On the ``prediction'' problem
type: Item
instance of: scholarly article
title: On the ``prediction'' problem
zbMATH Open document ID: 0999.42008
DOI: 10.1016/S1631-073X(02)02205-7
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2002-11-26T00:00:00Z
review text: By weight the author means any measurable function \(w\) such that \(0\leq w(t)\leq 1\) for \(t\in\mathbb{T}\). The set of all such functions endowed with an \(L^1\) distance forms a complete metric space \(W\). The following theorem is proved. (i) There exists a weight \(w>0\) a.e. such that any function \(f\in L^2(w,\mathbb{T})\) can be represented as a series of the form \(f= \sum_{n>0} c(n) e^{int}\), which converges in the norm. (ii) For any \(p>2\), the coefficients \(c(n)\) can be chosen in \(\ell_p\) with an arbitrarily small norm. (iii) The set of such weights is residual in the space \(W\). This result is discussed in the context of the classical Szegő-Kolmogorov ``prediction'' theorem.
reviewed by: Ferenc Móricz
Mathematics Subject Classification ID: 42A99
zbMATH DE Number: 1744345
zbMATH Keywords: weight
zbMATH Keywords: Szegő-Kolmogorov ``prediction'' theorem
author: Alexander Olevskiĭ
MaRDI profile type: Publication
cites work: Q3874458
cites work: Menshov representation spectra
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description: scientific article; zbMATH DE number 1744345