Nonparametric denoising signals of unknown local structure, II: Nonparametric function recovery

We consider the problem of recovering of continuous multi-dimensional functions f   from the noisy observations over the regular grid m−1Zdm−1Zd, m∈N∗m∈N∗. Our focus is at the adaptive estimation in the case when the function can be well recovered using a linear filter, which can depend on the unknown function itself. In the companion paper, Juditsky and Nemirovski (2009) [26], we have shown in the case when there exists an adapted time-invariant filter, which locally recovers “well” the unknown signal, there is a numerically efficient construction of an adaptive filter which recovers the signals “almost as well”. In the current paper we study the application of the proposed estimation techniques in the function estimation setting. Namely, we propose an adaptive estimation procedure for “locally well-filtered” signals (some typical examples being smooth signals, modulated smooth signals and harmonic functions) and show that the rate of recovery of such signals in the ℓpℓp-norm on the grid is essentially the same as that rate for regular signals with nonhomogeneous smoothness.