Functional equations generating space-densifying curves
type: Item
instance of: scholarly article
title: Functional equations generating space-densifying curves
zbMATH Open document ID: 0954.39016
DOI: 10.1016/S0898-1221(00)00085-7
publication date: 2000-09-20T00:00:00Z
review text: The first part of the paper offers the result that there are infinitely many linearly independent continuous solutions of the functional equation (1) \( f(x)=-f(2x)\) for \(x\) in \([0,1]\) or in \(\mathbb{R}\) or in \(\mathbb{C}\), with values in \(\mathbb{R}\) or in \(\mathbb{C}\) or in a subset of a metric space, the graphs of which are \(\alpha\)-dense there. In Example 1 it is verified that a class of functions satisfies (1) and the conclusion is drawn that ``therefore, this class is equivalent to the set of solutions to functional equation (1)''. The general continuous solution to this functional equation can clearly be chosen arbitrarily on \(]1/2,1]\) or \([-1,-1/2[ \cup ]1/2,1]\) or on \(\{z |1/2<|z|\leq 1\}\), respectively, as long as it is continuous there and \(\lim_{x\to 1/2} f(x)= - f(1)\) for the first domain, with the additional condition \(\lim_{x\to -1/2} f(x)=-f(-1)\) for the second and with \(\lim_{r\to 1/2} f(r e^{it})=-f(e^{it})\) for the third domain, further \(f(0)=0\), and then \(f\) can be extended to its whole domain by equation (1) itself. Similar remarks apply to the functional equation \(f(x)=-f(cx)\), also mentioned in the paper. The second part of the paper contains similar statements about the functional equation \(f(x)+f(2x)+f(3x)=0\), to which also noncontinuous solutions are offered.
reviewed by: János Aczél
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 26A30
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 39B32
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 1471218
zbMATH Keywords: functional equations
zbMATH Keywords: continuous solutions
zbMATH Keywords: Hamel bases
zbMATH Keywords: additive functions
zbMATH Keywords: space densifying
zbMATH Keywords: space-filling curves
zbMATH Keywords: noncontinuous solutions
author: G. Mora
author: Yves Cherruault
author: Abdelkader Ziadi
MaRDI profile type: Publication
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full work available at URL: s0898-1221(00)00085-7
OpenAlex ID: W2052332777
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1471218