About the Uniform Hölder Continuity of Generalized Riemann Function

Bastin, Françoise; Nicolay, Samuel; Simons, Laurent

doi:10.1007/s00009-014-0501-3

Request a copy

Article (Scientific journals)

About the Uniform Hölder Continuity of Generalized Riemann Function

Bastin, Françoise; Nicolay, Samuel; Simons, Laurent

2016 • In Mediterranean Journal of Mathematics, 13 (1), p. 101-117

Peer Reviewed verified by ORBi

Permalink
https://hdl.handle.net/2268/175527

DOI
10.1007/s00009-014-0501-3

Files (1)Send to Details Statistics Bibliography Similar publications

Files

Full Text

10.1007_s00009-014-0501-3.pdf

Publisher postprint (651.13 kB)

Request a copy

All documents in ORBi are protected by a user license.

Send to

RIS BibTex APA Chicago Permalink X Linkedin

Details

Keywords :

Hölder continuity; Continuous Wavelet Transform; Riemann function

Abstract :

[en] In this paper, we study the uniform H\"{o}lder continuity of the generalized Riemann function~$R_{\alpha,\beta}$ (with $\alpha>1$ 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~$R_{\alpha,\beta}$ as $\beta$ tends to infinity.

Disciplines :

Mathematics

Author, co-author :

Bastin, Françoise ; Université de Liège - ULiège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes

Nicolay, Samuel ; Université de Liège - ULiège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes

Simons, Laurent ; Université de Liège - ULiège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes

Language :

English

Title :

About the Uniform Hölder Continuity of Generalized Riemann Function

Publication date :

2016

Journal title :

Mediterranean Journal of Mathematics

ISSN :

1660-5446

eISSN :

1660-5454

Publisher :

Springer

Volume :

Issue :

Pages :

101-117

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

http://link.springer.com/article/10.1007/s00009-014-0501-3

Available on ORBi :

since 24 December 2014

Statistics

Number of views

166 (41 by ULiège)

Number of downloads

20 (18 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

Bibliography

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)
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)
Chamizo F., Ubis A.: Multifractal Behavior of Polynomial Fourier Series. Adv. Math. 250, 1–34 (2014)
Chamizo F., Ubis A.: Some Fourier series with gaps. J. Anal. Math. 101, 179–197 (2007)
Daubechies I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)
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)
Gerver J.L.: On Cubic Lacunary Fourier Series. Trans. Am. Math. Soc. 355(11), 4297–4347 (2003)
Gerver J.L.: The Differentiability of the Riemann Function at Certain Rational Multiples of π. Am. J. Math. 92, 33–55 (1970)
Hardy G.H.: Weierstrass’s Non-Differentiable Function. Trans. Am. Math. Soc. 17, 301–325 (1916)
Hardy G.H., Littlewood J.E.: Some Problems of Diophantine Approximation (II). Acta Math. 37(1), 193–239 (1914)
Holschneider M.: Wavelets, an Analysis Tool, Oxford Mathematical Monographs. Oxford Science Publications (1995)
Holschneider M., Tchamitchian P.: Pointwise Analysis of Riemann’s “Nondifferentiable” Function. Invent. Math. 105, 157–175 (1991)
Itatsu, S.: Differentiability of Riemann’s Function. In: Proceedings of the Japan Academy, Series A vol. 57(10), pp. 492–495 (1981)
Jaffard S.: Pointwise and Directional Regularity of Nonharmonic Fourier Series. Appl. Comput. Harmon. Anal. 28, 251–266 (2010)
Jaffard S.: The Spectrum of Singularities of Riemann’s Function. Rev. Math. Iberoam. 12(2), 441–460 (1996)
Jaffard, S., Meyer, Y.: Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions. Mem. Am. Math. Soc. (Book 587) (1996)
Jaffard, S., Meyer, Y., Ryan, R.D.: Wavelets: Tools for Science and Technologies, SIAM, (2001)
Jaffard, S., Nicolay, S.: Space-Filling Functions and Davenport Series. Recent Developments in Fractals and Related Fields, 19–34 (2010)
Johnsen J.: Simple Proofs of Nowhere-Differentiability for Weierstrass’s Function and Cases of Slow Growth. J. Fourier Anal. Appl. 16, 17–33 (2010)
Luther W.: The Differentiability of Fourier Gap Series and “Riemann’s Example” of a Continuous, Nondifferentiable Function. J. Approx. Theory 48, 303–321 (1986)
Mohr E.: Wo ist die Riemannsche Funktion nicht differenzierbar?. Annali di Matematica Pura ed Applicata 123(1), 93–104 (1980)
Queffelec H.: Dérivabilité de certaines sommes de séries de Fourier lacunaires. Comptes rendus de l’Académie des Sciences de Paris 273, 291–293 (1971)
Shidfar A., Sabetfakhri K.: On the Continuity of van der Waerden’s Function in the Hölder Sense. Am. Math. Mon. 93, 375–376 (1986)
Smith A.: The Differentiability of Riemann’s Function. Proc. Am. Math. Soc. 34(2), 463–468 (1972)
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)
Torrésani, B.: Analyse continue par ondelettes, CNRS Editions, Paris (1995)
Tricot, C.: Courbes et dimension fractale. Springer, Berlin (1999)
Ubis, A.: Questions of Arithmetic and Harmonic Analysis, PhD-Thesis, Madrid (2006)
Young, R.: An Introduction to Nonharmonic Fourier Series, Academic Press, San Diego (2001)