Singular functions with applications to fractal dimensions and generalized Takagi functions
type: Item
instance of: scholarly article
title: Singular functions with applications to fractal dimensions and generalized Takagi functions
zbMATH Open document ID: 1259.28008
published in: Acta Applicandae Mathematicae
publication date: 2012-12-27T00:00:00Z
review text: The authors study the family of singular functions \(\{S_a\}\) -- depending on the real parameter \(a\) -- introduced by Césaro in 1906 and Hellinger in 1907, with the aid of a generalized dyadic system (recall that a singular function is a monotone increasing, continuous real function whose derivative is zero almost everywhere). In particular, following \textit{M. Hata} and \textit{M. Yamaguti} [Japan J. Appl. Math. 1, No. 1, 183--199 (1984; Zbl 0604.26004)], they study the connection between \(\{S_a\}\) and the class of generalized Takagi functions \(\{T_a\}\) -- a class of monotonic type on no-interval and almost everywhere differentiable functions tightly related to the classic continuous nowhere differentiable Takagi function. As an application to fractal dimension, they compute the Hausdorff dimension of certain subsets of the unit interval related to these functions.
reviewed by: Enrico Zoli
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 6118579
zbMATH Keywords: singular function
zbMATH Keywords: Takagi function
zbMATH Keywords: generalized dyadic system
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Schauder basis
zbMATH Keywords: (simply) normal number
author: Enrique de Amo
author: Manuel Díaz Carrillo
author: Juan Fernández-Sánchez
MaRDI profile type: Publication
full work available at URL: s10440-011-9665-z
OpenAlex ID: W1963886846
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DOI: 10.1007/S10440-011-9665-Z
description: scientific article; zbMATH DE number 6118579