Stability of wavelet-like expansions under chromatic aberration
Let \({\mathcal D}\) denote the collection of dyadic cubes in \(\mathbb{R}^d\), let \({\mathcal F}\) be a finite subset of \({\mathcal D}\), and let \(\{\phi^{(Q)}\}_{Q \in {\mathcal D}}\) and \(\{\psi^{(Q)}\}_{Q \in {\mathcal D}}\) be two families of functions, both indexed in \(\mathbb{R}^d\), that satisfy certain uniform decay, cancellation and smoothness conditions that make them ``wavelet--like''. Given a function \(h\), the author defines a certain normalized translation operator \(h_Q\) and, for weights \(w\) in the Muckenhoupt class \(A_p\), studies the \(L^p(w)\) stability of operators of the form \[ T(f) := \sum_{Q \in {\mathcal F}} \langle f, \phi^Q_Q \rangle \psi^Q_Q \] under small perturbations in the arguments of the functions \(\phi^Q\) and \(\psi^Q\).
scientific article; zbMATH DE number 5549415