We study approximation of multivariate functions defined over Rd. We assume that all rth order partial derivatives of the functions considered are continuous and uniformly bounded. Approximation algorithms U(f) only use the values of f or its partial derivatives up to order r. We want to recover the function f with small error measured in a weighted Lq norm with a weight function �. We study the worst case (information) complexity which is equal to the minimal number of function and derivative evaluations needed to obtain error �. We provide necessary and sufficient conditions in terms of the weight � and the parameters q and r for the weighted approximation problem to have finite complexity. We also provide conditions guaranteeing that the complexity is of the same order as the complexity of the classical approximation problem over a finite domain. Since the complexity of the weighted integration problem is equivalent to the complexity of the weighted approximation problem with q=1, the results of this paper also hold for weighted integration. This paper is a continuation of 7], where weighted approximation over R was studied.
[1]
H. Woxniakowski.
Information-Based Complexity
,
1988
.
[2]
Francisco Curbera,et al.
Optimal Integration of Lipschitz Functions with a Gaussian Weight
,
1998,
J. Complex..
[3]
Grzegorz W. Wasilkowski,et al.
Complexity of Weighted Approximation over R
,
2000
.
[4]
Philip Rabinowitz,et al.
Methods of Numerical Integration
,
1985
.
[5]
L. Schumaker.
Spline Functions: Basic Theory
,
1981
.
[6]
E. Novak.
Deterministic and Stochastic Error Bounds in Numerical Analysis
,
1988
.
[7]
Peter Mathé.
Asymptotically Optimal Weighted Numerical Integration
,
1998,
J. Complex..
[8]
Henryk Wozniakowski,et al.
Weighted Tensor Product Algorithms for Linear Multivariate Problems
,
1999,
J. Complex..
[9]
Henryk Wozniakowski,et al.
Explicit Cost Bounds of Algorithms for Multivariate Tensor Product Problems
,
1995,
J. Complex..
[10]
A. Pinkus.
n-Widths in Approximation Theory
,
1985
.