Some prevalent results about strongly monoHölder functions

[en] We study the typical behaviour of strongly monoHölder functions from the prevalence point of view. To this end we first prove wavelet-based criteria for strongly monoHölder functions. We then use the notion of prevalence to show that the functions of $C^\alpha (R^d)$ are almost surely strongly monoHölder with Hölder exponent $\alpha$. Finally, we prove that for any $\alpha\in (0, 1)$ on a prevalent set of $C^\alpha (R^d)$ the Hausdorff dimension of the graph is equal to $d +1-\alpha$.

Disciplines :

Mathematics

Author, co-author :

Clausel, Marianne; Université Paris Est > UMR 8050 du CNRS > Laboratoire d'analyse et de mathématiques appliquées

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

Language :

English

Title :

Some prevalent results about strongly monoHölder functions

Publication date :

2010

Journal title :

Nonlinearity

ISSN :

0951-7715

Publisher :

Institute of Physics

Volume :

Pages :

2101-2116

Peer reviewed :

Peer Reviewed verified by ORBi

Available on ORBi :

since 30 July 2010

Statistics

Number of views

111 (32 by ULiège)

Number of downloads

38 (19 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

OpenAlex citations

publications

supporting

mentioning

contrasting

Smart Citations

Citing PublicationsSupportingMentioningContrasting

View Citations

See how this article has been cited at scite.ai

scite shows how a scientific paper has been cited by providing the context of the citation, a classification describing whether it supports, mentions, or contrasts the cited claim, and a label indicating in which section the citation was made.

Bibliography

Adler R J 1981 The Geometry of Random Fields (New York: Wiley)
Abry P, Lashermes B and Jaffard S 2006 Wavelets Leaders in Multifractal Analysis, Wavelets Analysis and Applications (Applied and Numerical Harmonic Analysis) ed T Qian et al (Basle: Birkauser) pp 219-64
Berry MV and Lewis ZV 1980 On The Weierstrass Mandelbrot fractal function Proc. R. Soc. Lond.A370 459-86
Christensen J 1972 On sets of Haar measure zero in Abelian Polish groups Israel J. Math. 13 255-60
Clausel M 2008 Quelques notions d'irrégularité uniforme et ponctuelle : le point de vue ondelettes PhD Thesis Université Paris XII
Clausel M and Nicolay S 2010Wavelet techniques for pointwise anti-Hölderian irregularity Construct. Approx. submitted
Daubechies I 1988 Orthonormal bases of compactly supported wavelets Commun. Pure Appl. Math. 41 909-96
Daubechies I and Lagarias J 1992 Two scale difference equations II local regularity, infinite products of matrices and fractals SIAM J. Math. Anal. 23 1031-79
Geman D and Horowitz J 1980 Occupation densities Anal. Prob. 8 1-67
DeVore R A and Sharpley R C 1984 Maximal functions measuring smoothness Mem. Am. Math. Soc. 47
Falconer K 1990 Fractal Geometry: Mathematical Foundations and Applications (New York: Wiley)
Heurteaux Y 2003 Weierstrass function with random phases Trans. Am. Math. Soc. 355 3065-77
Hunt B R 1998 The Hausdorff dimension of graphs ofWeierstrass functions Proc. Am. Math. Soc. 126 791-800
Jaffard S 2004 Wavelet techniques in multifractal analysis, fractal geometry and applications Proc. Symp. Pure Math. 72 91-151
Jaffard S and Nicolay S 2009 Pointwise smoothness of space-filling functions Appl. Comput. Harmon. Anal. 26 181-99
Kahane J P 1986 Geza Freud and lacunary Fourier series J. Approx. Theory 46 51-7
Khintchine A 1924 Ein satz der Wahrscheinlichkeitsrechnung Fundam. Math. 6 9-20
Krantz S G 1983 Lipschitz spaces, smoothness of functions, and approximation theory Expo. Math. 1 193-260
Ledrappier F 1992 On the dimension of some graphs Comtemp. Math. 135 285-93
Mallat S 1998 A Wavelet Tour of Signal Processing (New York: Academic Press)
Mandelbrot B B 1977 Fractals: Form, Chance and Dimension (San Francisco: Freeman)
Mc Mullen C 1984 The Hausdorff dimension of general Sierpinki carpets Nagoya Math. J. 96 1-9
Meyer Y 1990 Ondelettes et Opérateurs (Paris: Hermann)
Mauldin R D and Williams S C 1986 On the Hausdorff dimensions of some graphs Trans. Am. Math. Soc. 298 793-803
Roueff F 2003 Almost sure Hausdorff dimension of graphs of random wavelet series J. Fourier Anal. Appl. 9 237-60
Stein E M 1970 Singular Integrals and Differentiabilities Properties of Functions (Princeton, NJ: University Press)
Tricot C 1992 Courbes et Dimension Fractale (Berlin: Springer)
WeissM1959 On the lawof iterated logarithm for lacunary trigonometric series Trans. Am. Math. Soc. 91 444-69

Similar publications

Sorry the service is unavailable at the moment. Please try again later.

Name	Provider / Domaine	Expiration	Description
JSESSIONID	Oracle Corporation www.uliege.be	Session	General purpose platform session cookie, used by sites written in JSP. Usually used to maintain an anonymous user session by the server.
CookieScriptConsent	CookieScript .uliege.be	1 year	This cookie is used by Cookie-Script.com service to remember visitor cookie consent preferences. It is necessary for Cookie-Script.com cookie banner to work properly.

Name	Provider / Domaine	Expiration	Description
_pk_id	InnoCraft Ltd .uliege.be	1 year	Used to store a few details about the user such as the unique visitor ID
_pk_ses	InnoCraft Ltd .uliege.be	30 minutes	Short lived cookies used to temporarily store data for the visit
_pk_ref	InnoCraft Ltd .uliege.be	6 months	Used to store the attribution information, the referrer initially used to visit the website