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Zeitschrift für Analysis und ihre Anwendungen


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Volume 17, Issue 4, 1998, pp. 997–1020
DOI: 10.4171/ZAA/863

Published online: 1998-12-31

Cantor Sets and Integral-Functional Equations

Lothar Berg[1] and Manfred Krüppel[2]

(1) Universität Rostock, Germany
(2) Universität Rostock, Germany

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 $a > 1$.

Keywords: Integral-functional equations, generating functions, Cantor sets, singular functions, relations containing polynomials, iterated kernels, approximation by splines

Berg Lothar, Krüppel Manfred: Cantor Sets and Integral-Functional Equations. Z. Anal. Anwend. 17 (1998), 997-1020. doi: 10.4171/ZAA/863