Vector-stability of refinable vectors

Abstract

The stability is an expected property for refinable functions, which is widely considered in the study of refinement equations. Instead of studying the stability of entries of refinable vectors, we study the stability of refinable vectors themselves where they are considered as elements of super Hilbert spaces. We call this kind of stability the vector-stability. We give a necessary and sufficient condition for refinable vectors to be vector-stable. We also give an example to illustrate the difference between two types of stability.