Peter Alfeld Department of Mathematics College of Science University of Utah
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This is an annotated and chronologically ordered list of research papers that pertain directly to the subject of these pages, i.e., multivariate splines defined on triangulations, the Bernsteinézier form, and minimal determining sets. It does not include literature on other notions of splines, splines on special triangulations, approximation order, subdivision schemes, parametric splines, and a host of other topics. Nor does it include expository or survey articles.

Click here for a vast on-line and searchable bibliography on approximation theory in general, maintained by Carl de Boor and Larry Schumaker.

In the discussions below, S is the spline space of functions that are globally r times differentiable and on each triangle can be represented as a polynomial of degree d.

  1. 1973. G. Strang, Piecewise polynomials and the finite element method, Bull. Amer. Math. Soc. 79, 1128--1137. Strang made a conjecture on the dimension of S for the case of C1 cubics (r=1, d=3) that turned out to be wrong but started the whole subject.

  2. 1975. Morgan, J., and R. Scott, A Nodal Basis for C1 Piecewise Polynomials of Degree n >=5 , Math. Comp. 29, 736-740. In this fundamental paper Morgan and Scott settle the dimension of S for r=1 and d>=5. The Bernstein-Bézier form was unknown at the time but much later progress depended substantially on translating this paper into the language of the Bernstein-Bézier form.

  3. 1979. Schumaker, L.L., Lower bounds for the dimension of spaces of piecewise polynomials in two variables, in W. Schempp and Zeller, K. (ed.), Multivariate Approximation Theory, Birkhauser Verlag, 396--412. This paper gives very general lower bounds on the dimension of S. Using the Bernstein-Bézier form it is easy to find a determining set which gives an upper bound on the dimension. In later work, for values of d sufficiently much larger than r, those upper bounds were shown to equal Schumaker's lower bounds, thus establishing the exact dimension.

  4. 1979. Farin, G., Subsplines ueber Dreiecken, Dissertation, Braunschweig, Germany. Farin pioneered the use of the Bernstein-Bézier form. A revision of his thesis later appeared under the title Bézier polynomials over triangles and the construction of piecewise Cr-polynomials as report TR/92, Dept. Mathematics, Brunel University, Uxbridge, Middlesex, UK, 1980.

  5. 1984. Schumaker, L.L., Bounds on the dimension of spaces of multivariate piecewise polynomials, Rocky Mountain J. of Mathematics 14, 251-264. This augments the previous paper by giving (disparate) upper bounds on the dimension, and extended the upper and lower bounds to rectilinear partitions (which are more general than triangulations.

  6. 1987. Alfeld, P. and Schumaker, L.L., The dimension of bivariate spline spaces of smoothness r for degree d >=4r+1, Constructive Approximation 3, 189--197 We introduced the concept of a minimal determining set (although it is called an annihilating set in this paper.) The construction is explicit, except that only the number of points (rather than their precise selection) is established in the 2r disks around interior vertices of the triangulation.

  7. 1987. Alfeld, P., Piper, B., and Schumaker, L.L., Minimally Supported Bases for Spaces of Bivariate Piecewise Polynomials of Smoothness r and Degree d >=4r+1$, Computer Aided Geometric Design 4, 105--124 This augments the preceding paper by specifying the points in the 2r disks around interior vertices, but only for r=1,2,3.

  8. 1987. Alfeld, P., Piper, B., and Schumaker, L.L., An Explicit Basis for C1 Quartic Bivariate Splines, SIAM J. Num.Anal. 24, 891--911 We specify an explicit minimal determining set in the case r=1 and d=4, and in the process establish that for r>3 singular vertices are the only configurations that cause non-generic dimensions.

  9. 1988. Billera, L.,, Homology of smooth splines: generic triangulations and a conjecture by Strang, Trans. A.M.S. 310, 325--340. Using a sophisticated body of machinery Billera derives a linear system that describes C1 splines.

  10. 1988. Schumaker, L.L., Dual bases for spline spaces on cells, Computer Aided Geom. Design 5, 277--284. Schumaker gives an explicit minimal determining set for spline spaces defined on the star of a vertex, for all values of d and r.

  11. 1990. Alfeld, P., and Schumaker, L.L., 1990, On the Dimension of Bivariate Spline Spaces of Smoothness r and Degree d=3r+1, Numer. Math. 57, 651-661 We specify explicit minimal determining sets essentially in the generic case.

  12. 1991. Whiteley, W., A matrix for splines, in Progress in Approximation Theory P. Nevai and A. Pinkus (eds.), Academic Press, Boston, 821--828. Using extremely sophisticated techniques Whiteley analyzes the matrix derived by Billera and establishes the generic dimension for the case r=1 and all d (and thus in particular for the cases d=2 and d=3 where no previous such result was available.)

  13. 1991. Hong, D., Spaces of bivariate spline functions over triangulations, J. Approx. Th. Applic. 7, 56--75. Hong constructs a minimal determining set for the case that d &gt&=3r+2.

  14. 1991. Ibrahim, A., and Schumaker, L.L., Super spline spaces of smoothness r and degree d >= 3r+2 Constr. Approx., 7, 401--423, The result derived here is slightly more general than Hong's result in that it applies to "super " splines, i.e., splines that are smoother than elsewhere at the vertices of the triangulation.

  15. 1993. Alfeld, P., Schumaker L.L., and Whiteley, W. The generic dimension of the space of C1 splines of degree d >= 8 on tetrahedral decompositions, SIAM JNA, v. 30, pp. 889--920, 1993 Combining the MDS technology and the vertex splitting techniques we obtain generic dimension formulas for r=1 and d >= 8 in the trivariate case.

  16. 1996. Alfeld, P., Upper and Lower Bounds on the Dimension of Multivariate Spline Spaces, SIAM JNA, v. 33, No. 2, pp. 571--588,. The paper gives upper and lower bounds for spline spaces in an arbitrary number of variables.

  17. 1998. Alfeld, P., and Schumaker, L.L., Non-Existence of Star-Supported Spline Bases, to appear in the SIAM J. on Mathematical Analysis. In this forthcoming paper we show that when d < 3r+2 then S does not in general possess a basis of splines each of which is supported only on the star of a vertex. (It had bee shown earlier by Hong, Ibrahim and Schumaker that if d > 3r+1 such bases always exist).