Greedy Algorithms with Regard to Multivariate Systems with Special Structure

Abstract. The question of finding an optimal dictionary for nonlinear m -term approximation is studied in this paper. We consider this problem in the periodic multivariate (d variables) case for classes of functions with mixed smoothness. We prove that the well-known dictionary Ud which consists of trigonometric polynomials (shifts of the Dirichlet kernels) is nearly optimal among orthonormal dictionaries. Next, it is established that for these classes near-best m -term approximation, with regard to Ud , can be achieved by simple greedy-type (thresholding-type) algorithms. The univariate dictionary U is used to construct a dictionary which is optimal among dictionaries with the tensor product structure.