  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 show that, for a properly chosen prior on the number
of non-zero entries of the mean vector, the corresponding adaptive estimator is
simultaneously asymptotically minimax in a wide range of sparse and dense
$l_p$-balls.
  The proposed procedure is then applied in a wavelet context to derive
adaptive global and level-wise wavelet estimators of the unknown response
function in the Gaussian white noise model. These estimators are then proven to
be, respectively, asymptotically near-minimax and minimax in a wide range of
Besov balls. These results are also extended to the estimation of derivatives
of the response function.
  Simulated examples are conducted to illustrate the performance of the
proposed level-wise wavelet estimator in finite sample situations, and to
compare it with several existing
