Journal of the Mathematical Society of Japan

On a flexible class of continuous functions with uniform local structure

Pieter C. ALLAART

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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.

Article information

Source
J. Math. Soc. Japan Volume 61, Number 1 (2009), 237-262.

Dates
First available in Project Euclid: 9 February 2009

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

Subjects
Primary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
Secondary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 26A30: Singular functions, Cantor functions, functions with other special properties 60G50: Sums of independent random variables; random walks

Keywords
Takagi function nowhere-differentiable function modulus of continuity law of the iterated logarithm binomial measure Gray code

Citation

ALLAART, Pieter C. On a flexible class of continuous functions with uniform local structure. J. Math. Soc. Japan 61 (2009), no. 1, 237--262. doi:10.2969/jmsj/06110237. http://projecteuclid.org/euclid.jmsj/1234189035.


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