Cantor Sets and Integral-Functional Equations

  • Lothar Berg

    Universität Rostock, Germany
  • Manfred Krüppel

    Universität Rostock, Germany


In this paper, we continue our considerations in [1] on a homogeneous integral-functional equation with a parameter a>1a > 1. In the case of a>2a > 2 the solution ϕ\phi satisfies relations containing polynomials. By means of these polynomial relations the solution can explicitly be computed on a Cantor set with Lebesgue measure 1. Thus the representation of the solution ϕ\phi is immediately connected with the exploration of some Cantor sets, the corresponding singular functions of which can be characterized by a system of functional equations depending on aa. In the limit case a=2a = 2 we get a formula for the explicit computation of ϕ\phi in all dyadic points. We also calculate the iterated kernels and approximate ϕ\phi by splines in the general case a>1a > 1.

Cite this article

Lothar Berg, Manfred Krüppel, Cantor Sets and Integral-Functional Equations. Z. Anal. Anwend. 17 (1998), no. 4, pp. 997–1020

DOI 10.4171/ZAA/863