Summary
We study the fractal dimensions of continuous function graphs and more general fractal parameters. They are all obtained from the L p-norms of some well-built operators. We give general results about these norms in the continuous and the discrete cases. For a function that is uniformly Hölderian, they allow us to estimate in a very easy way a large family of dimensional indices, like the box dimension and regularization dimension.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
F. Chamizo and A. Córdoba. The fractal dimension of a family of Riemann’s graphs. C.R. Acad. Sci. Paris. 317 (1993) 455–460.
Y. Demichel. Analyse fractale d’une famille de fonctions aléatoires : les fonctions de bosses. Ph.D. Thesis. (2006).
Y. Demichel and K.J. Falconer. The Hausdorff dimension of pulse sum graphs. Math. Proc. Camb. Phil. Soc. 142 (2007) 145–155.
Y. Demichel and C. Tricot. Analysis of the fractal sum of pulses. Math. Proc. Camb. Phil. Soc. 141 (2006) 355–370.
S. Dubuc and C. Tricot. Variation d’une fonction et dimension de son graphe. C.R. Acad. Sci. Paris. 306 (1988) 531–533.
K.J. Falconer. Fractal Geometry, Mathematical Foundations and Applications. second edition (Wiley, New York, 2003).
G.H. Hardy. Weierstrass’s non-differentiable function. Trans. Amer. Math. Soc. 17 (1916) 301–325.
Y. Heurteaux. Weierstrass functions with random phases. Trans. Am. Math. Soc. 355 (2003) 3065–3077.
J. Istas and G. Lang. Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré, Prob. Stat. 33(4) (1997) 407–436.
S. Jaffard. The spectrum of singularities of Riemann’s function. Rev. Mat. Iber. 12 (1996) 441–460.
S. Jaffard. Multifractal formalism for functions. SIAM J. Math. Anal. 24 (1997) 944–970.
S. Jaffard. Old friends revisited: the multifractal nature of some classical functions. J. Fourier Anal. Appl. 3-1 (1997) 1–22.
F. Roueff and J. Lévy Véhel. A regularization approach to fractional dimension estimation. Proc. Fractals’ 98, Malta (1998).
C. Tricot. Function norms and fractal dimension. SIAM J. Math. Anal. 28 (1997) 189–212.
C. Tricot. Courbes et dimension fractale. Seconde édition (Springer, Berlin, 1999).
K. Weierstrass. On continuous functions of a real argument that do not have a well-defined differential quotient. Mathematische Werke, Vol. II, (Mayer and Müller, Berlin, 1895) 71–74.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Demichel, Y. (2010). L p-Norms and Fractal Dimensions of Continuous Function Graphs. In: Barral, J., Seuret, S. (eds) Recent Developments in Fractals and Related Fields. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4888-6_10
Download citation
DOI: https://doi.org/10.1007/978-0-8176-4888-6_10
Published:
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4887-9
Online ISBN: 978-0-8176-4888-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)