Number systems, α-splines and refinement

This paper is concerned with the smooth refinable function on a plane relative with complex scaling factor α∈Q[i]⊂C. Characteristic functions of certain self-affine tiles related to a given scaling factor are the simplest examples of such refinable function. We study the smooth refinable functions obtained by a convolution power of such charactericstic functions. Dahlke, Dahmen, and Latour obtained some explicit estimates for the smoothness of the resulting convolution products. In the case α=1+i, we prove better results. We introduce α-splines in two variables which are the linear combination of shifted basic functions. We derive basic properties of α-splines and proceed with a detailed presentation of refinement methods. We illustrate the application of α-splines to subdivision with several examples. It turns out that α-splines produce well-known subdivision algorithms which are based on box splines: Doo-Sabin, Catmull-Clark, Loop, Midedge and some 2,3-subdivision schemes with good continuity. The main geometric ingredient in the definition of α-splines is the fundamental domain F (a fractal set or a self-affine tile). The properties of the fractal F obtained in number theory are important and necessary in order to determine two basic properties of α-splines: partition of unity and the refinement equation.