Multivariate Birkhoff Interpolation Book

1	Introduction	1
2	Univariate Interpolation	4
2.1	Introduction and definitions	4
2.2	Main theorems	6
3	Basic Properties of Birkhoff interpolation	9
3.1	Introduction and definitions	9
3.2	Properties of the spaces P[subscript s]	13
3.3	The Polya condition	16
3.4	Regular incidence matrices	17
3.5	Properties of the determinant	20
4	Singular Interpolation Schemes	23
4.1	Introduction and definitions	23
4.2	Hermite interpolation of type total degree in R[superscript d]	26
4.3	Uniform Hermite interpolation of type total degree in R[superscript 2], R[superscript 3], and R[superscript 4]	32
4.4	Hermite interpolation of tensor-product type	34
4.5	Number-theoretic considerations	37
4.6	Numerical results	46
4.7	Slicing the pie the other way	49
5	Shifts and Coalescences	50
5.1	Taylor expansion of the Vandermonde determinant	50
5.2	Definition of shifts	50
5.3	Existence of shifts	52
5.4	Numbers of shifts	55
5.5	Coefficients of the Taylor expansion	57
5.6	Coalescences	60
6	Decomposition Theorems	62
6.1	Introduction	62
6.2	Decomposition theorems without knots	62
6.3	Decomposition theorems with nodes	64
6.4	Comparison with other approaches	68
7	Reduction	72
7.1	Introduction	72
7.2	The reduction theorem	72
8	Examples	75
8.1	Introduction	75
8.2	Interpolation on rectangles	78
8.3	Triangular elements	86
9	Uniform Hermite Interpolation of Tensor-product Type	90
9.1	Introduction	90
9.2	The Polya condition	90
9.3	Basic theorems	92
9.4	Application of the basic theorems	95
9.5	Interpolation with derivatives of low order	96
9.6	Non-uniform Hermite interpolation of tensor-product type	99
10	Uniform Hermite Interpolation of Type Total Degree	103
10.1	Introduction	103
10.2	The Polya condition	104
10.3	Number-theoretic considerations	106
10.4	Interpolation and singularities	108
10.5	Minimality of triangles	111
10.6	An extension theorem	114
10.7	Interpolation of first derivatives	116
10.8	Interpolation of second and third derivatives	119
10.9	An interpolation in R[superscript 3]	126
10.10	A conjecture	127
10.11	An alternate proof of almost regularity for [actual symbol not reproducible]	129
10.12	The general case	137
11	Vandermonde determinants	139
11.1	Introduction	139
11.2	The determinant of Lagrange interpolation	140
11.3	Determinants of the decomposition theorem	144
11.4	Related results	145
11.5	Hack's interpolation scheme	147
11.6	Determinants of two particular problems	153
12	A theorem of Severi	156
12.1	Introduction and the theorem of Severi	156
12.2	Smaller interpolation spaces	157
12.3	Lagrange Interpolation	159
13	Kergin Interpolation via Birkhoff Interpolation	162
13.1	Introduction	162
13.2	Kergin's interpolant	162
13.3	An alternative proof of regularity	169
	A Appendix - A Bibliography on Multivariate Interpolation	171
	References	183
	Glossary of notation	190