Cubic Spline Prewavelets on the Four-Directional Mesh

In this paper,wedesign differentiable, two-dimensional, piecewise polynomial cubic prewavelets of particularly small compact support. They are given in closed form, and provide stable, orthogonal decompositions of L(R). In particular, the splines we use in our prewavelet constructions give rise to stable bases of spline spaces that contain all cubic polynomials, whereas the more familiar box spline constructions cannot reproduce all cubic polynomials, unless resorting to a box spline of higher polynomial degree. Date received: January 5, 2002. Final version received: October 29, 2002. Date accepted: November 19, 2002. Communicated by Ronald A. DeVore. Online publication January 17, 2003. AMS classification: 41A15, 41A63, 65D07, 65D15, 65T60. OF2 M. D. Buhmann, O. Davydov, and T. N. T. Goodman