The best [formula omitted]-term approximations on generalized Besov classes [formula omitted] with regard to orthogonal dictionaries

Abstract

In this paper, we investigate nonlinear m -term approximation with regard to orthogonal dictionaries. We consider this problem in the periodic multivariate case for generalized Besov classes M B q , θ Ω under the condition Ω ( t ) = ω ( t 1 ⋅ ⋯ ⋅ t d ) where ω ( t ) ∈ Ψ l ∗ is a univariate function. We prove that the well-known dictionary U d which consists of trigonometric polynomials (shifts of the Dirichlet kernels) is nearly optimal among orthogonal dictionaries. Moreover, it is established that for these classes near-best m -term approximation, with regard to U d , can be achieved by simple greedy-type algorithms.