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 $\phi$ 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 $\phi$ 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 $a$. In the limit case $a = 2$ we get a formula for the explicit computation of $\phi$ in all dyadic points. We also calculate the iterated kernels and approximate $\phi$ by splines in the general case $a > 1$.