Nonlinear approximation with adaptive dictionaries

It is well known that the study of the Kolmogorov widths of a function class, which is the image of the unit ball of the $L_q$ space of an integral operator $J_K$ with the kernel $K$, is closely connected with the study of sparse approximations of the kernel $K$ with respect to the classical bilinear dictionary. Recently, it was discovered that if instead of the Kolmogorov widths we study the errors of optimal linear sampling recovery of the same classes, then we need to study sparse approximations of the kernel $K$ with respect to an adaptive dictionary, which is determined by the kernel $K$. In this paper we study this important problem of nonlinear approximation with respect to an adaptive dictionary. Also, in this paper we continue to develop the following general approach, which is related to the above nonlinear approximation problem. We study asymptotic behavior of the errors of sampling recovery not for an individual smoothness class, how it is usually done, but for the collection of classes, which are defined by integral operators with kernels coming from a given class of functions. Earlier, such approach was realized for the Kolmogorov widths and very recently for the entropy numbers.