How fast and in what sense(s) does the Calderón reproducing formula converge? (Q1958521)

The author investigates questions about the convergence of the Calderón reproducing formula. He considers the functions \[ f_{(A)}(x)=\int_A(f*\psi_y(t))\phi_y(x-t)\frac{dt\,dy}{y}, \] where \(A\subseteq\mathbb{R}^d\times(0,\infty)\) is a measurable set with compact closure and \(f\in L^1_{\text{loc}}(\mathbb{R}^d),\) and proves that \(f_{(A_j)}\rightarrow f\) in \(L^p\) for very general sequences of sets \(\{A_j\},\) for all \(1<p<\infty,\) and for pairs \(\phi,\psi\) that satisfy weak decay and smoothness conditions. In fact, \(f_{(A_j)}\rightarrow f\) in \(L^p(w)\) whenever \(w\) belongs to the Muckenhoupt \(A_p\) class and \(f\in L^p(w).\) The author also obtains information about the speed of convergence. This is a well-written paper with a good introduction to the philosophy of the Calderón reproducing formula.