Abstract: The purpose of this paper is to investigate the characterization of a compactly supported spline function φdefined on simplex in Rⁿ. We give the equivalent condition of φbeing M-refinable, where M. the dilation matrix of the refinement function, is an n×ninteger matrix with the norms of all its eigenvalues greaterthan 1. Furthermore, we give a necessary and sufficient condition of the shifts of φ forming Riesz basis.

Key words: compact support, M-refinable, spline funcion, Riesz basis