Approximation by Spline Functions

Gunther Nurnberger, "Approximation by Spline Functions"
Springer-Verlag Berlin and Heidelberg GmbH Co. K | 1989 | ISBN: 3540516182 | 254 pages | Djvu | 4,8 mb
Contents
Chapter I. Polynomials and Chebyshev Spaces
1. Interpolation by Chebyshev Spaces 1
1.1. Lagrange Interpolation by Chebyshev Spaces .. 1
1.2. Hermite Interpolation by Extended Chebyshev Spaces 4
1.3. Characterization of Extended Complete Chebyshev Spaces . 7
1.4. Further Properties of Chebyshev Spaces . 11 F
1.5. Variation Diminishing Property of Order Complete
Chebyshev Spaces .. 15
2. Interpolation by Polynomials and Divided Differences 16
2.1. Divided Differences . 17
2.2. Newton Form of Interpolating Polynomials .. 21
2.3. Nearly Optimal Interpolation Points .. 25
3. Best Uniform Approximation by Chebyshev Spaces 29
3.1. Best Approximation in Normed Linear Spaces . 29
3.2. Characterization of Best Uniform Approximations 32
3.3. Global Unicity and Strong Unicity of Best Uniform
Approximations . 36
3.4. Algorithm 48
3.5. Approximation Power of Polynomials .. V .. 54
4. Best Ll—Approximation by Chebyshev Spaces .. 56
4.1. Global Unicity of Best L1—Approximations .. 56
4.2. Interpolation at Canonical Points .. 60
5. Best One—Sided L1—Approximation by Chebyshev Spaces and
Quadrature Formulas I . 65
5.1. Unicity of Best One—Sided L1—Approximations . 65
5.2. Gauss Quadrature Formulas for Chebyshev Spaces 69
6. Best Lg—Approximation .. 76

mptur II. Hplimm mul Wenk ("llmlnyulwv Spaces
1. Weak (ilwbyulwv Spawn . 80
1.1. llanic l’rop¢·rti¢·n .. .. . 80
1.2. Best Uniform Approximation by Weak Chebyshev Spaces . 88
1.3. Spline Spaces .. 93
2. B—Splines 95 p .
2.1. Basic Properties .. 95
2.2. B—Spline Basis . 98
2.3. Recurrence Relations . 99
2.4. Variation Diminishing Property .. 106 é
3. Interpolation by Splines . 107
3.1. Lagrange and Hermite Interpolation by Splines . .` 107 i
3.2. Interpolation by Complete Splines, Periodic Splines
and Natural Splines .. 115
3.3. Quasi-Interpolation . i . 127 ii
4. Best Uniform Approximation by Splines 131 C
4.1. Characterization, Unicity and Strong Unicity of Best A
Uniform Approximations 132 Y
4.2. Algorithm (Fixed Knots).` .. 143 i`
4.3. Algorithm (Free Knots) . 150
4.4. Approximatgrf Power of Splines .. 159 i
5. Continuity of the Set Valued Metric Projection for Spline Spaces . . . 161
5.1. Upper Semicontinuity 162
5.2. Lower Semicontinuity . 163 i
5.3. Continuous Selections 164 A
6. Best Ll—Approximation by Weak Chebyshev Spaces 168 .
6.1. Unicity of Best L1—Approximations .. 169 ri
6.2. Interpolation at Canonical Points . 171 .
7. Best One-Sided L1—Approximation by Weak Chebyshev I
Spaces and Quadrature Formulas 174 i
/�?7.1. Unicity of Best One-Sided L1—Approximations 174 C
7.2. Gauss Quadrature Formulas for Weak Chebyshev Spaces . I . . . 176 �?
8. Approximation of Linear Functionals and Splines 180
9. Spaces of Splines with Multiple Knots .. 187 0

Appendix ·
1. Splincs with Frcc Knots . 190
2. Splines in Two Variables 195
2.1. Tensor Product and Blending .. 195
2.2. Finite Element Functions 200
2.3. Spline Functions .. 205
3. Spline Collocation and Differential Equations . 218
References . 223
Index .. Jl 240


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