Hausdorff dimension of the graphs of the classical Weierstrass functions
By W. Shen. 2015
111 citations recorded by the source corpus.
We show that the graph of the classical Weierstrass function $$\sum _{n=0}^\infty \lambda ^n \cos (2\pi b^n x)$$∑n=0∞λncos(2πbnx) has Hausdorff dimension $$2+\log \lambda /\log b$$2+logλ/logb, for every integer $$b\ge 2$$b≥2 and every $$\lambda \in (1/b,1)$$λ∈(1/b,1). Replacing $$\cos (2\pi x)$$cos(2πx) by a general non-constant $$C^2$$C2 periodic function, we obtain the same result under a further assumption that $$\lambda b$$λb is close to 1.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1268195301016767551ce75d83112189c2a18c0b
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- 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].”