Construction of Fractal Bases for Spaces of Functions

  • María A. NavascuésEmail author
  • María V. Sebastián
  • Arya K. B. Chand
  • Saurabh Katiyar
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 655)


The construction of fractal versions of classical functions as polynomials, trigonometric maps, etc. by means of a particular Iterated Function System of the plane is tackled. The closeness between the classical function and its fractal analogue provides good properties of approximation and interpolation to the latter. This type of methodology opens the use of non-smooth and fractal functions in approximation. The procedure involves the definition of an operator mapping standard functions into their dual fractals. The transformation is linear and bounded and some bounds of its norm are established. Through this operator we define families of fractal functions that generalize the classical Schauder systems of Banach spaces and the orthonormal bases of Hilbert spaces. With an appropriate election of the coefficients of Iterated Function System we define sets of fractal maps that span the most important spaces of functions as \({\mathcal {C}}[a,b]\) or \({\mathcal {L}}^p[a,b]\).


Fractal interpolation functions Bases of functional spaces Approximation Interpolation Fractals 

AMS subject classifications:

28A80 41A10 58C05 65D05 26A27. 


  1. 1.
    Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ciesielski, Z.: Hölder conditions for realization of Gaussian processes. Trans A.M.S. 99, 403–413 (1961)zbMATHGoogle Scholar
  3. 3.
    Enflo, P.: A counterexample to the approximation property in Banach spaces. Acta Math. 130, 309–317 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Faber, G.: Ueber die orthogonalfunktionen des Hern Haar Jahresber. Deutsch. Math. Verein. 19, 104–112 (1910)zbMATHGoogle Scholar
  5. 5.
    Johnson, W.B., Rosenthal, H.P., Zippin, M.: On bases, finite dimensional decompositions and weaker structures in Banach spaces. Israel J. Math. 9, 488–506 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lebedev, L.P., Vorovich, I.I., Gladwell, G.M.L.: Functional Analysis. Applications in Mechanics and Inverse Problems, 2nd edn. Kluwer Academic Publishers, Dordrecht (2002)zbMATHGoogle Scholar
  7. 7.
    Navascués, M.A.: Fractal approximation. Complex Anal. Oper. Th. 4(4), 953–974 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Navascués, M.A.: Fractal bases of Lp spaces. Fractals 20(2), 141–148 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Navascués, M.A.: Affine fractal functions as bases of continuous functions. Quaestiones Math. 37, 1–14 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Navascués, M.A., Sebastián, M.V.: Construction of affine fractal functions close to classical interpolants. J. Comp. Anal. Appl. 9(3), 271–283 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, I, Functional Analysis, 2nd edn. Academic Press, New York (1980)zbMATHGoogle Scholar
  12. 12.
    Schauder, J.: Eine eigenschaft des haarschen orthogonalsystems. Math. Z. 28, 317–320 (1928)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  • María A. Navascués
    • 1
    Email author
  • María V. Sebastián
    • 2
  • Arya K. B. Chand
    • 3
  • Saurabh Katiyar
    • 4
  1. 1.Dpto. de Matemática AplicadaEscuela de Ingeniería y Arquitectura, Universidad de ZaragozaZaragozaSpain
  2. 2.Centro Universitario de la Defensa Academia General MilitarZaragozaSpain
  3. 3.Department of MathematicsIIT MadrasChennaiIndia
  4. 4.Department of MathematicsIIT MadrasChennaiIndia

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