Wavelet-Like Transforms for Admissible Semi-Groups; Inversion Formulas for Potentials and Radon Transforms

Abstract

We introduce a family of wavelet-like transforms associated to certain admissible semigroups of operators acting on L_p-spaces, and prove the corresponding reproducing formula of Calderòn’s type. The new transforms constitute a unified approach to inversion of a wide class of integral operators in analysis and applications. We illustrate the general theory by considering Riesz and Bessel potentials (associated to the ordinary and the generalized translation), and the k-plane Radon transform on ℝⁿ.