De Rham's singular function, two-scale difference equations and Appell polynomials
type: Item
instance of: scholarly article
title: De Rham's singular function, two-scale difference equations and Appell polynomials
zbMATH Open document ID: 0954.39017
DOI: 10.1007/BF03322429
published in: Results in Mathematics
publication date: 2000-08-31T00:00:00Z
review text: The de Rham function, which is the main subject of this paper, is the (up to a multiplicative constant) general unbounded solution of the pair of functional equations \(\phi(t/2)=a\phi(t), \phi((1+t)/2)=a+(1-a)\phi(t) (t\in[0,1])\). For \(a\in ]0,1] \setminus \{1/2\}\) it is continuous and strictly monotonic but its derivative is almost everywhere 0. The authors offer further properties of this function, in particular its Hölder-continuity, and connections to Appell polynomials, to difference and integral-functional equations, and to wavelet theory.
reviewed by: János Aczél
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 26A16
Mathematics Subject Classification ID: 26A30
Mathematics Subject Classification ID: 33C65
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 1498576
zbMATH Keywords: de Rham's singular function
zbMATH Keywords: almost everywhere zero derivatives
zbMATH Keywords: functional equations
zbMATH Keywords: difference equation
zbMATH Keywords: functional-integral equation
zbMATH Keywords: Appell polynomials
zbMATH Keywords: wavelet
author: Lothar Berg
author: Manfred Krüppel
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03322429
OpenAlex ID: W2073399025
description: scientific article; zbMATH DE number 1498576