Refinement equations with nonnegative coefficients
type: Item
instance of: scholarly article
title: Refinement equations with nonnegative coefficients
zbMATH Open document ID: 0974.42023
DOI: 10.1007/BF02510118
published in: The Journal of Fourier Analysis and Applications
publication date: 2001-12-02T00:00:00Z
full work available at URL: https://eudml.org/doc/59631
review text: In this paper the author analyzes the \(n\)-scale refinement equation \(\Phi(x)= \sum_k p_k \Phi(nx-k)\), where \(n\geq 2\) is an integer, the coefficients \(\{ p_k\}\) are nonnegative, \(\sum_k p_k=1\) and \(\sum_k |k|p_k<\infty\). Under these conditions, a unique solution (up to normalization and addition of a constant) with bounded variation exists, it is real-valued, nondecreasing and either absolutely continuous or purely singular continuous. To decide the continuity type, a tree of order \(n\) is constructed and a sharp criterion involving the zeros of the associated mask \(m_0(\xi)=\sum_k p_k e^{-ik\xi}\) is obtained. The same criterion is applied to the problem of absolute continuity of the distribution function of some random series. Furthermore, the author obtains a complete classification of refinement equations with positive coefficients (in the case of finitely many nonzero \(p_k\)'s) with respect to the existence of continuous or compactly supported solutions.
reviewed by: Radu Balan
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42A32
zbMATH DE Number: 1445626
zbMATH Keywords: refinement equation
zbMATH Keywords: Fourier series
zbMATH Keywords: tree
zbMATH Keywords: random series
author: Vladimir Yu. Protasov
MaRDI profile type: Publication
cites work: Bounded semigroups of matrices
cites work: Stationary subdivision
cites work: A stability criterion for biorthogonal wavelet bases and their related subband coding scheme
cites work: Characterizations of Scaling Functions: Continuous Solutions
cites work: Orthonormal bases of compactly supported wavelets
cites work: Two-Scale Difference Equations. I. Existence and Global Regularity of Solutions
cites work: Two-Scale Difference Equations II. Local Regularity, Infinite Products of Matrices and Fractals
cites work: Probabilistic method for a class of functional-differential equations
cites work: Generalized refinement equations and subdivision processes
cites work: Symmetric iterative interpolation processes
cites work: A 4-point interpolatory subdivision scheme for curve design
cites work: Analysis of uniform binary subdivision schemes for curve design
cites work: Spatially chaotic configurations and functional equations with rescaling
cites work: Arithmetic Properties of Bernoulli Convolutions
cites work: The finiteness conjecture for the generalized spectral radius of a set of matrices
cites work: On Some Sharp Regularity Estimations of $L^2 $-Scaling Functions
cites work: Characterization of $L^p $-Solutions for the Two-Scale Dilation Equations
cites work: Uniform refinement of curves
cites work: Absolute continuity of Bernoulli convolutions, a simple proof
cites work: Q4396587
cites work: The generalized joint spectral radius. A geometric approach
cites work: A Complete Solution Characterizing Smooth Refinable Functions
cites work: Q3272731
cites work: Q5332701
cites work: Q3895980
cites work: On the random series \(\sum\pm\lambda^ n\) (an Erdös problem)
cites work: Q5687392
cites work: Q3128965
cites work: Q4176223
description: scientific article; zbMATH DE number 1445626