Existence results for refinement equations
type: Item
instance of: scholarly article
title: Existence results for refinement equations
zbMATH Open document ID: 0945.39012
DOI: 10.1007/PL00000125
author: Anca Deliu
author: M. C. Spruill
published in: Aequationes Mathematicae
publication date: 2000-10-08T00:00:00Z
review text: The authors consider the functional equation \[ f(x)= \sum^N_{n=0} c_nf(kx-n), \quad c_n\geq 0,\;\sum^N_{n=0}c_n=k, \] and seek simple conditions on \(c_n\) under which this equation has nontrivial \(L_1\)-solutions. They introduce the random expression \[ Y=\sum_{n\geq 1} C_nk^{-n} \] where \(C_1, C_2,\dots\) are independent identically distributed random variables distributed as \(C\), where \(P[C=j] =c_j/k\), \(j=0,1, \dots,\mathbb{N}\). The cumulative distribution function \(F\) of \(Y\) satisfies \[ F(x)= {1\over k}\sum^N_{n=0} c_i F(kx-n). \] The authors show that there is a solution to the considered functional equation if and only if the measure whose cumulative distribution function \(F\) is absolutely continuous with respect to Lebesgue measure, and in that case, \(f\) can be taken as the probability density of \(Y\). ``Utilizing this characterization a simple description in terms of the coefficients is given of all refinement equations with no more than six coefficients possessing an integrable solution''.
reviewed by: Borislav Crstici
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 26A30
Mathematics Subject Classification ID: 26A46
zbMATH DE Number: 1436162
zbMATH Keywords: dilation equation
zbMATH Keywords: characteristic function
zbMATH Keywords: functional equation
zbMATH Keywords: cumulative distribution function
zbMATH Keywords: refinement equations
zbMATH Keywords: integrable solution
MaRDI profile type: Publication
full work available at URL: pl00000125
OpenAlex ID: W2012089791
description: scientific article; zbMATH DE number 1436162