  We consider the problem of pointwise estimation of multi-dimensional signals
$s$, from noisy observations $(y_\tau)$ on the regular grid $\bZd$. Our focus
is on the adaptive estimation in the case when the signal can be well recovered
using a (hypothetical) linear filter, which can depend on the unknown signal
itself.
  The basic setting of the problem we address here can be summarized as
follows: suppose that the signal $s$ is "well-filtered", i.e. there exists an
adapted time-invariant linear filter $q^*_T$ with the coefficients which vanish
outside the "cube" $\{0,..., T\}^d$ which recovers $s_0$ from observations with
small 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 the
following questions: -- is it possible to construct an adaptive estimator of
the value $s_0$, which relies upon observations and recovers $s_0$ with
basically the same estimation error as the unknown filter $q^*_T$? -- how rich
is the family of well-filtered (in the above sense) signals?
  We show that the answer to the first question is affirmative and provide a
numerically efficient construction of a nonlinear adaptive filter. Further, we
establish a simple calculus of "well-filtered" signals, and show that their
family is quite large: it contains, for instance, sampled smooth signals,
sampled modulated smooth signals and sampled harmonic functions.
