\nu\over \Gamma(\nu+1)} {_1 F_1} \left(k+1; \nu+1;- { yt\over k} \right) F(y)dy \] \((k\in\mathbb{N}\), \(0\notin\mathbb{N}\), \(t\in (0,\infty)\); \(\nu\geq -1/2)\), the authors in this paper obtain a new inversion formula for (*) on certain weighted \(L_p\)-spaces proving that the following formula \(\lim_{k\to\infty} H^\nu_{k,t} [h_\nu f]= f(t)\) \((f \in L_p)\) holds. The operator \(H^\nu_{k,t}\) plays in the above Hankel transformation a role similar to the one played by the related inversion operator \(F_{k,t}\) for the Fourier transformation [see \textit{P. G. Rooney}, Canadian Math. Bull. 3, 157-165 (1960; Zbl 0097.31102)] and also by the well-known Post-Widder operator for the Laplace transformation. By means of \(H^\nu_{k,t}\), the range of (*) is also characterized on the weighted \(L_p\)-spaces.