逼近理论和方法【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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1 The approximation problem and existence of best approximations1

1.1 Examples of approximation problems1

1.2 Approximation in a metric space3

1.3 Approximation in a normed linear space5

1.4 The Lp-norms6

1.5 A geometric view of best approximations9

2 The uniqueness of best approximations13

2.1 Convexity conditions13

2.2 Conditions for the uniqueness of the best approximation14

2.3 The continuity of best approximation operators16

2.4 The 1-,2-and ∞-norms17

3 Approximation operators and some approximating functions22

3.1 Approximation operators22

3.2 Lebesgue constants24

3.3 Polynomial approximations to differentiable functions25

3.4 Piecewise polynomial approximations28

4 Polynomial interpolation33

4.1 The Lagrange interpolation formula33

4.2 The error in polynomial interpolation35

4.3 The Chebyshev interpolation points37

4.4 The norm of the Lagrange interpolation operator41

5 Divided differences46

5.1 Basic properties of divided differences46

5.2 Newton's interpolation method48

5.3 The recurrence relation for divided differences49

5.4 Discussion of formulae for polynomial interpolation51

5.5 Hermite interpolation53

6 The uniform convergence of polynomial approximations61

6.1 The Weierstrass theorem61

6.2 Monotone operators62

6.3 The Bernstein operator65

6.4 The derivatives of the Bernstein approximations67

7 The theory of minimax approximation72

7.1 Introduction to minimax approximation72

7.2 The reduction of the error of a trial approximation74

7.3 The characterization theorem and the Haar condition76

7.4 Uniqueness and bounds on the minimax error79

8 The exchange algorithm85

8.1 Summary of the exchange algorithm85

8.2 Adjustment of the reference87

8.3 An example of the iterations of the exchange algorithm88

8.4 Applications of Chebyshev polynomials to minimax approximation90

8.5 Minimax approximation on a discrete point set92

9 The convergence of the exchange algorithm97

9.1 The increase in the levelled reference error97

9.2 Proof of convergence99

9.3 Properties of the point that is brought into reference102

9.4 Second-order convergence105

10 Rational approximation by the exchange algorithm111

10.1 Best minimax rational approximation111

10.2 The best approximation on a reference113

10.3 Some convergence properties of the exchange algorithm116

10.4 Methods based on linear programming118

11 Least squares approximation123

11.1 The general form of a linear least squares calculation123

11.2 The least squares characterization theorem125

11.3 Methods of calculation126

11.4 The recurrence relation for orthogonal polynomials131

12 Properties of orthogonal polynomials136

12.1 Elementary properties136

12.2 Gaussian quadrature138

12.3 The characterization of orthogonal polynomials141

12.4 The operator Rn143

13 Approximation to periodic functions150

13.1 Trigonometric polynomials150

13.2 The Fourier series operator Sn152

13.3 The discrete Fourier series operator156

13.4 Fast Fourier transforms158

14 The theory of best L1 approximation164

14.1 Introduction to best L1 approximation164

14.2 The characterization theorem165

14.3 Consequences of the Haar condition169

14.4 The L1 interpolation points for algcbraic polynomials172

15 An example ot L1 approximation and the discrete case177

15.1 A useful example of L1 approximation177

15.2 Jackson's first theorem179

15.3 Discrete L1 approximation181

15.4 Linear programming methods183

16 The order of convergence of polynomial approximations189

16.1 Approximations to non-differentiable functions189

16.2 The Dini-Lipschitz theorem192

16.3 Some bounds that depend on higher derivatives194

16.4 Extensions to algebraic polynomials195

17 The uniform boundedness theorem200

17.1 Preliminary results200

17.2 Tests for uniform convergence202

17.3 Application to trigonometric polynomials204

17.4 Application to algebraic polynomials208

18 Interpolation by piecewise polynomials212

18.1 Local interpolation methods212

18.2 Cubic spline interpolation215

18.3 End conditions for cubic spline interpolation219

18.4 Interpolating splines of other degrees221

19 B-splines227

19.1 The parameters of a spline function227

19.2 The form of B-splines229

19.3 B-splines as basis functions231

19.4 A recurrence relation for B-splines234

19.5 The Schoenberg-Whitney theorem236

20 Convergence properties of spline approximations241

20.1 Uniform convergence241

20.2 The order of convergence when f is differentiable243

20.3 Local spline interpolation246

20.4 Cubic splines with constant knot spacing248

21 Knot positions and the calculation of spline approximations254

21.1 The distribution of knots at a singularity254

21.2 Interpolation for general knots257

21.3 The approximation of functions to prescribed accuracy261

22 The Peano kernel theorem268

22.1 The error of a formula for the solution of differential equations268

22.2 The Peano kernel theorem270

22.3 Application to divided differences and to polynomial interpolation274

22.4 Application to cubic spline interpolation277

23 Natural and perfect splines283

23.1 A variational problem283

23.2 Properties of natural splines285

23.3 Perfect splines290

24 Optimal interpolation298

24.1 The optimal interpolation problem298

24.2 L1 approximation by B-splines301

24.3 Properties of optimal interpolation307

Appendix A The Haar condition313

Appendix B Related work and references317

Index333