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

초록

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 lambda is an element of [0, infinity) and 0 less than or equal to alpha < infinity, let phi(x, lambda) be a solution of the Sturm-Liouville equation d(2)y/dx(2) - q(x)y = -lambday, y(0)=sin alpha, y'(0) = -cos alpha, 0 less than or equal to x < infinity. We define a test-function space A such that for each lambda is an element of [0,infinity), phi((.), lambda) is an element of A and hence for f is an element of A*, we define the phi-transforim of f by F(lambda) = <f(x),phi(x,lambda)>. This paper studies properties of the phi-transform of f, in particular its inversion formula.

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