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.
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