Summary
We study adaptive information for approximation of linear problems in a separable Hilbert space equipped with a probability measure μ. It is known that adaption does not help in the worst case for linear problems. We prove that adaption also doesnot help on the average. That is, there exists nonadaptive information which is as powerful as adaptive information. This result holds for “orthogonally invariant” measures. We provide necessary and sufficient conditions for a measure to be orthogonally invariant. Examples of orthogonally invariant measures include Gaussian measures and, in the finite dimensional case, weighted Lebesgue measures.
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This research was supported in part by the National Science Foundation under Grant MCS-7823676
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Wasilkowski, G.W., Woźniakowski, H. Can adaption help on the average?. Numer. Math. 44, 169–190 (1984). https://doi.org/10.1007/BF01410103
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DOI: https://doi.org/10.1007/BF01410103