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arXiv · 1312.2812

Measure and Hausdorff dimension of randomized Weierstrass-type functions

Julia Romanowska
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Abstract

In this paper we consider functions of the type $$f(x) = \sum_{n=0}^\infty a_n g(b_nx+θ_n),$$ where $(a_n)$ are independent random variables uniformly distributed on $(-a^n, a^n)$ for some $0 1$, $a^2b> 1$ and $g$ is a $C^1$ periodic real function with finite number of critical points in every bounded interval. We prove that the occupation measure for $f$ has $L^2$ density almost surely. Furthermore, the Hausdorff dimension of the graph of $f$ is almost surely equal to $D = 2+ \log{a}/\log{b}$ provided $ b = \lim_{n\rightarrow \infty}b_{n+1}/b_n>1$ and $ab>1$.

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Julia Romanowska. 2013-12-10. Measure and Hausdorff dimension of randomized Weierstrass-type functions. https://doi.org/10.1088/0951-7715%2F27%2F4%2F787

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