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

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