Eigenvalues of scaling operators and a characterization of $B$-splines

A finitely supported sequence $a$ that sums to $2$ defines a scaling operator $T_a f = \sum_{k\in \mathbb Z} a(k)f(2 \cdot -k)$ on functions $f,$ a transition operator $S_a v = \sum_{k\in \mathbb Z} a(k) (2 \cdot -k)$ on sequences $v,$ and a unique compactly supported scaling function $\phi$ that satisfies $\phi = T_a \phi$normalized with $\widehat \phi (0) = 1.$ 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 finitely supported sequences, and that the two sets of eigenvalues are equal if and only if the corresponding scaling function $\phi$is a uniform $B$-spline.