Cantor Sets and Integral-Functional Equations

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 $\varphi$ 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 $\varphi$ 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 $\varphi$ in all dyadic points. We also calculate the iterated kernels and approximate $\varphi$ by splines in the general case $a>1$.