On the dimension of the graph of the classical Weierstrass function
By Krzysztof Bara'nski, Balázs Bárány, J. Romanowska. 2013
83 citations recorded by the source corpus.
Abstract This paper examines dimension of the graph of the famous Weierstrass non-differentiable function W λ , b ( x ) = ∑ n = 0 ∞ λ n cos ( 2 π b n x ) for an integer b ≥ 2 and 1 / b λ 1 . We prove that for every b there exists (explicitly given) λ b ∈ ( 1 / b , 1 ) such that the Hausdorff dimension of the graph of W λ , b is equal to D = 2 + log λ log b for every λ ∈ ( λ b , 1 ) . We also show that the dimension is equal to D for almost every λ on some larger interval. This partially solves a well-known thirty-year-old conjecture. Furthermore, we prove that the Hausdorff dimension of the graph of the function f ( x ) = ∑ n = 0 ∞ λ n ϕ ( b n x ) for an integer b ≥ 2 and 1 / b λ 1 is equal to D for a typical Z -periodic C 3 function ϕ.
Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/5a6c728e7bb1cba7f86b89f9f893b4ad1082bbb6
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In Algebra and Number Theory
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Hausdorff dimension of the graphs of the classical Weierstrass functions (2015, background): “For any Z-periodic, non-constant C2 function φ : R → R and any integer b ≥ 2 there exists K0 = K0(φ, b) > 1 such that if 1 λb K0, then the graph of fφλ,b has Hausdorff dimension D. Recently, Barańsky, Bárány and Romanowska [2], based on results of Ledrappier [10] and Tsujii [16], proved that for”
- Assouad Dimension and Fractal Geometry (2020, methodology): “This was resolved by Shen [251] in full generality and for certain ranges of parameters a little earlier by Barański, Bárány and Romanowska [16].”