Eigenvalues of Scaling Operators and a Characterization of B-splines

A finitely supported sequence a that sums to 2 defines a scaling operator Taf = ∑ k∈Z a(k)f(2 ·−k) on functions f, a transition operator Sav = ∑ k∈Z a(k)(2 ·−k) on sequences v, and a unique compactly supported scaling function φ that satisfies φ = Taφ normalized with φ̂(0) = 1. It is shown that the eigenvalues of Ta on the space of compactly supported square-integrable functions are a subset of the nonzero eigenvalues of the transition operator Sa on the space of finitely supported sequences, and that the two sets of eigenvalues are equal if and only if the corresponding scaling function φ is a uniform B-spline. Proceeding of the American Mathematical Society, 134(2006), 1051-1057.