Stability of Haar decompositions

TL;DR

This paper proves the $L^2$ stability of Haar decompositions under small affine distortions and extends results on frames of smooth functions close to Haar functions, with applications to dyadic averages.

Contribution

It establishes a general $L^2$ stability result for Haar decompositions under affine distortions and generalizes previous work on frames of smooth functions near Haar functions.

Findings

Haar decompositions are stable under small affine distortions in $L^2$.

Constructs frames of smooth functions close to Haar functions in multiple dimensions.

Provides optimal estimates on the sensitivity of dyadic averages to local affine changes.

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.