The Hankel-Clifford transformation on certain spaces of ultradistributions
This work deals with the generalized Hankel-Clifford transformation on certain spaces of ultradistributions. Following mainly the works of \textit{J. M. Méndez} [La transformation integral de Hankel-Clifford, Secretariado de Publicaciones de la Universidad de La Laguna, Coleccion Monografias \(n^ 08\) (1981)] and \textit{J. M. González} [Ph. D. Thesis, Faculty of Mathematics, University of La Laguna, (1985)] the author defines testing function spaces \({}^ pS^{\beta}_{\mu,\alpha}\) and he calls the elements of its dual as ultradistributions. Méndez called a integral transform given by the following pair \[ F(y)=h_{\mu}\{f(x)\}(y)=y^{\mu}\int^{\infty}_{0}C_{\mu}(xy)f(x)d x \] \[ f(x)=h_{\mu}\{F(y)\}(x)=x^{\mu}\int^{\infty}_{0}C_{\mu}(xy)f(y)d y \] as the Hankel-Clifford transformation. The kernel of this transform \(C_{\mu}\) is the Bessel Clifford function of the first kind and order \(\mu\). The author maintains that the Hankel-Clifford transform \(h_{\mu}\) (\(\mu\geq 0)\) is a continuous linear operator in spaces of \({}^ pS^{\beta}_{\mu,\alpha}\). The generalized Hankel-Clifford transform \(h'_{\mu}\) is then defined as a continuous linear mapping between the dual spaces. In the end, the developed theory is applied to find classes of existence of classical and generalized solutions for a Cauchy problem containing the Bessel-type operator \(B_{\mu}=Dx^{\mu +1}Dx^{-\mu}\).
generalized Hankel-Clifford transformation on certain spaces of ultradistributions
Bessel Clifford function of the first kind
classical and generalized solutions for a Cauchy problem containing the Bessel-type operator
scientific article; zbMATH DE number 4131034