## Abstract

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.

Original language | English |
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Pages (from-to) | 57-77 |

Number of pages | 21 |

Journal | Journal of Approximation Theory |

Volume | 130 |

Issue number | 1 |

DOIs | |

State | Published - Sep 2004 |

### Bibliographical note

Funding 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: greg@cs.uky.edu (G.W. Wasilkowski).

## Keywords

- Approximation
- Integration
- Multivariate problems
- Tractability

## ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Mathematics (all)
- Applied Mathematics