Structure of refinable splines

A refinable spline is a compactly supported refinable function that is piecewise polynomial. Refinable splines, such as the well known B-splines, play a key role in computer aided geometric designs. Refinable splines have been studied in several papers, most noticeably in [W. Lawton, S.L. Lee, Z. Shen, Characterization of compactly supported refinable splines, Adv. Comput. Math. 3 (1995) 137–145] for integer dilations and [X. Dai, D.-J. Feng, Y. Wang, Classification of refinable splines, Constr. Approx. 24 (2) (2006) 187–200] for real dilations. There are general characterizations in these papers, but these characterizations are not explicit enough to tell us the structures of refinable splines. In this paper, we give complete characterization of the structure of refinable splines.