Paley–Wiener and Boas theorems for singular Sturm–Liouville integral transforms

Abstract

This paper deals with a class of integral transforms arising from a singular Sturm–Liouville problem y″−q(x)y=−λy, x∈(a,b), in the limit-point case at one end or both ends of the interval ( a, b). The paper completely solves the problem of characterization of the image of a function that has compact support (Paley–Wiener theorem) and also of a function that vanishes on some interval (Boas problem) under this class of transforms. The characterizations are obtained with no restriction on q( x) other than being locally integrable.