Journal of Fourier Analysis and Applications

, Volume 16, Issue 1, pp 17–33 | Cite as

Simple Proofs of Nowhere-Differentiability for Weierstrass’s Function and Cases of Slow Growth

  • Jon JohnsenEmail author


Using a few basics from integration theory, a short proof of nowhere-differentiability of Weierstrass functions is given. Restated in terms of the Fourier transformation, the method consists in principle of a second microlocalisation, which is used to derive two general results on existence of nowhere differentiable functions. Examples are given in which the frequencies are of polynomial growth and of almost quadratic growth as a borderline case.


Nowhere-differentiability Weierstrass function Lacunary Fourier series Second microlocalisation 

Mathematics Subject Classification (2000)



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  1. 1.
    Baouche, A., Dubuc, S.: La non-dérivabilité de la fonction de Weierstrass. Enseign. Math. 38, 89–94 (1992) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Chamizo, F., Ubis, A.: Some Fourier series with gaps. J. Anal. Math. 101, 179–197 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Duistermaat, J.J.: Self-similarity of “Riemann’s nondifferentiable function”. Nieuw Arch. Wiskd. (4) 9(3), 303–337 (1991) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Freud, G.: Über trigonometrische Approximation und Fouriersche Reihen. Math. Z. 78, 252–262 (1962) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hardy, G.H.: Weierstrass’s non-differentiable function. Trans. Am. Math. Soc. 17(3), 301–325 (1916) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Holschneider, M.: Wavelets: An Analysis Tool. Oxford Mathematical Monographs. Oxford University Press, London (1995) zbMATHGoogle Scholar
  7. 7.
    Hykšová, M.: Bolzano’s inheritance research in Bohemia. In: Mathematics Throughout the Ages, Holbæk, 1999/Brno, 2000. Děj. Mat./Hist. Math., vol. 17, pp. 67–91. Prometheus, Prague (2001) Google Scholar
  8. 8.
    Jaffard, S.: Old friends revisited: the multifractal nature of some classical functions. J. Fourier Anal. Appl. 3, 1–22 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Johnsen, J.: Type 1,1-operators defined by vanishing frequency modulation. In: Rodino, L., Wong, M.W. (eds.) New Developments in Pseudo-Differential Operators. Operator Theory: Advances and Applications, vol. 189, pp. 201–246. Birkhäuser, Basel (2008) CrossRefGoogle Scholar
  10. 10.
    Lebesgue, H.: Sur la méthode de M. Goursat pour la résolution de l’équation de Fredholm. Bull. Soc. Math. Fr. 36, 3–19 (1908) zbMATHMathSciNetGoogle Scholar
  11. 11.
    Luther, W.: The differentiability of Fourier gap series and “Riemann’s example” of a continuous, nondifferentiable function. J. Approx. Theory 48, 303–321 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    McCarthy, J.: An everywhere continuous nowhere differentiable function. Am. Math. Mon. 60, 709 (1953) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Meyer, Y.: Wavelets: Algorithms and Applications. SIAM, Philadelphia (1993) zbMATHGoogle Scholar
  14. 14.
    Richards, I., Youn, H.: Theory of Distributions. A Nontechnical Introduction. Cambridge University Press, Cambridge (1990) Google Scholar
  15. 15.
    Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966). Revised and enlarged edn. zbMATHGoogle Scholar
  16. 16.
    Stein, E.M., Shakarchi, R.: Fourier Analysis. Princeton Lectures in Analysis, vol. 1. Princeton University Press, Princeton (2003) zbMATHGoogle Scholar
  17. 17.
    Takagi, T.: A simple example of the continous function without derivative. Proc. Phys.-Math. Soc. Jpn. 1, 176–177 (1903) Google Scholar
  18. 18.
    Takagi, T.: The Collected Papers of Teiji Takagi, pp. 5–6. Iwanami Shoteu, Tokyo (1973). S. Kuroda (ed.) Google Scholar
  19. 19.
    Thim, J.: Continuous nowhere differentiable functions. Master’s Thesis, Luleå University of Technology, Sweden (2003) Google Scholar
  20. 20.
    Wen, L.: A nowhere differentiable continuous function constructed by infinite products. Am. Math. Mon. 109, 378–380 (2002) zbMATHCrossRefGoogle Scholar
  21. 21.
    Zygmund, A.: Trigonometric Series, 2nd edn., vols. I, II. Cambridge University Press, Cambridge (1959) zbMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborg ØstDenmark

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