Results in Mathematics

, Volume 47, Issue 1–2, pp 93–114 | Cite as

Eigenfunctions of Two-Scale Difference Equations with Dilation Parameter and Infinite Products

  • Manfred Krüppel


This paper deals with two-scale difference equations having an arbitrary dilation parameter and a formal power series as symbol. We investigate the equation concerning the existence of nonzero compactly supported distributional solutions. In order to include also continuous solutions it is advantageous to consider the two-scale difference equation as eigenvalue problem where the solutions are either compactly supported or integrals of compactly supported distributions. Such solutions are called eigenfunctions. As main result we determine the necessary and sufficient condition for the existence of eigenfunctions that the symbol must be a rational function with a special structure depending on the dilation parameter. We show that the eigenfunctions can be expressed by means of a finite sum of shifted eigenfunctions belonging to the case with a polynomial symbol (characteristic polynomial), which is well investigated.

Key words

Two-scale difference equations dilation parameter distributional solutions eigenfunctions rational symbols infinite products with rational factors 

AMS subject classification

39A13 39B32 46F10 44A10 26C15 12D05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. Antosik, J. Mikusiński, R. Sikorski, Theory of Distributions. The Sequential Approach, Elsevier Sc. Pub. Comp., Amsterdam, 1973.Google Scholar
  2. [2]
    P. Auscher, Ondelettes fractales et applications, Ph.D. Thesis, Univ. Paris-Dauphine, 1989.Google Scholar
  3. [3]
    P. Auscher, Wavelet bases for L2(ℝ) with rational dilation factor, in Ruskai et al. (1992) 439–452.Google Scholar
  4. [4]
    L. Berg, M. Krüppel, Eigenfunctions of two-scale difference equations and Appell polynomials, Z. Anal. Anw. 20 (2001) 457–488.MATHGoogle Scholar
  5. [5]
    L. Berg, M. Krüppel, Eigenfunctions of two-scale difference equations with rational symbol, Result.Math. 44 (2003) 226–241.MATHCrossRefGoogle Scholar
  6. [6]
    L. Berg, G. Plonka, Some notes on two-scale difference equations, in: Th.M. Rassias (Ed), Functional Equations and Inequalities, Mathematics and Its Applications 518, Kluwer Acad. Publ., Dordrecht-Boston-London 2000, 7–29.CrossRefGoogle Scholar
  7. [7]
    A.S. Cavaretta, W. Dahmen, C.A. Micchelli, Stationary Subdivision, Mem. Amer. Math. Soc. 453 (1991) 1–186.Google Scholar
  8. [8]
    C.K. Chui, An Introduction to Wavelets, Academic Press, Boston etc., 1992.MATHGoogle Scholar
  9. [9]
    I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.Google Scholar
  10. [10]
    I. Daubechies, J. Lagarias, Two-scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991) 1388–1410.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    G. Deslauriers, S. Dubuc, Interpolation dyadique, in Fractals: dimensions non entières et applications, G. Cherbit, ed., Masson, Paris (1987) 44–55.Google Scholar
  12. [12]
    M. Kuczma, Functional Equations in a Single Variable. (PAN Monografie Mat.: Vol 46). Warsaw: Polish Sci. Publ., 1968.Google Scholar
  13. [13]
    S. Lang, Algebra, Addison-Wesley, Reading Massachusetts, 1971.Google Scholar
  14. [14]
    I.J. Schoenberg, Cardinal Spline Interpolation, SIAM, Philadelphia, 1973.MATHCrossRefGoogle Scholar
  15. [15]
    W. Sierpiński, Sur un système d’équations fonctionelles définissant une fonction avec un ensemble dense d’intervalles d’invariabilité. Bull. Inter. Acad. Sci. Cracovie, Cl. Sci. Math. Nat. Sér. A (1911), 577–582.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität RostockRostockGermany

Personalised recommendations