# Refinement type equations: sources and results

• Volume: 99, Issue: 1, page 87-110
• ISSN: 0137-6934

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## Abstract

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It has been proved recently that the two-direction refinement equation of the form $f\left(x\right)={\sum }_{n\in }{c}_{n,1}f\left(kx-n\right)+{\sum }_{n\in ℤ}{c}_{n,-1}f\left(-kx-n\right)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f\left(x\right)={\sum }_{n\in ℤ}cₙf\left(kx-n\right)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f\left(x\right)={\int }_{ℝ}c\left(y\right)f\left(kx-y\right)dy$ has also various interesting applications. This equation is a special case of the continuous refinement type equation of the form $f\left(x\right)={\int }_{\Omega }|K\left(\omega \right)|f\left(K\left(\omega \right)x-L\left(\omega \right)\right)dP\left(\omega \right)$, which has been studied recently in connection with probability theory. The purpose of this paper is to give a survey on the above refinement type equations. We begin with a brief introduction of types of refinement equations. In the first part we present several problems from different areas of mathematics which lead to the problem of the existence/nonexistence of integrable solutions of refinement type equations. In the second part we discuss and collect recent results on integrable solutions of refinement type equations, including some necessary and sufficient conditions for the existence/nonexistence of integrable solutions of the two-direction refinement equation. Finally, we say a few words on the existence of extremely non-measurable solutions of the two-direction refinement equation.

## How to cite

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Rafał Kapica, and Janusz Morawiec. "Refinement type equations: sources and results." Banach Center Publications 99.1 (2013): 87-110. <http://eudml.org/doc/282085>.

@article{RafałKapica2013,
abstract = {It has been proved recently that the two-direction refinement equation of the form $f(x) = ∑_\{n∈ \}c_\{n,1\}f(kx-n) + ∑_\{n∈ℤ\}c_\{n,-1\}f(-kx-n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f(x) = ∑_\{n∈ℤ\}cₙf(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f(x) = ∫_\{ℝ\}c(y)f(kx-y)dy$ has also various interesting applications. This equation is a special case of the continuous refinement type equation of the form $f(x) = ∫_\{Ω\} |K(ω)| f(K(ω)x-L(ω)) dP(ω)$, which has been studied recently in connection with probability theory. The purpose of this paper is to give a survey on the above refinement type equations. We begin with a brief introduction of types of refinement equations. In the first part we present several problems from different areas of mathematics which lead to the problem of the existence/nonexistence of integrable solutions of refinement type equations. In the second part we discuss and collect recent results on integrable solutions of refinement type equations, including some necessary and sufficient conditions for the existence/nonexistence of integrable solutions of the two-direction refinement equation. Finally, we say a few words on the existence of extremely non-measurable solutions of the two-direction refinement equation.},
author = {Rafał Kapica, Janusz Morawiec},
journal = {Banach Center Publications},
keywords = {refinement equations; distributional solutions; -solutions; extremely non-measurable solutions; Fourier transforms; Foias operators; self-similar measures; perpetuities; random affine maps; distributional fixed points; refinable splines; wavelets; special functions},
language = {eng},
number = {1},
pages = {87-110},
title = {Refinement type equations: sources and results},
url = {http://eudml.org/doc/282085},
volume = {99},
year = {2013},
}

TY - JOUR
AU - Rafał Kapica
AU - Janusz Morawiec
TI - Refinement type equations: sources and results
JO - Banach Center Publications
PY - 2013
VL - 99
IS - 1
SP - 87
EP - 110
AB - It has been proved recently that the two-direction refinement equation of the form $f(x) = ∑_{n∈ }c_{n,1}f(kx-n) + ∑_{n∈ℤ}c_{n,-1}f(-kx-n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f(x) = ∑_{n∈ℤ}cₙf(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f(x) = ∫_{ℝ}c(y)f(kx-y)dy$ has also various interesting applications. This equation is a special case of the continuous refinement type equation of the form $f(x) = ∫_{Ω} |K(ω)| f(K(ω)x-L(ω)) dP(ω)$, which has been studied recently in connection with probability theory. The purpose of this paper is to give a survey on the above refinement type equations. We begin with a brief introduction of types of refinement equations. In the first part we present several problems from different areas of mathematics which lead to the problem of the existence/nonexistence of integrable solutions of refinement type equations. In the second part we discuss and collect recent results on integrable solutions of refinement type equations, including some necessary and sufficient conditions for the existence/nonexistence of integrable solutions of the two-direction refinement equation. Finally, we say a few words on the existence of extremely non-measurable solutions of the two-direction refinement equation.
LA - eng
KW - refinement equations; distributional solutions; -solutions; extremely non-measurable solutions; Fourier transforms; Foias operators; self-similar measures; perpetuities; random affine maps; distributional fixed points; refinable splines; wavelets; special functions
UR - http://eudml.org/doc/282085
ER -

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