Characterization of compactly supported refinable splines whose shifts form a Riesz basis

Based on (J. Approx. Theory 86 (1996) 240), we prove that the integer shifts of a multivariate blockwise polynomial φ(x) which is compactly supported and m-refinable form a Riesz basis if and only if φ(x)=cB(x-n-lm-1|v1,v2,…,vk). Here n,l∈Zs, c≠0 is a constant, B(x|v1,v2,…,vk) is a multivariate box spline and the matrix (v1,v2,…,vk) is unimodular.