Connections between the Support and Linear Independence of Refinable Distributions Original Research Article

The purpose of this paper is to study the relationships between the support of a refinable distributionφand the global and local linear independence of the integer translates ofφ. It has been shown elsewhere that a compactly supported distributionφhas globally independent integer translates if and only ifφhas minimal convex support. However, such a distribution may have “holes” in its support. By insisting thatφ∈L2(R) and generates a multiresolution analysis, Lemarié and Malgouyres have ensured that no such holes can occur. In this article we generalize this result to refinable distributions. We also give a result on the local linear independence of the integer translates ofφ. We work with integer dilation factorN⩾2 throughout this paper.