Average case \(L_\infty\)-approximation in the presence of Gaussian noise
The paper considers the average case setting, where both the function and the noise are assumed to be random. The problem is to approximate a real function \(f\in F^r\), where \[ F^r=\{f\in C^r ([0,1]):f^{(j)}(0)=0,\quad 0 \leq j\leq r\} \] and where \(f\) is a realization of the stochastic process corresponding to the \(r\)-fold Wiener measure \(w^r\) on \(F^r,r\geq 0\). The approximation is given as a function \(\varphi(y):[0,1]\rightarrow {\mathbb R}\) with \(y=N(f)+ x\) and \(N(f)=(f(t_1),\ldots , f(t_n))\). Let \({\mathcal N}_n\) be the set of all vectors \(N\) consisting of at most \(n\) functions. Further, let \(x\) have the zero mean \(n\)-dimensional normal distribution with covariance matrix \(\sigma^2I_n\), \(x \sim\Pi_\sigma^n\). For the minimal average \(L_p\)-error \[ e_\infty^p(r,\sigma;n):=\inf_{N\in {\mathcal N}_n}\inf_{\varphi} \Biggl(\int_{F^r}\int_{{\mathbb R}^n}(\| f-\varphi (N(f)+x)\| _\infty)^p\Pi_\sigma^n(dx) w^r (df)\Biggr)^{1/p} \] it is shown that \[ e_\infty^p(r, \sigma;n) \sim n^{-(r +1/2)}\ln^{1/2} n +\left({\sigma\over \sqrt n}\right)^{(2 r +1)/(2r + 2)} \ln^{1/2} n. \] The optimal \(L^\infty\)-approximation can be obtained by piecewise polynomial interpolation and by smoothing spline approximation.
scientific article; zbMATH DE number 1197069