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Gunther Nurnberger, "Approximation by Spline Functions"

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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