Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Weierstrass functions with random phases


Author: Yanick Heurteaux
Journal: Trans. Amer. Math. Soc. 355 (2003), 3065-3077
MSC (2000): Primary 26A27, 28A80, 37A05; Secondary 60F20
DOI: https://doi.org/10.1090/S0002-9947-03-03221-5
Published electronically: March 19, 2003
MathSciNet review: 1974675
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Consider the function \[ f_\theta (x)=\sum _{n=0}^{+\infty }b^{-n\alpha }g(b^nx+\theta _n),\] where $b>1$, $0<\alpha <1$, and $g$ is a non-constant 1-periodic Lipschitz function. The phases $\theta _n$ are chosen independently with respect to the uniform probability measure on $[0,1]$. We prove that with probability one, we can choose a sequence of scales $\delta _k\searrow 0$ such that for every interval $I$ of length $|I|=\delta _k$, the oscillation of $f_\theta$ satisfies $\operatorname {osc}(f_\theta ,I)\geq C|I|^\alpha$. Moreover, the inequality $\operatorname {osc}(f_\theta ,I)\geq C|I|^{\alpha +\varepsilon }$ is almost surely true at every scale. When $b$ is a transcendental number, these results can be improved: the minoration $\operatorname {osc}(f_\theta ,I)\geq C|I|^\alpha$ is true for every choice of the phases $\theta _n$ and at every scale.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 26A27, 28A80, 37A05, 60F20

Retrieve articles in all journals with MSC (2000): 26A27, 28A80, 37A05, 60F20


Additional Information

Yanick Heurteaux
Affiliation: Laboratoire de Mathématiques pures, Université Blaise Pascal, F-63177 Aubière cedex, France
Email: Yanick.Heurteaux@math.univ-bpclermont.fr

Keywords: Weierstrass functions, almost periodic functions, oscillations, fractal dimension
Received by editor(s): July 8, 2002
Published electronically: March 19, 2003
Article copyright: © Copyright 2003 American Mathematical Society