We give a survey of some resent results on the minimax estimation and detection of a multivariate function in the white Gaussian noise model. We study the "curse of dimensionality" phenomenon for a function belonging to a ball in various functional spaces: Sobolev spaces, tensor product Sobolev spaces, and spaces of analytic functions. Typically, the rates of the quadratic risk in the estimation problem and the separation rates in the detection problem become catastrophically bad when the number of variables is larger then log(1/ε), where ε is the noise intensity.
We show that the curse of dimensionality is "lifted" for the balls in anisotropic Sobolev spaces and in weighted tensor product spaces. The spaces of the last type were introduced by Sloan and Woznjakovski (1998) in the context of numerical integration problem.
The methods are based on some new probabilistic tools for studying approximation characteristics of the balls in the spaces under consideration.