The Takaga operator, Bernoulli sequences, smoothness conditions and fractal curves
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- by Anca Deliu and Peter Wingren PDF
- Proc. Amer. Math. Soc. 121 (1994), 871-881 Request permission
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.References
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Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 871-881
- MSC: Primary 28A80
- DOI: https://doi.org/10.1090/S0002-9939-1994-1196164-2
- MathSciNet review: 1196164