Abstract
We study weighted approximation and integration of Gaussian stochastic processes X defined over ℝ+ whose rth derivatives satisfy a Hölder condition with exponent β in the quadratic mean. We assume that the algorithms use samples of X at a finite number of points. We study the average case (information) complexity, i.e., the minimal number of samples that are sufficient to approximate/integrate X with the expected error not exceeding E. We provide sufficient conditions in terms of the weight and the parameters r and β for the weighted approximation and weighted integration problems to have finite complexity. For approximation, these conditions are necessary as well. We also provide sufficient conditions for these complexities to be proportional to the complexities of the corresponding problems defined over [0, 1], i.e., proportional to E-1/α where α = r+β for the approximation and α=r+β+1/2 for the integration.
| Original language | English |
|---|---|
| Pages (from-to) | 517-544 |
| Number of pages | 28 |
| Journal | Journal of Complexity |
| Volume | 18 |
| Issue number | 2 |
| DOIs | |
| State | Published - 2002 |
Bibliographical note
Funding Information:We are grateful to S. Kwapień for his remarks concerning (17). We thank an anonymous referee for detailed remarks and suggestions. The first author was partially supported by the State Committee for Scientific Research of Poland (KBN). The second author was supported in part by the Alexander von Humboldt Foundation. The third author was partially supported by the National Science Foundation under Grant CCR-0095709.
Keywords
- Average error
- Complexity
- Gaussian process
- Weighted approximation
- Weighted integration
ASJC Scopus subject areas
- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- General Mathematics
- Control and Optimization
- Applied Mathematics