Abstract

We report on solving the inverse problem of finding the kernel of an asymptotic singular integral operator under which monomial signals that are compactly supported over the closed unit interval [0, 1^–] are asymptotic generalized fixed points over the semi-closed unit interval (0, 1]. This operator defines a certain Even-Hilbert Riemann–Lebesgue transformation with a kernel that is double parameterized over a certain momentum Hilbert space. The inversion of this singular transformation is proved to be in the form of an associated Odd-Hilbert Riemann–Lebesgue transformation. The paper contains also proofs for a number of operational properties of this transform, with an identified area for potential applicability in solving certain functional initial-value problems.