A Constructive Approach to Generalized Integral Transforms

Results of recent work on distributional extensions of normal operators are used to show that distributional versions of a large number of classical integral transforms can be developed in a systematic way by noting that they can be regarded as functions of a particular normal operator. As examples, we consider logarithmic fractional integrals, Erd´elyi]Kober operators, radially symmetric Riesz potentials, the semi-infinite Hilbert transform, and the Hankel transform. We show how particular spaces of test and generalized functions which have been used for developing distributional versions of various integral transforms arise in a natural way.