We study the minimal number n (ε, d) of information evaluations needed to compute a worst case ε -approximation of a linear multivariate problem. This problem is defined over a weighted Hilbert space of functions f of d variables. One information evaluation of f is defined as the evaluation of a linear continuous functional or the value of f at a given point. Tractability means that n (ε, d) is bounded by a polynomial in both ε-1 and d. Strong Tractability means that n (ε, d) is bounded by a polynomial only in ε-1. We consider weighted reproducing kernel Hilbert spaces with finite-order weights. This means that each function of d variables is a sum of functions depending only on q* variables, where q* is independent of d. We prove that finite-order weights imply strong tractability or tractability of linear multivariate problems, depending on a certain condition on the reproducing kernel of the space. The proof is not constructive if one uses values of f.
|Number of pages||21|
|Journal||Journal of Approximation Theory|
|State||Published - Sep 2004|
Bibliographical noteFunding Information:
G.W. Wasilkowski and H. Woźniakowski were partially supported by the National Science Foundation under Grants CCR-0095709 and DMS-0308713, respectively. ∗Corresponding author. Fax: +1-859-323-1971. E-mail address: email@example.com (G.W. Wasilkowski).
- Multivariate problems
ASJC Scopus subject areas
- Numerical Analysis
- Mathematics (all)
- Applied Mathematics