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逼近理论和方法pdf电子书版本下载
- （英）M· J· D· Powell（M·J·D·鲍威尔） 著
- 出版社： 北京;西安：世界图书出版公司
- ISBN：9787510086250
- 出版时间：2015
- 标注页数：339页
- 文件大小：55MB
- 文件页数：351页
- 主题词：逼近论－教材－英文
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图书目录
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