Abstract
We give an example of a Hilbert space embedding \(H \subset {\mathcal{l}}_{p}\), \(1 \leq p < \infty \), whose approximation numbers tend to zero much faster than its sampling numbers. The main result shows that optimal algorithms for approximation that use only function evaluation can be more than polynomially worse than algorithms using general linear information.
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Notes
- 1.
We use the Hölder inequality in the form \(\|{({x}_{n}{y}_{n})\|}_{p} \leq \| {({x}_{n})\|}_{q}\|{({y}_{n})\|}_{r}\), where \(1/p = 1/q + 1/r\). In our case we have q = 2 and \(r = \frac{1} {1/p-1/2}\).
References
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F. Y. Kuo, G. W. Wasilkowski, H. Woźniakowski: On the power of standard information for multivariate approximation in the worst case setting, J. Approx. Theory 158 (2009), pp. 97–125.
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G. W. Wasilkowski, H. Woźniakowski: On the power of standard information for weighted approximation, Found. Comput. Math. 1 (2001), pp. 417–434.
Acknowledgements
I want to thank Erich Novak and Aicke Hinrichs for valuable discussions and ideas which lead to the results in this paper.
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Tandetzky, R. (2012). Approximation of Functions from a Hilbert Space Using Function Values or General Linear Information. In: Plaskota, L., Woźniakowski, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2010. Springer Proceedings in Mathematics & Statistics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27440-4_38
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DOI: https://doi.org/10.1007/978-3-642-27440-4_38
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