Cubic Spline Prewavelets on the Four-Directional Mesh

Abstract.Dedicated to Professor M. J. D. Powell on the occasionof his sixty-fifth birthday and his retirement. In this paper, we design differentiable, two-dimensional, 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.

[1]  Ming-Jun Lai,et al.  Approximation order from bivariate C1-cubics on a four-directional mesh is full , 1994, Comput. Aided Geom. Des..

[2]  Rob Stevenson.  Piecewise linear (pre-)wavelets on non-uniform meshes , 1998 .

[3]  Zuowei Shen,et al.  Compactly supported tight affine spline frames in L2(Rd) , 1998, Math. Comput..

[4]  Wolfgang Dahmen,et al.  C 1 -hierarchical bases , 1994 .

[5]  P. Oswald,et al.  Hierarchical conforming finite element methods for the biharmonic equation , 1992 .

[6]  Malcolm A. Sabin,et al.  Piecewise Quadratic Approximations on Triangles , 1977, TOMS.

[7]  P. G. Ciarlet,et al.  Basic error estimates for elliptic problems , 1991 .

[8]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[9]  C. Micchelli,et al.  On multivariate -splines , 1989 .

[10]  Charles K. Chui,et al.  Wavelets and Frames on the Four-Directional Mesh , 1994 .

[11]  Wolfgang Dahmen,et al.  Element-by-Element Construction of Wavelets Satisfying Stability and Moment Conditions , 1999, SIAM J. Numer. Anal..

[12]  Costanza Conti,et al.  A new subdivision method for bivariate splines on the four-directional mesh , 2000 .

[13]  Hans-Peter Seidel,et al.  Multiresolution analysis over triangles, based on quadratic Hermite interpolation , 2000 .

[14]  R. DeVore,et al.  On the construction of multivariate (pre)wavelets , 1993 .

[15]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[16]  Michael S. Floater,et al.  Piecewise linear prewavelets on arbitrary triangulations , 1999, Numerische Mathematik.

[17]  C. Micchelli,et al.  Spline prewavelets for non-uniform knots , 1992 .

[18]  Oleg Davydov,et al.  Stable Local Bases for Multivariate Spline Spaces , 2001, J. Approx. Theory.

[19]  C. D. Boor,et al.  Box splines , 1993 .

[20]  Juan Manuel Peña,et al.  Least supported bases and local linear independence , 1994 .

[21]  Harry Yserentant,et al.  Hierarchical bases , 1992 .

[22]  Tom Lyche,et al.  Theory and Algorithms for Non-Uniform Spline Wavelets , 2001 .

[23]  Martin Greiner,et al.  Wavelets , 2018, Complex..

[24]  Wolfgang Dahmen,et al.  Translates of multivarlate splines , 1983 .

[25]  P. Zwart.  Multivariate Splines with Nondegenerate Partitions , 1973 .

[26]  E. H. Mansfield.  The Bending and Stretching of Plates , 1963 .