On probabilistic properties of Takagi's function
type: Item
instance of: scholarly article
title: On probabilistic properties of Takagi's function
zbMATH Open document ID: 0733.26004
published in: Journal of Mathematics of Kyoto University
publication date: 1990-01-01T00:00:00Z
review text: Takagi's nowhere differentiable continuous function (rediscovered by van der Waerden) \(f: {\mathbb{R}}\to {\mathbb{R}}\) is defined by \(f(x)=\sum^{\infty}_{n=1}2^{-n}\phi^{(n)}(x),\) where \(\phi (x)=2 dist(x,{\mathbb{Z}})\) and \(\phi^{(n)}\) represents the n-th iterate of \(\phi\). The author proves the following formula: \[ \lim_{h\downarrow 0}mes\{x\in (0,1):\;\frac{f(x+h)-f(x)}{h\sqrt{\log_ 2(1/h)}}<y\}=\frac{1}{\sqrt{2\pi}}\int^{y}_{-\infty}e^{-u^ 2/2}du. \] Previously, the author proved the same result for the Weierstrass' function [Mat. Zametki 36, No.1, 35-38 (1984; Zbl 0543.42007)].
Mathematics Subject Classification ID: 26A27
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 4212352
zbMATH Keywords: asymptotically normal
zbMATH Keywords: Takagi's nowhere differentiable continuous function
author: N. G. Gamkrelidze
reviewed by: Valeriu Anisiu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520068
OpenAlex ID: W1550324611
DOI: 10.1215/KJM/1250520068
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description: scientific article; zbMATH DE number 4212352