Abstract
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.
Similar content being viewed by others
References
Baouche, A., Dubuc, S.: La non-dérivabilité de la fonction de Weierstrass. Enseign. Math. 38, 89–94 (1992)
Chamizo, F., Ubis, A.: Some Fourier series with gaps. J. Anal. Math. 101, 179–197 (2007)
Duistermaat, J.J.: Self-similarity of “Riemann’s nondifferentiable function”. Nieuw Arch. Wiskd. (4) 9(3), 303–337 (1991)
Freud, G.: Über trigonometrische Approximation und Fouriersche Reihen. Math. Z. 78, 252–262 (1962)
Hardy, G.H.: Weierstrass’s non-differentiable function. Trans. Am. Math. Soc. 17(3), 301–325 (1916)
Holschneider, M.: Wavelets: An Analysis Tool. Oxford Mathematical Monographs. Oxford University Press, London (1995)
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)
Jaffard, S.: Old friends revisited: the multifractal nature of some classical functions. J. Fourier Anal. Appl. 3, 1–22 (1997)
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)
Lebesgue, H.: Sur la méthode de M. Goursat pour la résolution de l’équation de Fredholm. Bull. Soc. Math. Fr. 36, 3–19 (1908)
Luther, W.: The differentiability of Fourier gap series and “Riemann’s example” of a continuous, nondifferentiable function. J. Approx. Theory 48, 303–321 (1986)
McCarthy, J.: An everywhere continuous nowhere differentiable function. Am. Math. Mon. 60, 709 (1953)
Meyer, Y.: Wavelets: Algorithms and Applications. SIAM, Philadelphia (1993)
Richards, I., Youn, H.: Theory of Distributions. A Nontechnical Introduction. Cambridge University Press, Cambridge (1990)
Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966). Revised and enlarged edn.
Stein, E.M., Shakarchi, R.: Fourier Analysis. Princeton Lectures in Analysis, vol. 1. Princeton University Press, Princeton (2003)
Takagi, T.: A simple example of the continous function without derivative. Proc. Phys.-Math. Soc. Jpn. 1, 176–177 (1903)
Takagi, T.: The Collected Papers of Teiji Takagi, pp. 5–6. Iwanami Shoteu, Tokyo (1973). S. Kuroda (ed.)
Thim, J.: Continuous nowhere differentiable functions. Master’s Thesis, Luleå University of Technology, Sweden (2003)
Wen, L.: A nowhere differentiable continuous function constructed by infinite products. Am. Math. Mon. 109, 378–380 (2002)
Zygmund, A.: Trigonometric Series, 2nd edn., vols. I, II. Cambridge University Press, Cambridge (1959)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by David Walnut.
Rights and permissions
About this article
Cite this article
Johnsen, J. Simple Proofs of Nowhere-Differentiability for Weierstrass’s Function and Cases of Slow Growth. J Fourier Anal Appl 16, 17–33 (2010). https://doi.org/10.1007/s00041-009-9072-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-009-9072-2