On modified Takagi functions of two variables
type: Item
instance of: scholarly article
title: On modified Takagi functions of two variables
zbMATH Open document ID: 0609.26007
author: Yoshiki Tsujii
published in: Journal of Mathematics of Kyoto University
publication date: 1985-01-01T00:00:00Z
review text: In this paper explicit examples are constructed of continuous functions of two variables whose non-trivial contours have union of Lebesgue measure zero. This follows the work of the reviewer [Z. Wahrscheinlichkeitstheor. Verw. Geb. 52, 267-276 (1980; Zbl 0431.60056)], who established this result for multiparameter Brownian motions [and incidentally the Baire category analogue in Bull. Lond. Math. Soc. 14, 30-32 (1982; Zbl 0485.54023)]. The construction uses a modification of the Takagi function, a beautifully simple early example of a nondifferentiable function. The resulting function is shown to have graph of Hausdorff dimension 2, which is the smallest possible value. Thus the function is much smoother than Brownian motion.
Mathematics Subject Classification ID: 26B35
Mathematics Subject Classification ID: 26A27
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 3985546
zbMATH Keywords: Cantor function
zbMATH Keywords: continuous functions of two variables
zbMATH Keywords: contours
zbMATH Keywords: multiparameter Brownian motions
zbMATH Keywords: modification of the Takagi function
zbMATH Keywords: nondifferentiable function
zbMATH Keywords: graph of Hausdorff dimension 2
reviewed by: Wilfrid S. Kendall
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250521072
OpenAlex ID: W1484520943
DOI: 10.1215/KJM/1250521072
description: scientific article; zbMATH DE number 3985546