Stable Summation of Orthogonal Series with Noisy Coefficients

Abstract

We study the recovery of continuous functions from Fourier coefficients with respect to certain given orthonormal systems, blurred by noise. For deterministic noise this is a classical ill-posed problem. Emphasis is laid on a priori smoothness assumptions on the solution, which allows to apply regularization to reach the best possible accuracy. Results are obtained for systems obeying norm growth conditions. In the white noise setting mild additional assumptions have to be made to have accurate bounds. We finish our study with the recovery of functions from noisy coefficients with respect to the Haar system.