The Complexity Of Function Approximation On Sobolev Spaces With Common Mixed Smoothness By Linear Monte Carlo Methods

In this paper,the information-based complexity of the approximation problem on the class WpA(Td)of functions with common mixed smoothness in the Lq(Td)-metric has been studied. And by applying V. E. Maiorov’s discretization and some properties of pseudo-s-scale,the exact asymptotic order of n-linear Monte Carlo approximation error is determined for1<p,q<∞.NamelyLet1<p,q<∞,A is a finite subset of Rd,and0∈int N(A(μ)).Then where A(ε):=A-ε1,ε≥0,μ:=max{0,1/p-1/q,1/p-1/2},and v satisfy the definition of(1.3).