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Average CaseL∞-Approximation in the Presence of Gaussian Noise

✍ Scribed by Leszek Plaskota

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
266 KB
Volume
93
Category
Article
ISSN
0021-9045

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✦ Synopsis

We consider the average case L -approximation of functions from C r ([0, 1]) with respect to the r-fold Wiener measure. An approximation is based on n function evaluations in the presence of Gaussian noise with variance _ 2 >0. We show that the n th minimal average error is of order n &(2r+1)Â(4r+4) ln 1Â2 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 for L q -approximation with q< and _ 0, and for L -approximation with _=0.

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Stabilization of linear systems in state space in the presence of parametric perturbations is considered. The perturbed system is represented by a matrix differential equation with the elements of the matrices given by Gaussian processes with known mean and covariance. Using methods from stochastic