An inversion theorem for integral transforms related to singular Sturm-Liouville problems on the half line

We extend the classical theory of singular Sturm-Liouville boundary value problems on the half line, as developed by Titshmarsh and Levitan to generalized functions in order to obtain a general approach to handle many integral transforms, such as the sine, cosine, Weber, Hankel, and the K-transforms, in a unified way. This approach will lead to an inversion formula that holds in the sense of generalized functions. More precisely, for [0,) and 0<, let (x,) be a solution of the Sturm-Liouville equation We define a test-function space A such that for each [0,), (.,) A and hence for f A*, we define the -transform of f by F()= f(x),(x,). This paper studies properties of the -transform of f, in particular its inversion formula.