Stability of Haar decompositions

Abstract

We prove a general result implying the $L^2$ stability of Haar decompositions of $L^2({\bf R}^d)$ functions when the Haar functions are distorted by arbitrary, independent, affine changes of variable that are close to the identity. We apply our method to get fully $d$-dimensional generalizations of results of Aimar, Bernardis, Gorosito, Govil, and Zalik, on constructing frames of smooth functions which are, in many natural senses, arbitrarily close to the Haar functions. We also obtain a best-possible estimate on the $L^2$ sensitivity of dyadic averages of functions to small distortions caused by local affine changes of variable.