Cantor sets and integral-functional equations
type: Item
instance of: scholarly article
title: Cantor sets and integral-functional equations
zbMATH Open document ID: 0926.45002
DOI: 10.4171/ZAA/863
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1999-11-21T00:00:00Z
review text: In an earlier paper [ibid. 17, No. 1, 159-181 (1998; Zbl 0897.45004)] the authors have shown that the Fredholm integral equation \[ \phi(t) = (L\phi)(t) = b \int_{at - a + 1}^{at} \phi(\tau) d\tau, \tag{\(*\)} \] where \(a > 1\) and \(b = a/(a-1)\), has a compactly supported solution, which is unique up to a constant scaling factor. Here they primarily investigate the case \(a \geq 2\) closer. In this case the solution is piecewise polynomial. It can be computed recursively, and the authors give explicit formulas for these polynomials. Moreover, for arbitrary \(a > 1\), the iterated kernels of the integral equation \((*)\) are computed, and new spline approximations for the solutions are given.
reviewed by: Olof Staffans
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 65R20
zbMATH DE Number: 1270174
zbMATH Keywords: Cantor sets
zbMATH Keywords: integral-functional equations
zbMATH Keywords: Fredholm integral equation
zbMATH Keywords: iterated kernels
zbMATH Keywords: spline approximation
author: Lothar Berg
author: Manfred Krüppel
MaRDI profile type: Publication
cites work: On the solution of an integral-functional equation with a parameter
cites work: Q4519454
cites work: Q5422500
cites work: Mellin transforms and asymptotics: Digital sums
cites work: Q5585336
cites work: Q5787490
cites work: Q5592596
cites work: Q4358539
cites work: The dynamical system generated by the \(3n+1\) function
cites work: Balls in constrained urns and Cantor-like sets
full work available at URL: https://doi.org/10.4171/zaa/863
OpenAlex ID: W1979442699
description: scientific article; zbMATH DE number 1270174