On a flexible class of continuous functions with uniform local structure
Pieter C. ALLAART
Source: J. Math. Soc. Japan Volume 61, Number 1
(2009), 237-262.
Abstract
This paper considers a class of continuous functions constructed as a series of iterates of the “tent map” multiplied by variable signs. This class includes Takagi's nowhere-differentiable function, and contains the functions studied by Hata and Yamaguti [Japan J. Appl. Math., 1 (1984), 183-199] and Kono [Acta Math. Hungar., 49 (1987), 315-324] as a proper subclass. A complete description is given of the differentiability properties of the functions in this class, and several statements are proved concerning their uniform and local moduli of continuity. The results are applied to generation of random functions.
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Keywords: Takagi function; nowhere-differentiable function; modulus of continuity; law of the iterated logarithm; binomial measure; Gray code
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Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1234189035
Digital Object Identifier: doi:10.2969/jmsj/06110237
Mathematical Reviews number (MathSciNet): MR2272878
Zentralblatt MATH identifier: 1161.26003
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