A constructive approach to generalized integral transforms (Q1815449)

In this interesting work, the authors show that distributional versions of several classical integral transforms can be related (and developed) to functions of a particular normal operator which is defined by means of a spectral integral. By suitably defining, the test function space and corresponding space of generalized functions, a particular normal operator \({\mathcal D}\) is considered and its important spectral properties discussed by invoking the Mellin transforms. A large number of operators defined in the Hilbert space \(L^2_\mu= \{\phi: x^{-\mu}\phi(x)\in L^2(0,\infty)\}\), \(\mu\in\mathbb{R}\), are shown to be expressible as functions of \({\mathcal D}\). As applications, some useful examples involving logarithmic fractional integrals, Erdélyi-Kober fractional integrals, radially symmetric Riesz potentials, the semi-infinite Hilbert transform and Hankel transform are discussed.