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 online 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.

1973.
G. Strang, Piecewise polynomials and the finite
element method, Bull. Amer. Math. Soc. 79,
11281137. Strang made a conjecture on the
dimension of S for the case of
C^{1} cubics (r=1, d=3)
that turned out to be wrong but started the whole
subject.

1975.
Morgan, J., and R. Scott,
A Nodal Basis for C^{1} Piecewise
Polynomials of Degree n >=5 , Math. Comp.
29, 736740. In this fundamental paper Morgan and
Scott settle the dimension of S for
r=1 and d>=5. The
BernsteinBézier form was unknown at the time
but much later progress depended substantially on
translating this paper into the language of the
BernsteinBézier form.

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, 396412.
This paper gives very general lower bounds on the dimension
of S. Using the BernsteinBézier
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.

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

1984.
Schumaker, L.L., Bounds on the dimension of spaces
of multivariate piecewise polynomials, Rocky
Mountain J. of Mathematics 14, 251264. 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.

1987.
Alfeld, P. and Schumaker, L.L., The dimension of
bivariate spline spaces of smoothness r for degree d
>=4r+1, Constructive Approximation 3,
189197 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.

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,
105124 This augments the preceding paper by
specifying the points in the 2r disks around
interior vertices, but only for
r=1,2,3.

1987.
Alfeld, P., Piper, B., and Schumaker, L.L., An
Explicit Basis for C^{1} Quartic Bivariate
Splines, SIAM J. Num.Anal. 24, 891911 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 nongeneric dimensions.

1988.
Billera, L.,, Homology of smooth splines: generic
triangulations and a conjecture by Strang,
Trans. A.M.S. 310, 325340. Using a sophisticated
body of machinery Billera derives a linear system that
describes C^{1} splines.

1988.
Schumaker, L.L., Dual bases for spline spaces on
cells, Computer Aided Geom. Design 5,
277284. Schumaker gives an explicit minimal
determining set for spline spaces defined on the star of
a vertex, for all values of d and r.

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, 651661 We
specify explicit minimal determining sets essentially in
the generic case.

1991.
Whiteley, W., A matrix for splines, in
Progress in Approximation Theory P. Nevai and A. Pinkus
(eds.), Academic Press, Boston, 821828. 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.)

1991.
Hong, D., Spaces of bivariate spline functions over
triangulations, J. Approx. Th. Applic. 7,
5675. Hong constructs a minimal determining set
for the case that d >&=3r+2.

1991.
Ibrahim, A., and Schumaker, L.L., Super
spline spaces of smoothness r and degree d >= 3r+2
Constr. Approx., 7, 401423,
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.

1993.
Alfeld, P., Schumaker L.L., and Whiteley, W. The
generic dimension of the space of C^{1} splines
of degree d >= 8 on tetrahedral decompositions,
SIAM JNA, v. 30, pp. 889920, 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.

1996.
Alfeld, P., Upper and Lower Bounds on the Dimension
of Multivariate Spline Spaces, SIAM JNA, v. 33,
No. 2, pp. 571588,. The paper gives upper and
lower bounds for spline spaces in an arbitrary number of
variables.

1998.
Alfeld, P., and Schumaker, L.L.,
NonExistence of StarSupported 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).
[25Mar1999]