Paley-Wiener-type theorem for a class of integral transforms arising from a singular Dirac system

A characterization of weighted L2(I) spaces in terms of their images under various integral trans-formations is derived, where I is an interval (finite or infinite). This characterization is then used to derive Paley-Wiener- type theorems for these spaces. Unlike the classical Paley-Wiener theorem, our theorems use real variable techniques and do not require analytic continuation to the com-plex plane. The class of integral transformations considered is related to singular Sturm-Liouville boundary-value problems on a half line and on the whole line. 1