Eigenvalues of scaling operators and a characterization of B-splines

A finitely supported sequence a that sums to 2 defines a scaling operator T(a)f = Sigma(k is an element of Z) a(k)f(2a-k) on functions f, a transition operator S(a)v = Sigma(k is an element of Z) a(k)(2 center dot- k) on sequences v, and a unique compactly supported scaling function phi that satisies phi = T-a phi normalized with (phi) over cap (0) = 1. It is shown that the eigenvalues of T-a on the space of compactly supported square- integrable functions are a subset of the nonzero eigenvalues of the transition operator S-a on the space of. nitely supported sequences, and that the two sets of eigenvalues are equal if and only if the corresponding scaling function f is a uniform B- spline.