Geometry of Müntz spaces and related questions
type: Item
instance of: scholarly article
title: Geometry of Müntz spaces and related questions
zbMATH Open document ID: 1094.46003
DOI: 10.1007/11551621
author: N. E. Zubov
published in: Lecture Notes in Mathematics
publication date: 2006-03-15T00:00:00Z
review text: For a given Banach space \(E\) of functions on \([0,1]\), a sequence of the form \(\{t^{\lambda_n}\} \subset E\), \(0=\lambda_0<\lambda_1< \ldots\), is called a \textit{Müntz sequence}, and the subspace spanned in \(E\) by such a sequence is called a \textit{Müntz space}. According to the famous Müntz theorem, if \(E=C[0,1]\) or \(E=L_p[0,1]\), \(1 \leq p < \infty\), and \(\sum \lambda_n^{-1} < \infty\), the corresponding Müntz space does not coincide with \(E\). In this case, the study of the properties of this Müntz space (which is the main objective of the book under review) appears to be an interesting and highly non-trivial task. The Müntz sequence turns out to be minimal, but not necessarily uniformly minimal or a basis. For a Müntz sequence in \(E=C[0,1]\) or \(E=L_p[0,1]\), \(1 \leq p < \infty\), to be uniformly minimal is equivalent to be a basic sequence and is, in turn, equivalent to lacunarity in the following sense: \(\inf_k \lambda_{k+1}/\lambda_k > 1\). In this case (and, more generally, for quasi-lacunary sequences of \(\lambda_k\)), the Müntz space is isomorphic to \(c_0\) if \(E=C[0,1]\) or to \(\ell_p\) if \(E=L_p[0,1]\). There are at least two isomorphism classes of Müntz spaces in \(E=C[0,1]\) and infinitely many isometry classes. The book consists of two parts. The first one, ``Subspaces and sequences in Banach spaces'', has five chapters: Disposition of subspaces; Sequences in normed spaces; Isomorphisms, isometries and embedddings; Spaces of universal disposition; Approximation properties. Although this part gives some necessary background to the study of Müntz spaces, the main goal is to give an introduction to the disposition of subspaces language and technique in the framework of Banach space theory and to present, among other things, some of his results obtained in the sixties and seventies and published in Russian. The second part, ``On the geometry of Müntz sequences'', consists of seven chapters: Coefficient estimates and the Müntz the…
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 5012644
zbMATH Keywords: Müntz sequence
zbMATH Keywords: Müntz polynomial
zbMATH Keywords: supercomplete sequence
zbMATH Keywords: minimal sequence
zbMATH Keywords: basis
zbMATH Keywords: space of universal disposition
zbMATH Keywords: uniformly convex space
zbMATH Keywords: Müntz space
reviewed by: Vladimir Kadets
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/11551621
OpenAlex ID: W615361247
description: scientific article; zbMATH DE number 5012644