Ultranormal case of the Bernstein problem
type: Item
instance of: scholarly article
title: Ultranormal case of the Bernstein problem
zbMATH Open document ID: 0908.17023
DOI: 10.1007/BF02466006
publication date: 1998-10-11T00:00:00Z
review text: Let \(V\) be a quadratic stochastic mapping on the simplex \(\Delta^{n-1}\). The mapping \(V\) is called a Bernstein mapping if \(V^2=V\). The Bernstein problem is to describe all Bernstein mappings explicitly. The case \(n\leq 2\) is trivial. This problem was solved by Bernstein for the case \(n=3\), and he obtained some results for \(n\geq 4\). In earlier papers of the author the Bernstein problem was solved for all \(n\) under the regularity assumption, i.e. \(Vx\) depends only on the values \(f(x)\) (where \(f\) runs over all invariant linear forms). In this work a stochastic quadratic mapping \(V\) and the corresponding Bernstein algebra \(A_V\) is considered, which has the ultranormality property. The algebra \(A_V\) is said to be normal if the quadratic forms \((Vx)_i\) and \((Vx)_j\) \((i\neq j)\) are linearly independent and, for no pair \(k,s\) \((k\neq s)\), all \((Vx)_i\) depend on \(x_k+x_s\) and \(x_j\) only \((j\neq k\), \(j\neq s)\); \(A_V\) is called ultranormal if all coordinate subalgebras are normal. It is proved that every ultranormal stochastic Bernstein algebra is regular. This implies the solution of the Bernstein problem in the ultranormal case.
Mathematics Subject Classification ID: 17D92
Mathematics Subject Classification ID: 92D10
zbMATH DE Number: 1179568
zbMATH Keywords: quadratic stochastic mapping
zbMATH Keywords: regularity
zbMATH Keywords: ultranormal stochastic Bernstein algebra
zbMATH Keywords: Bernstein problem
author: Yu. I. Lubich
reviewed by: A. A. Rakhimov
MaRDI profile type: Publication
cites work: Q3724143
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 1179568