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We consider the problem of pointwise estimation of multi-dimensional signals$s$, from noisy observations $(y_\tau)$ on the regular grid $\bZd$. Our focusis on the adaptive estimation in the case when the signal can be well recoveredusing a (hypothetical) linear filter, which can depend on the unknown signalitself.  The basic setting of the problem we address here can be summarized asfollows: suppose that the signal $s$ is "well-filtered", i.e. there exists anadapted time-invariant linear filter $q^*_T$ with the coefficients which vanishoutside the "cube" $\{0,..., T\}^d$ which recovers $s_0$ from observations withsmall mean-squared error. We suppose that we do not know the filter $q^*$,although, we do know that such a filter exists. We give partial answers to thefollowing questions: -- is it possible to construct an adaptive estimator ofthe value $s_0$, which relies upon observations and recovers $s_0$ withbasically the same estimation error as the unknown filter $q^*_T$? -- how richis the family of well-filtered (in the above sense) signals?  We show that the answer to the first question is affirmative and provide anumerically efficient construction of a nonlinear adaptive filter. Further, weestablish a simple calculus of "well-filtered" signals, and show that theirfamily is quite large: it contains, for instance, sampled smooth signals,sampled modulated smooth signals and sampled harmonic functions.

Nonparametric Denoising of Signals with Unknown Local Structure, I: Oracle Inequalities