The Calderón Reproducing Formula, Windowed X-Ray Transforms, and Radon Transforms in L^p-Spaces

The generalized Calderón reproducing formula involving "wavelet measure" is established for functions f ∈ L^p (ℝⁿ). The special choice of the wavelet measure in the reproducing formula gives rise to the continuous decomposition of f into wavelets, and enables one to obtain inversion formulae for generalized windowed X-ray transforms, the Radon transform, and k-plane transforms. The admissibility conditions for the wavelet measure μ are presented in terms of μ itself and in terms of the Fourier transform of μ.