Classification of Refinable Splines in ℝd

Abstract

A refinable spline in ℝ^d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ^d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374-381, 2007), Lawton et al. (Comput. Math. 3, 137-145, 1995). In higher dimensions Sun (J. Approx. Theory 86, 240-252, 1996) characterized those splines when the dilation matrices are of the form A = mI, where m ∈ ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ^d for arbitrary dilation matrices A ∈ M<inf>d</inf>(ℤ). © 2008 Springer Science+Business Media, LLC.