Abstract
We study approximation of linear functionals on separable Banach spaces equipped with a Gaussian measure. We study optimal information and optimal algorithms in average case, probabilistic, and asymptotic settings, for a general error criterion. We prove that adaptive information is not more powerful than nonadaptive information and that μ-spline algorithms, which are linear, are optimal in all three settings. Some of these results hold for approximation of linear operators. We specialize our results to the space of functions with continuous rth derivatives, equipped with a Wiener measure. In particular, we show that the natural splines of degree 2r + I yield the optimal algorithms. We apply the general results to the problem of integration.
Original language | English |
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Pages (from-to) | 12-43 |
Number of pages | 32 |
Journal | Journal of Complexity |
Volume | 2 |
Issue number | 1 |
DOIs | |
State | Published - Mar 1986 |
Bibliographical note
Funding Information:*This research was supported in part by the National Science Foundation under Grant DCR-82-14322. ‘Current address: AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, N. J. 07974.
ASJC Scopus subject areas
- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- General Mathematics
- Control and Optimization
- Applied Mathematics