Non-linear Approximations Using Sets of Finite Cardinality or Finite Pseudo-dimension

Abstract

We investigate optimal non-linear approximations of multivariate periodic functions with mixed smoothness. In particular, we study optimal approximation using sets of finite cardinality (as measured by the classical entropy number), as well as sets of finite pseudo-dimension (as measured by the non-linear widths introduced by Ratsaby and Maiorov). Approximation error is measured in the Lq(Td)-sense, where Td is the d-dimensional torus. The functions to be approximated are in the unit ball SBrp, θ of the mixed smoothness Besov space or in the unit ball SWrp of the mixed smoothness Sobolev space. For 1<p, q<∞, 0<θ⩽∞ and r>0 satisfying some restrictions, we establish asymptotic orders of these quantities, as well as construct asymptotically optimal approximation algorithms. We particularly prove that for either r>1/p and θ⩾p or r>(1/p−1/q)+ and θ⩾min{q, 2}, the asymptotic orders of these quantities for the Besov class SBrp, θ are both n−r(logn)(d−1)(r+1/2−1/θ).