Componentwise polynomial solutions and distribution solutions of refinement equations

In this paper, we present an example of a refinement equation such that up to a multiplicative constant it has a unique compactly supported distribution solution while it can simultaneously have a compactly supported componentwise constant function solution that is not locally integrable. This leads to the conclusion that in general the componentwise polynomial solution cannot be globally identified with the unique compactly supported distribution solution of the same refinement equation. We further show that any compactly supported componentwise polynomial solution to a given refinement equation with the dilation factor 2 must coincide, after a proper normalization, with the unique compactly supported distribution solution to the same refinement equation. This is a direct consequence of a general result stating that any compactly supported componentwise polynomial refinable function with the dilation factor 2, without assuming that the refinable function is locally integrable in advance, must be a finite linear combination of the integer shifts of some B-spline.