Average CaseL∞-Approximation in the Presence of Gaussian Noise |
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Authors: | Leszek Plaskota |
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Institution: | Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Polandf1 |
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Abstract: | We consider the average caseL∞-approximation of functions fromCr(0, 1]) with respect to ther-fold Wiener measure. An approximation is based onnfunction evaluations in the presence of Gaussian noise with varianceσ2>0. We show that the n th minimal average error is of ordern−(2r+1)/(4r+4) ln1/2 n, and that it can be attained either by the piecewise polynomial approximation using repetitive observations, or by the smoothing spline approximation using non-repetitive observations. This completes the already known results forLq-approximation withq<∞ andσ0, and forL∞-approximation withσ=0. |
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