Characterization of compactly supported refinable splines

We prove that a compactly supported spline function of degree k satisses the scaling equation (x) = P N n=0 c(n)(mx?n) for some integer m 2, if and only if (x) = P n p(n)B k (x?n) where p(n) are the coeecients of a polynomial P(z) such that the roots of P(z)(z ? 1) k+1 are mapped into themselves by the mapping z ! z m , and B k is the uniform B-spline of degree k. Furthermore, the shifts of form a Riesz basis if and only if P is a monomial.