Prewavelets with Splines
by
Martin Buhmann
University of Giessen, Germany

We design continuously differentiable, twodimensional, piecewise polynomial cubic prewavelets of particularly small compact support. They are given in closed form, and provide stable, orthogonal decompositions of L²(R²). 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. Prewavelets are at the core of modern methods for numerical analysis due to their usefulness for numerical PDE solvers, as well as for signal processing, data compression and transmission, and other applications of harmonic analysis.

Date received: February 6, 2002