We consider the problem of estimating the unknown response function in the
multichannel deconvolution model with a boxcar-like kernel which is of
particular interest in signal processing. It is known that, when the number of
channels is finite, the precision of reconstruction of the response function
increases as the number of channels $M$ grow (even when the total number of
observations $n$ for all channels $M$ remains constant) and this requires that
the parameter of the channels form a Badly Approximable $M$-tuple.

We consider the problem of estimating the unknown response function in the
Gaussian white noise model. We first utilize the recently developed Bayesian
maximum a posteriori "testimation" procedure of Abramovich et al. (2007) for
recovering an unknown high-dimensional Gaussian mean vector. The existing
results for its upper error bounds over various sparse $l_p$-balls are extended
to more general cases.

We consider an unknown response function $f$ defined on $\Delta=[0,1]^d$,
$1\le d\le\infty$, taken at $n$ random uniform design points and observed with
Gaussian noise of known variance.

Ramachandran (1969, Theorem 8) has shown that for any univariate infinitely
divisible distribution and any positive real number $\alpha$, an absolute
moment of order $\alpha$ relative to the distribution exists (as a finite
number) if and only if this is so for a certain truncated version of the
corresponding L$\acute{\rm e}$vy measure. A generalized version of this result
in the case of multivariate infinitely divisible distributions, involving the
concept of g-moments, is given by Sato (1999, Theorem 25.3).