  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.
