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Journal of Fourier Analysis and Applications

, Volume 18, Issue 2, pp 386–409 | Cite as

On the Hausdorff Dimension of Continuous Functions Belonging to Hölder and Besov Spaces on Fractal d-Sets

  • Abel Carvalho
  • António CaetanoEmail author
Article

Abstract

The Hausdorff dimension of the graphs of the functions in Hölder and Besov spaces (in this case with integrability p≥1) on fractal d-sets is studied. Denoting by s∈(0,1] the smoothness parameter, the sharp upper bound min{d+1−s,d/s} is obtained. In particular, when passing from ds to d<s there is a change of behaviour from d+1−s to d/s which implies that even highly nonsmooth functions defined on cubes in ℝ n have not so rough graphs when restricted to, say, rarefied fractals.

Keywords

Hausdorff dimension Box counting dimension Fractals d-Sets Continuous functions Weierstrass function Hölder spaces Besov spaces Wavelets 

Mathematics Subject Classification (2000)

26A16 26B35 28A78 28A80 42C40 46E35 

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References

  1. 1.
    Bricchi, M.: Tailored function spaces and related h-sets. PhD thesis, Friedrich-Schiller-Universität Jena (2001) Google Scholar
  2. 2.
    Carvalho, A.: Box dimension, oscillation and smoothness in function spaces. J. Funct. Spaces Appl. 3(3), 287–320 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Deliu, A., Jawerth, B.: Geometrical dimension versus smoothness. Constr. Approx. 8, 211–222 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Falconer, K.J.: Fractal Geometry. Wiley, Chichester (1990) zbMATHGoogle Scholar
  5. 5.
    Hunt, B.: The Hausdorff dimension of graphs of Weierstrass functions. Proc. Am. Math. Soc. 126, 791–800 (1998) zbMATHCrossRefGoogle Scholar
  6. 6.
    Jonsson, A., Wallin, H.: Function Spaces on Subsets of ℝn. Math. Reports, vol. 2. Harwood Academic, Reading (1984) Google Scholar
  7. 7.
    Kahane, J.-P.: Some Random Series of Functions, 2nd edn. Cambridge University Press, Cambridge (1985) zbMATHGoogle Scholar
  8. 8.
    Kamont, A., Wolnik, B.: Wavelet expansions and fractal dimensions. Constr. Approx. 15(1), 97–108 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Moura, S.: Function spaces of generalised smoothness. Diss. Math. 398, 88 (2001) MathSciNetGoogle Scholar
  10. 10.
    Moura, S.: Function spaces of generalised smoothness, entropy numbers, applications. PhD thesis, University of Coimbra (2001) Google Scholar
  11. 11.
    Roueff, F.: Dimension de Hausdorff du graphe d’une fonction continue: une étude analytique et statistique. PhD thesis, Ecole Nat. Supér. Télécom. (2000) Google Scholar
  12. 12.
    Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992) zbMATHCrossRefGoogle Scholar
  13. 13.
    Triebel, H.: Fractals and Spectra. Birkhäuser, Basel (1997) zbMATHCrossRefGoogle Scholar
  14. 14.
    Triebel, H.: Theory of Function Spaces III. Birkhäuser, Basel (2006) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Centro I&D Matemática e AplicaçõesUniversidade de AveiroAveiroPortugal
  2. 2.Centro I&D Matemática e Aplicações, Departamento de MatemáticaUniversidade de AveiroAveiroPortugal

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