Skip to main content
SpringerLink
Log in

Pointwise analysis of Riemann's “nondifferentiable” function

  • Published:
Inventiones mathematicae Aims and scope

Summary

We will show how to analyse the local regularity of functions with the help of the wavelet transform. These results will be applied to the function of Riemann, where we show the existence of a dense set of points where this function is differentiable. On another dense set we show the existence of local singularities of cusp type. On a third set we show differentiability to the right (left). On the remaining set the functions will be shown to be not differentiable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Weierstrass, K.: Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten differentialquotienten besitzen. Mathematische Werke II, 71–74. Königl. Akad. Wiss.: 1872

  2. Hardy, G.H.: Wererstrass's nondifferentiable function, Trans. Am. Math. Soc.17, 301–325 (1916)

    Google Scholar 

  3. Hardy, G.H., Littlewood, E.: Some problems of diophantine approximation II. Acta Math.37, 194–238 (1914)

    Google Scholar 

  4. Gerver, J.: The Differentiability of the Riemann function at certain rational multiples of π. Am. J. Math. XCII., Nr. 1 (1970)

  5. Gerver, J.: More on the differentiability of the Riemann function. Am. J. Math.93, 33–41 (1970)

    Google Scholar 

  6. Queffelec, M.: Dérivabilité de certains sommes de séries de Fourier lacunnaires. Note au C.R. Séan. Acad. Sci.273, Sér. A, 291–293 (2.8.1971)

    Google Scholar 

  7. Queffelec, M.: Dérivabilité de certains sommes de séries de Fourier lacunnaires. Centre de Orssay, Université Paries-Sud (Thèse 3e cycle) juin 1971

  8. Holschneider, M.: On the Wavelet transform of Fractal Objects. J. Stat. Phys.5/6, 963–993 (1988)

    Google Scholar 

  9. Calderon, A.P., Zygmund, A.: Local properties of solutions of elliptic partial differential equations. Stud. Math.20, 171–225 (1961)

    Google Scholar 

  10. Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton University Press 1979

    Google Scholar 

  11. Jaffard, S.: Exposants de Holder en des points donnes et coefficients d'ondelettes. C. R. Acad. Sci., Paris, Sér. I.308, 7 (1989)

    Google Scholar 

  12. Grossmann, A., Morlet, J.: In: Streit, L. (ed.) Mathematics and Physics, Lectures on recent results. Singapore: World Scientific Publishing 1987

    Google Scholar 

  13. Lemarie, P.G., Meyer, Y.: Ondelettes et bases Hilbertiennes. Rev. Mat. Iberoam.1, 1286 (1987)

    Google Scholar 

  14. Arneodo, A., Grasseau, G., Holschneider, M.: On the Wavelet transform of Multifractals. Phys. Rev. Lett.61, Nr. 20 (1988)

    Google Scholar 

  15. Coifman, R., Meyer, Y.: Au délà des opérateurs pseudo-différentiels. Astérisque57 (1978)

  16. Paul, T.: Ondelettes et mécanique quantique. Thesis. Marseile (1985)

  17. Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory. Berlin Heidelberg New York: Springer (1976)

    Google Scholar 

  18. Gunning, R.C.: Lectures on Modular Forms. Princeton: Princeton University Press 1962

    Google Scholar 

  19. Mumford, D.: Tata Lectures on Theta 1. Boston: Birkhäuser (1983)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. C.N.R.S.-Luminy-Case 907, Centre de Physique Théorique, F-13288, Marseille Cedex 9, France

    M. Holschneider & Ph. Tchamitchian

  2. CEREMADE, Université de Paris-Dauphine, Paris, France

    M. Holschneider

  3. Faculté des Sciences et Techniques de Saint-Jerôme, Marseille

    Ph. Tchamitchian

Authors
  1. M. Holschneider
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ph. Tchamitchian
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Oblatum 27-III-1990 & 20-XI-1990

Laboratoire Propre LP-7061, Centre National de la Recherche Scientifique Juin 1989

Rights and permissions

Reprints and permissions

About this article

Cite this article

Holschneider, M., Tchamitchian, P. Pointwise analysis of Riemann's “nondifferentiable” function. Invent Math 105, 157–175 (1991). https://doi.org/10.1007/BF01232261

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01232261

Keywords

Search

Navigation