Efficient estimation and detection of functions from anisotropic Sobolev classes (jointly with Yu. I. Ingster)

We consider the problems of estimating and detecting an unknown function $f$ depending on a multidimensional variable (for instance, an image) observed in the Gaussian white noise. It is assumed that $f$ belongs to anisotropic Sobolev class. The case of a function of infinitely many variables is also considered. An asymptotic study (as the noise level tends to zero) of the estimation and detection problems is done. In connection with estimating and detecting unknown signal, the problems of rate and sharp optimality (minimaxity) are investigated. In the case when $d$ is fixed and finite the sharp optimality problems are solved. In the infinite-dimensional case log-asymptotics of the risk in the estimation problem and of the separation rate in the detection problem are obtained.