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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Weierstrass functions with random phases
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by Yanick Heurteaux PDF
Trans. Amer. Math. Soc. 355 (2003), 3065-3077 Request permission

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.
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Additional Information
  • Yanick Heurteaux
  • Affiliation: Laboratoire de Mathématiques pures, Université Blaise Pascal, F-63177 Aubière cedex, France
  • Email: Yanick.Heurteaux@math.univ-bpclermont.fr
  • Received by editor(s): July 8, 2002
  • Published electronically: March 19, 2003
  • © Copyright 2003 American Mathematical Society
  • 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
  • MathSciNet review: 1974675