L. Plaskota

Average case $L_\infty$ approximation in the presence of gaussian noise

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Abstract.   We consider the average case $L_\infty$ approximation of functions from $C^r([0,1])$ with respect to the $r$-fold Wiener measure. An approximaion is based on n function evaluations in the presence of gaussian noise with variance $\sigma^2>0$. We show that the $n$th minimal average error is of order $n^{-(2r+1)/(4r+4)}\ln^{1/2}n$, 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 for $L_q$ approximation with $q<\infty$ and $\sigma>=0$, and for $L_\infty$ approximation with $\sigma=0$.

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