Characterization of compactly supported refinable splines whose shifts form a Riesz basis (Q1772704)

The authors prove that a compactly supported \(m\)-refinable blockwise polynomial \(\phi\) has stable shifts, i.e., \(\{\phi(\cdot-k): k\in {\mathbb Z}^s\}\) is a Riesz basis, if and only if \(\phi(x)=c B(x-n-l/(m-1)| v_1, \dots, v_k)\) for some \(c\neq 0\), \(n , l\in {\mathbb Z}^s\), and \(B(x|v_1, \dots, v_k)\) is a multivariate box spline with the matrix \((v_1, \dots, v_k)\) being unimodular. This work is based on the characterization of a compactly supported \(m\)-refinable blockwise polynomial by \textit{Q. Sun} [J. Approximation Theory 86, No.~2, 240--252 (1996; Zbl 0870.42008)].