Asymptotic properties of BB-splines, Eulerian numbers and cube slicing

In this paper, we clarify the connections among BB-splines, Eulerian numbers and cube slicing. We first show that the asymptotic formula for Eulerian numbers can be considered as a special case of the asymptotic properties of BB-splines. The volume of cube slicing can be considered as a value of box spline functions. Based on the connection, a very simple proof for Laplace and Pólya’s formulas, which were settled by probabilistic methods, is given by spline theory.