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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Takaga operator, Bernoulli sequences, smoothness conditions and fractal curves

Authors: Anca Deliu and Peter Wingren
Journal: Proc. Amer. Math. Soc. 121 (1994), 871-881
MSC: Primary 28A80
MathSciNet review: 1196164
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Abstract: We map Lipschitz spaces and functions of bounded variation by the operator $ \sigma :\varphi (x) \to \sum\nolimits_0^\infty {{2^{ - n}}} \varphi ({2^n}x),x \in [0,1]$, and we estimate the Hausdorff measure of $ \sigma (\varphi )$. We furthermore introduce a class of continuous and nowhere differentiable functions on [0,1] which we call $ \mathcal{T}$. We make a refined analysis of the fractal and smoothness properties of the functions in $ \mathcal{T}$ and study the relationship between the two. We show that all the functions in $ \mathcal{T}$ have box dimension equal to $ \frac{1}{2}$, with respect to the dimension family $ \{ t/{(\log \frac{1}{t})^s}:s \in {\mathbb{R}^ + }\} $, but that their order of smoothness covers a wide range.

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Keywords: Nowhere differentiable, Hausdorff dimension
Article copyright: © Copyright 1994 American Mathematical Society