逼近理论和方法PDF电子书下载

1 The approximation problem and existence of best approximations 1

1.1 Examples of approximation problems 1

1.2 Approximation in a metric space 3

1.3 Approximation in a normed linear space 5

1.4 The Lp-norms 6

1.5 A geometric view of best approximations 9

2 The uniqueness of best approximations 13

2.1 Convexity conditions 13

2.2 Conditions for the uniqueness of the best approximation 14

2.3 The continuity of best approximation operators 16

2.4 The 1-,2-and ∞-norms 17

3 Approximation operators and some approximating functions 22

3.1 Approximation operators 22

3.2 Lebesgue constants 24

3.3 Polynomial approximations to differentiable functions 25

3.4 Piecewise polynomial approximations 28

4 Polynomial interpolation 33

4.1 The Lagrange interpolation formula 33

4.2 The error in polynomial interpolation 35

4.3 The Chebyshev interpolation points 37

4.4 The norm of the Lagrange interpolation operator 41

5 Divided differences 46

5.1 Basic properties of divided differences 46

5.2 Newton's interpolation method 48

5.3 The recurrence relation for divided differences 49

5.4 Discussion of formulae for polynomial interpolation 51

5.5 Hermite interpolation 53

6 The uniform convergence of polynomial approximations 61

6.1 The Weierstrass theorem 61

6.2 Monotone operators 62

6.3 The Bernstein operator 65

6.4 The derivatives of the Bernstein approximations 67

7 The theory of minimax approximation 72

7.1 Introduction to minimax approximation 72

7.2 The reduction of the error of a trial approximation 74

7.3 The characterization theorem and the Haar condition 76

7.4 Uniqueness and bounds on the minimax error 79

8 The exchange algorithm 85

8.1 Summary of the exchange algorithm 85

8.2 Adjustment of the reference 87

8.3 An example of the iterations of the exchange algorithm 88

8.4 Applications of Chebyshev polynomials to minimax approximation 90

8.5 Minimax approximation on a discrete point set 92

9 The convergence of the exchange algorithm 97

9.1 The increase in the levelled reference error 97

9.2 Proof of convergence 99

9.3 Properties of the point that is brought into reference 102

9.4 Second-order convergence 105

10 Rational approximation by the exchange algorithm 111

10.1 Best minimax rational approximation 111

10.2 The best approximation on a reference 113

10.3 Some convergence properties of the exchange algorithm 116

10.4 Methods based on linear programming 118

11 Least squares approximation 123

11.1 The general form of a linear least squares calculation 123

11.2 The least squares characterization theorem 125

11.3 Methods of calculation 126

11.4 The recurrence relation for orthogonal polynomials 131

12 Properties of orthogonal polynomials 136

12.1 Elementary properties 136

12.2 Gaussian quadrature 138

12.3 The characterization of orthogonal polynomials 141

12.4 The operator Rn 143

13 Approximation to periodic functions 150

13.1 Trigonometric polynomials 150

13.2 The Fourier series operator Sn 152

13.3 The discrete Fourier series operator 156

13.4 Fast Fourier transforms 158

14 The theory of best L1 approximation 164

14.1 Introduction to best L1 approximation 164

14.2 The characterization theorem 165

14.3 Consequences of the Haar condition 169

14.4 The L1 interpolation points for algcbraic polynomials 172

15 An example ot L1 approximation and the discrete case 177

15.1 A useful example of L1 approximation 177

15.2 Jackson's first theorem 179

15.3 Discrete L1 approximation 181

15.4 Linear programming methods 183

16 The order of convergence of polynomial approximations 189

16.1 Approximations to non-differentiable functions 189

16.2 The Dini-Lipschitz theorem 192

16.3 Some bounds that depend on higher derivatives 194

16.4 Extensions to algebraic polynomials 195

17 The uniform boundedness theorem 200

17.1 Preliminary results 200

17.2 Tests for uniform convergence 202

17.3 Application to trigonometric polynomials 204

17.4 Application to algebraic polynomials 208

18 Interpolation by piecewise polynomials 212

18.1 Local interpolation methods 212

18.2 Cubic spline interpolation 215

18.3 End conditions for cubic spline interpolation 219

18.4 Interpolating splines of other degrees 221

19 B-splines 227

19.1 The parameters of a spline function 227

19.2 The form of B-splines 229

19.3 B-splines as basis functions 231

19.4 A recurrence relation for B-splines 234

19.5 The Schoenberg-Whitney theorem 236

20 Convergence properties of spline approximations 241

20.1 Uniform convergence 241

20.2 The order of convergence when f is differentiable 243

20.3 Local spline interpolation 246

20.4 Cubic splines with constant knot spacing 248

21 Knot positions and the calculation of spline approximations 254

21.1 The distribution of knots at a singularity 254

21.2 Interpolation for general knots 257

21.3 The approximation of functions to prescribed accuracy 261

22 The Peano kernel theorem 268

22.1 The error of a formula for the solution of differential equations 268

22.2 The Peano kernel theorem 270

22.3 Application to divided differences and to polynomial interpolation 274

22.4 Application to cubic spline interpolation 277

23 Natural and perfect splines 283

23.1 A variational problem 283

23.2 Properties of natural splines 285

23.3 Perfect splines 290

24 Optimal interpolation 298

24.1 The optimal interpolation problem 298

24.2 L1 approximation by B-splines 301

24.3 Properties of optimal interpolation 307

Appendix A The Haar condition 313

Appendix B Related work and references 317

Index 333