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. In this paper, we start with an example showing that a compactly supported componentwise polynomial solution of a refinement equation may not coincide globally with its compactly supported distribution solution in general. However, this is not the case when the dilation factor is 2. In fact, we show that any compactly supported componentwise polynomial solution of a refinement equation with the dilation factor 2 can be globally identified with its compactly supported distribution solution as a consequence of a general result. As in [2], a componentwise polynomial is defined as follows: Definition 1. A compactly supported function φ defined on R is a componentwise polynomial if there exists an open set G such that the Lebesgue measure of R\G is zero and the restriction of φ on any connected open component of G coincides with some polynomial. It is clear that a compactly supported spline is a componentwise polynomial, since the open set G in Definition 1 is a union of finitely many connected open intervals. A componentwise polynomial has an analytic expression up to a set of measure zero, since it is a polynomial on each connected component of G. The concept of componentwise polynomials was first introduced and studied in [1, 21] under the term of local polynomials. In [2], a few examples of compactly supported componentwise polynomial refinable functions are given. In particular, examples of componentwise constant refinable functions, which satisfy either orthogonality or interpolation property and which are continuous and symmetric, are given in [2]. Additional examples of componentwise linear refinable functions that are differentiable and symmetric are also given in [2]. Next, we present another example of a componentwise constant that is a compactly supported measurable function solution of a refinement equation but it cannot be regarded globally as a compactly supported distribution solution of the same refinement equation. 2000 Mathematics Subject Classification. 42C40.