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.
© de Gruyter 2011