Stability of wavelet-like expansions under chromatic aberration

We prove that wavelet and wavelet-like expansions of functions are L p -stable under small (but otherwise arbitrary and independent) errors in translation and dilation of the constituent reproducing kernels. These perturbations are frequency-dependent, which is why we call them “chromatic aberration.” We show that, if these errors have sizes no bigger than η, then the L p distance between the “true” and “perturbed” output functions is bounded by a constant times η τ ‖ f ‖ p , where τ is a positive number depending on the family of kernels in question. We show that this result also holds in L p ( w ) if w is a Muckenhoupt A p weight.