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Proceedings of the American Mathematical Society

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On maxima of Takagi-van der Waerden functions


Author: Yoshikazu Baba
Journal: Proc. Amer. Math. Soc. 91 (1984), 373-376
MSC: Primary 26A27
DOI: https://doi.org/10.1090/S0002-9939-1984-0744632-0
MathSciNet review: 744632
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Abstract: Generalizing Takagi's function $ {F_2}\left( x \right)$ and van der Waerden's function $ {F_{10}}\left( x \right)$, we introduce a class of nowhere differentiable continuous functions $ {F_r}\left( x \right)$, $ r \geqslant 2$. Some properties of $ {F_r}\left( x \right)$ concerning especially maxima are discussed. When $ r$ is even, the Hausdorff dimension of the set of $ {x^,}$'s giving the maxima of $ {F_r}\left( x \right)$ is proved to be $ 1/2$.


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

  • [1] T. Takagi, A simple example of the continuous function without derivative, Proc. Phys.-Math. Soc. Tokyo Ser. II 1 (1903), 176-177.
  • [2] B. L. van der Waerden, Ein einfaches Beispiel einer nichtdifferenzierbaren stetigen Funktion, Math. Z. 32 (1930), 474-475. MR 1545179
  • [3] B. Martynov, On maxima of the van der Waerden function, Kvant, June 1982, 8-14. (Russian)
  • [4] M. Yamaguti and M. Hata, Weierstrass's function and chaos, Hokkaido Math. J. (to appear). MR 719972 (84k:58121)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0744632-0
Keywords: Nowhere differentiable continuous function, Hausdorff dimension
Article copyright: © Copyright 1984 American Mathematical Society

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