Stability of wavelet-like expansions under chromatic aberration

We prove that wavelet and wavelet-like expansions of functions are Lp-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 Lp distance between the “true” and “perturbed” output functions is bounded by a constant times ητ‖f‖p, where τ is a positive number depending on the family of kernels in question. We show that this result also holds in Lp(w) if w is a Muckenhoupt Ap weight.