On superpositions of continuous functions defined on the Baire space
type: Item
instance of: scholarly article
title: On superpositions of continuous functions defined on the Baire space
zbMATH Open document ID: 1005.26009
DOI: 10.1007/BF02674199
published in: Mathematical Notes
publication date: 2000-10-16T00:00:00Z
review text: The paper deals with the following question: if \(M(n)\) is a class of functions of \(n\) variables and \([M(n)]\) is the set of functions representable as superpositions of functions from \(M(n)\), is it true that \(M(n+1)\subset [M(n)]\)? Using the method based on Boolean functions introduced by the same author [Probl. Kibern. 37, 5-17 (1980; Zbl 0472.26006)], it is proved that for functions mapping \(I^n\) into \(I\), where \(I= \prod^\infty_{n=1} \{0,1\}\) is equipped with the metric of the Baire space, the above inclusion depends essentially on some kind of the modulus \(\varphi\) of continuity. The answer is positive in the case when \(\varphi(t)\geq (1+ a)t\) (Theorem 1) and negative if the last condition is violated (Theorem 2 and Theorem 3).
Mathematics Subject Classification ID: 26B40
zbMATH DE Number: 1517553
zbMATH Keywords: superposition
zbMATH Keywords: modulus of continuity
zbMATH Keywords: Baire space
author: S. S. Marchenkov
reviewed by: Władysław Wilczyński
MaRDI profile type: Publication
cites work: Q3232381
cites work: Q3263814
cites work: Q3263816
cites work: Q5825842
cites work: Q5544663
cites work: On representation of functions by means of superpositions and related topics
cites work: Q3926809
cites work: Q4396564
full work available at URL: https://doi.org/10.1007/bf02674199
OpenAlex ID: W2095401689
description: scientific article; zbMATH DE number 1517553