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About the Uniform Hölder Continuity of Generalized Riemann Function

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Abstract

In this paper, we study the uniform Hölder continuity of the generalized Riemann function \({R_{\alpha,\beta} \,\,{\rm (with}\,\, \alpha > 1 \,\,{\rm and}\,\, \beta > 0}\)) defined by

$$R_{\alpha,\beta}(x) = \sum_{n=1}^{+\infty} \frac{\sin(\pi n^\beta x)}{n^\alpha},\quad x \in \mathbb{R},$$

using its continuous wavelet transform. In particular, we show that the exponent we find is optimal. We also analyse the behaviour of \({R_{\alpha,\beta} \,\,{\rm as}\,\, \beta}\) tends to infinity.

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References

  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, vol. 55. New York (1972)

  2. Boichu, D.: Analyse 2-microlocale et développement en série de chirps d’une fonction de Riemann et de ses généralisations, Colloquium Mathematicum vol. LXVII(2) (1994)

  3. Chamizo F., Ubis A.: Multifractal Behavior of Polynomial Fourier Series. Adv. Math. 250, 1–34 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chamizo F., Ubis A.: Some Fourier series with gaps. J. Anal. Math. 101, 179–197 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Daubechies I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

  6. du Bois-Reymond P.: Versuch einer Klassifikation der willkürlichen Funktionen reeller Argumente nach ihren Änderungen in den kleinsten Intervallen. J. für die reine und andewandte Mathematik 79, 21–37 (1875)

    Google Scholar 

  7. Gerver J.L.: On Cubic Lacunary Fourier Series. Trans. Am. Math. Soc. 355(11), 4297–4347 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gerver J.L.: The Differentiability of the Riemann Function at Certain Rational Multiples of π. Am. J. Math. 92, 33–55 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hardy G.H.: Weierstrass’s Non-Differentiable Function. Trans. Am. Math. Soc. 17, 301–325 (1916)

    MATH  Google Scholar 

  10. Hardy G.H., Littlewood J.E.: Some Problems of Diophantine Approximation (II). Acta Math. 37(1), 193–239 (1914)

    Article  MathSciNet  Google Scholar 

  11. Holschneider M.: Wavelets, an Analysis Tool, Oxford Mathematical Monographs. Oxford Science Publications (1995)

  12. Holschneider M., Tchamitchian P.: Pointwise Analysis of Riemann’s “Nondifferentiable” Function. Invent. Math. 105, 157–175 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Itatsu, S.: Differentiability of Riemann’s Function. In: Proceedings of the Japan Academy, Series A vol. 57(10), pp. 492–495 (1981)

  14. Jaffard S.: Pointwise and Directional Regularity of Nonharmonic Fourier Series. Appl. Comput. Harmon. Anal. 28, 251–266 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jaffard S.: The Spectrum of Singularities of Riemann’s Function. Rev. Math. Iberoam. 12(2), 441–460 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jaffard, S., Meyer, Y.: Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions. Mem. Am. Math. Soc. (Book 587) (1996)

  17. Jaffard, S., Meyer, Y., Ryan, R.D.: Wavelets: Tools for Science and Technologies, SIAM, (2001)

  18. Jaffard, S., Nicolay, S.: Space-Filling Functions and Davenport Series. Recent Developments in Fractals and Related Fields, 19–34 (2010)

  19. Johnsen J.: Simple Proofs of Nowhere-Differentiability for Weierstrass’s Function and Cases of Slow Growth. J. Fourier Anal. Appl. 16, 17–33 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Luther W.: The Differentiability of Fourier Gap Series and “Riemann’s Example” of a Continuous, Nondifferentiable Function. J. Approx. Theory 48, 303–321 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mohr E.: Wo ist die Riemannsche Funktion nicht differenzierbar?. Annali di Matematica Pura ed Applicata 123(1), 93–104 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  22. Queffelec H.: Dérivabilité de certaines sommes de séries de Fourier lacunaires. Comptes rendus de l’Académie des Sciences de Paris 273, 291–293 (1971)

    MathSciNet  MATH  Google Scholar 

  23. Shidfar A., Sabetfakhri K.: On the Continuity of van der Waerden’s Function in the Hölder Sense. Am. Math. Mon. 93, 375–376 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Smith A.: The Differentiability of Riemann’s Function. Proc. Am. Math. Soc. 34(2), 463–468 (1972)

    MATH  Google Scholar 

  25. Takagi, T.: A Simple Example of the Continuous Function without Derivative. In: Proceedings of the Physico-Mathematical Society of Japan, vol. 1, pp. 176–177 (1903)

  26. Torrésani, B.: Analyse continue par ondelettes, CNRS Editions, Paris (1995)

  27. Tricot, C.: Courbes et dimension fractale. Springer, Berlin (1999)

  28. Ubis, A.: Questions of Arithmetic and Harmonic Analysis, PhD-Thesis, Madrid (2006)

  29. Young, R.: An Introduction to Nonharmonic Fourier Series, Academic Press, San Diego (2001)

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  1. Institute of Mathematics, University of Liège, Grande Traverse 12 (Bât. 37), 4000, Liège, Belgium

    F. Bastin, S. Nicolay & L. Simons

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  1. F. Bastin
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  2. S. Nicolay
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  3. L. Simons
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Correspondence to L. Simons.

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Bastin, F., Nicolay, S. & Simons, L. About the Uniform Hölder Continuity of Generalized Riemann Function. Mediterr. J. Math. 13, 101–117 (2016). https://doi.org/10.1007/s00009-014-0501-3

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  • DOI: https://doi.org/10.1007/s00009-014-0501-3

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