Robust and efficient recovery of a signal passed through a filter and then contaminated by non-Gaussian noise

Consider a channel where a continuous periodic input signal is passed through a linear filter and then is contaminated by an additive noise, The problem is to recover this signal when we observe n repeated realizations of the output signal, Adaptive efficient procedures, that are asymptotically minimax ol-er all possible procedures, are known for the channels with Gaussian noise and no filter (the case of direct observation), Efficient procedures, based on smoothness of a recovered signal, are known for the case of Gaussian noise, Robust rate-optimal procedures are known as well. However, there is no results on robust and efficient data-driven procedures; moreover, the known results for the case of direct observation indicate that even a smalt deviation from Gaussian noise may lead to a drastic change, We show that for the considered case of indirect data and a particular class of so-called supersmooth filters there exists a procedure of recovery of an input signal that possesses the desired properties; namely, it is: adaptive to smoothness of input signal; robust to the distribution of a noise; globally and pointwise-efficient, that is, its minimax global and pointwise risks converge with the best constant and rate over all possible estimators as n --> infinity; universal in the sense that for a wide class of linear (not necessarily bounded) operators the efficient estimator is a plug-in one. Furthermore, we explain how to employ the obtained asymptotic results for the practically important case of small n (large noise).