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

Abstract

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 λ∈[0,∞) and 0≤α<∞, let ϕ(x,λ) be a solution of the Sturm-Liouville equation $$\frac{{d^2 y}}{{dx^2 }} - q(x)y = - \lambda y, y(0) = \sin \alpha , y'(0) = - \cos \alpha , \leqq x < \infty .$$ We define a test-function space ∈ A such that for each λ∈[0,∞), ϕ(.,λ)∈ A and hence for f∈ A*, we define the ϕ-transform of f by F(λ)= 〈f(x),ϕ(x,λ)〉. This paper studies properties of the ϕ-transform of f, in particular its inversion formula.