On Spaces of Type $H_\mu$ and Their Hankel Transformations

Source abstract

The topological and algebraic properties of the spaces of type $H_\mu$, that is, $H_{\mu ,\alpha }$, $H_\mu ^\beta$, and $H_{\mu ,\alpha }^\beta$, are investigated in this paper,where $\mu$ is any real number, $\alpha$ and $\beta$ are nonnegative real numbers. The conventional Hankel transformation $h_\mu$ for $\mu \geqq - 1/2$ is a continuous linear mapping from each of the spaces of type $H_\mu$ into certain other spaces of type $H_\mu$. This assertion is extended to any real number $\mu$ and to the generalized Hankel transformation $h'_\mu$. The nontriviality of the spaces of type $H_\mu$, the relation of certain entire functions with a space of type $H_\mu$, and the relations between the spaces of type S and type $H_\mu$ are proved in the Appendices.