Average case \(L_\infty\)-approximation in the presence of Gaussian noise
type: Item
instance of: scholarly article
title: Average case \(L_\infty\)-approximation in the presence of Gaussian noise
zbMATH Open document ID: 0913.41019
author: Leszek Plaskota
published in: Journal of Approximation Theory
publication date: 1999-05-31T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 1197069
zbMATH Keywords: \(L_\infty\)-approximation
zbMATH Keywords: Wiener measure
zbMATH Keywords: Gaussian noise
zbMATH Keywords: minimal average error
reviewed by: Gerlind Plonka
MaRDI profile type: Publication
full work available at URL: jath.1998.3175
OpenAlex ID: W2092016490
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DOI: 10.1006/JATH.1998.3175
description: scientific article; zbMATH DE number 1197069