Extremal homogeneous polynomials on real normed spaces
type: Item
instance of: scholarly article
title: Extremal homogeneous polynomials on real normed spaces
zbMATH Open document ID: 0923.46047
author: Pádraig Kirwan
author: Yannis Sarantopoulos
author: Andrew Tonge
published in: Journal of Approximation Theory
publication date: 1999-10-25T00:00:00Z
review text: Let \(P\) be a continuous \(m\)-homogeneous polynomial on a normed space \(E\) over either the real or the complex scalars, and let \(\check P\) denote the unique associated symmetric \(m\)-linear form [see, e.g., \textit{S. Dineen}, ``Complex analysis on infinite-dimensional spaces'', Springer-Verlag (1999)] for background information on polynomials, multilinear forms, and their norms. It is well-known that \(1\leq{\|\check P\|\over\| P\|}\leq{m^m\over m!}\), with equality in the left-hand side for every \(P\) if \(E\) is Hilbert space. In [Proc. Am. Math. Soc. 99, 340-346 (1987; Zbl 0618.46022)], \textit{I. Sarantopoulos} showed that if the scalar field is \(\mathbb{C}\), then there are \(P\)'s for which we have equality in the right-hand side and for which \(\check P\) attains its norm if and only if \(E\) contains an isometric copy of \(\ell^m_1\). Here, the authors investigate the situation in which the space \(E= \ell^m_1\) over the scalar field \(\mathbb{R}\). They show that, as expected, the Nachbin polynomial \(P(x)= x_1\cdots x_m\) is the unique extremal polynomial in case \(m=2\) or 3, but that many small perturbations of this polynomial also yield extremal polynomials when \(m\geq 4\).
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 1278840
zbMATH Keywords: polynomial on a normed space
zbMATH Keywords: isometric copy
zbMATH Keywords: Nachbin polynomial
zbMATH Keywords: extremal polynomial
reviewed by: Richard M. Aron
MaRDI profile type: Publication
OpenAlex ID: W1980536770
cites work: Q3943468
cites work: Q3244654
cites work: On a theorem of Grace
cites work: Q4241773
cites work: Extremal Multilinear Forms on Banach Spaces
cites work: Norm attaining polynomials
DOI: 10.1006/JATH.1996.3273
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description: scientific article; zbMATH DE number 1278840