On dimension and existence of local bases for multivariate spline spaces

Abstract We consider spaces of splines in k variables of smoothness r and degree d defined on a polytope in R k which has been divided into simplices. Bernstein-Bezier methods are used to develop a framework for analyzing dimension and basis questions. Dimension formulae and local bases are found for the case r = 0 and general k. The main result of the paper shows the existence of local bases for spaces of trivariate splines (where k = 3) whenever d > 8r.

[1]  Walter Whiteley.  The Combinatorics of Bivariate Splines , 1990, Applied Geometry And Discrete Mathematics.

[2]  C. D. Boor,et al.  B-Form Basics. , 1986 .

[3]  Hong Dong,et al.  Spaces of bivariate spline functions over triangulation , 1991 .

[4]  Carl de Boor,et al.  A Local Basis for Certain Smooth Bivariate PP Spaces , 1989 .

[5]  Larry L. Schumaker,et al.  Super spline spaces of smoothnessr and degreed≥3r+2 , 1991 .

[6]  P. Alfeld.  Scattered data interpolation in three or more variables , 1989 .

[7]  Dwight Diener,et al.  Instability in the dimension of spaces of bivariate piecewise polynomials of degree 2 r and smoothness order r , 1990 .

[8]  Larry L. Schumaker,et al.  The dimension of bivariate spline spaces of smoothnessr for degreed≥4r+1 , 1987 .

[9]  Charles L. Lawson,et al.  Properties of n-dimensional triangulations , 1986, Comput. Aided Geom. Des..

[10]  L. R. Scott,et al.  A nodal basis for ¹ piecewise polynomials of degree ≥5 , 1975 .

[11]  Larry L. Schumaker,et al.  On Spaces of Piecewise Polynomials in Two Variables , 1984 .

[12]  Larry L. Schumaker,et al.  Bounds on the dimension of spaces of multivariate piecewise polynomials , 1984 .

[13]  Peter Alfeld,et al.  A trivariate clough-tocher scheme for tetrahedral data , 1984, Comput. Aided Geom. Des..

[14]  Larry L. Schumaker,et al.  Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree d 4r + 1 , 1987, Comput. Aided Geom. Des..

[15]  L. Schumaker.  On the Dimension of Spaces Of Piecewise Polynomials in Two Variables , 1979 .

[16]  Larry L. Schumaker,et al.  Dual bases for spline spaces on cells , 1988, Comput. Aided Geom. Des..

[17]  Peter Alfeld,et al.  A recursion formula for the dimension of super spline spaces of smoothness r and degree d>r2k , 1989 .