Refinement type equations: sources and results
Volume 99 / 2013
Abstract
It has been proved recently that the two-direction refinement equation of the form \[ f(x)=\sum_{n\in\mathbb Z}c_{n,1}f(kx-n)+\sum_{n\in\mathbb Z}c_{n,-1}f(-kx-n) \] 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 $f(x)=\sum_{n\in\mathbb Z}c_nf(kx-n)$, which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f(x)=\int_{\mathbb R}c(y)f(kx-y)\,dy$ has also various interesting applications. This equation is a special case of the continuous refinement type equation of the form \[ f(x)=\int_{\Omega}|K(\omega)|f(K(\omega)x-L(\omega))\,dP(\omega), \] 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.
