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We consider the problem of recovering of continuous multi-dimensionalfunctions from the noisy observations over the regular grid. Our focus is atthe adaptive estimation in the case when the function can be well recoveredusing a linear filter, which can depend on the unknown function itself. In thecompanion paper "Nonparametric Denoising of Signals with Unknown LocalStructure, I: Oracle Inequalities" we have shown in the case when there existsan adapted time-invariant filter, which locally recovers "well" the unknownsignal, there is a numerically efficient construction of an adaptive filterwhich recovers the signals "almost as well". In the current paper we study theapplication of the proposed estimation techniques in the non-parametricregression 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 ofrecovery of such signals in the $\ell_p$-norm on the grid is essentially thesame as that rate for regular signals with nonhomogeneous smoothness.

Nonparametric denoising Signals of Unknown Local Structure, II: Nonparametric Regression Estimation