# Cantor Sets and Integral-Functional Equations

### Lothar Berg

Universität Rostock, Germany### Manfred Krüppel

Universität Rostock, Germany

## Abstract

In this paper, we continue our considerations in [1] on a homogeneous integral-functional equation with a parameter $a>1$. In the case of $a>2$ the solution $ϕ$ 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 $ϕ$ 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 $a$. In the limit case $a=2$ we get a formula for the explicit computation of $ϕ$ in all dyadic points. We also calculate the iterated kernels and approximate $ϕ$ by splines in the general case $a>1$.

## 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