Characterization of compactly supported refinable splines

Abstract

We prove that a compactly supported spline function φ of degree k satisfies the scaling equation {Mathematical expression} for some integer m ≥ 2, if and only if {Mathematical expression} where p(n) are the coefficients of a polynomial P(z) such that the roots of P(z)(z - 1)k+1 TM are mapped into themselves by the mapping z →zm, and Bk is the uniform B-spline of degree k. Furthermore, the shifts of φ form a Riesz basis if and only if P is a monomial. © 1995 J.C. Baltzer AG, Science Publishers.