The Calderón reproducing formula, windowed \(X\)-ray transforms, and Radon transforms in \(L^p\)-spaces (Q1271484)

In this paper the author studies the Calderón reproducing formula for different wavelet type transforms on \(L^p\)-spaces. For admissible measures \(\mu\), i.e., measures such that \[ k(x):={1\over {|\Sigma_{n-1}||x|^n}} \int_{|y|<|x|}d\mu(y)\in L^1, \] the reproducing formula takes the following form: \[ \lim_{\varepsilon\rightarrow 0,\rho\rightarrow \infty}\int_{SO(n)} d\gamma=\int_{\varepsilon}^{\rho}{dt\over t} \int_{R^n} f(\cdot -t\gamma y) d\mu(\gamma)=\Biggl(\int_{R^n} k(x) dx\Biggr) f(\cdot). \] For functions in \(L^p\), the convergence is obtained in both \(L^p\)-norm sense and pointwise almost everywhere sense; for \(f\in C_0\) (continuous and vanishing at \(\infty\)) the convergence is shown in sup-norm. Next the convergence conditions are particularized for windowed \(X\)-ray, Radon and \(k\)-plane transforms, respectively.