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Chapter 1．Problems of Polynomial Approximation 1

1．Examples of Polynomials of Best Approximation 1

2．Distribution of Alternation Points of Polynomials of Best Approximation 4

3．Distribution of Zeros of Polynomials of Best Approximation 11

4．Error of Approximation 20

5．Approximation on(-∞,∞)by Linear Combinations of Functions(x-c)-1 23

6．Weighted Approximation by Polynomials on(-∞,∞) 28

7．Spaces of Approximation Theory 33

8．Problems and Notes 37

Chapter 2．Approximation Problems with Constraints 39

1．Introduction 39

2．Growth Restrictions for the Coefficients 39

3．Monotone Approximation 43

4．Polynomials with Integral Coefficients 49

5．Determination of the Characteristic Sets 59

6．Markov-Type Inequalities 64

7．The Inequality of Remez 73

8．One-sided Approximation by Polynomials 76

9．Problems 81

10．Notes 82

Chapter 3．Incomplete Polynomials 85

1．Incomplete Polynomials 85

2．Incomplete Chebyshev Polynomials 89

3．Incomplete Trigonometric Polynomials 92

4．Sequences of Polynomials with Many Real Zeros 98

5．Problems 104

6．Notes 104

Chapter 4．Weighted Polynomials 105

1．Essential Sets of Weighted Polynomials 105

2．Weighted Chebyshev Polynomials 109

3．The Equilibrium Measure 117

4．Determination of Minimal Essential Sets 125

5．Weierstrass Theorems and Oscillations 131

6．Weierstrass Theorem for Freud Weights 134

7．Problems 141

8．Notes 141

Chapter 5．Wavelets and Orthogonal Expansions 145

1．Multiresolutions and Wavelets 145

2．Scaling Functions with a Monotone Majorant 151

3．Periodization 156

4．Polynomial Schauder Bases 160

5．Orthonormal Polynomial Bases 164

6．Problems and Notes 172

Chapter 6．Splines 175

1．General Facts 175

2．Splines of Best Approximation 181

3．Periodic Splines 189

4．Convergence of Some Spline Operators 196

5．Notes 202

Chapter 7．Rational Approximation 205

1．Introduction 205

2．Best Rational Approximation 210

3．Rational Approximation of｜x｜ 217

4．Approximation of ex on[-1,1] 221

5．Rational Approximation of e-x on[0,∞) 227

6．Approximation of Classes of Functions 231

7．Theorems of Popov 235

8．Properties of the Operator of Best Rational Approximation in C and Lp 242

9．Appro ？imation by Rational Functions with Arbitrary Powers 248

10．Problems 251

11．Notes 252

Chapter 8．Stahl's Theorem 255

1．Introduction and Main Result 255

2．A Dirichlet Problem on[1/2,1/ρn] 256

3．The Second Approach to the Dirichlet Problem 263

4．Proof of Theorem 1.1 271

5．Notes 276

Chapter 9．Padé Approximation 277

1．The Padé Table 277

2．Convergence of the Rows of the Padé Table 282

3．The Nuttall-Pommerenke Theorem 290

4．Problems 296

5．Notes 296

Chapter 10．Hardy Space Methods in Rational Approximation 299

1．Bernstein-Type Inequalities for Rational Functions 300

2．Uniform Rational Approximation in Hardy Spaces 308

3．Approximation by Simple Functions 314

4．The Jackson-Rusak Operator;Rational Approximation of Sums of Simple Functions 320

5．Rational Approximation on T and on[-1,1] 322

6．Relations Between Spline and Rational Approximation in the Spaces Lp,0<p<∞ 332

7．Problems 341

8．Notes 341

Chapter 11．Müntz Polynomials 345

1．Definitions and Simple Properties 345

2．Müntz-Jackson Theorems 347

3．An Inverse Müntz-Jackson Theorem 353

4．The Index of Approximation 360

5．Markov-Type Inequality for Müntz Polynomials 362

6．Problems 365

7．Notes 366

Chapter 12．Nonlinear Approximation 369

1．Definitions and Simple Properties 369

2．Varisolvent Families 371

3．Exponential Sums 376

4．Lower Bounds for Errors of Nonlinear Approximation 383

5．Continuous Selections from Metric Projections 386

6．Approximation in Banach Spaces:Suns and Chebyshev Sets 390

7．Problems 395

8．Notes 396

Chapter 13．Widths Ⅰ 399

1．Definitions and Basic Properties 399

2．Relations Between Different Widths 407

3．Widths of Cubes and Octahedra 410

4．Widths in Hilbert Spaces 412

5．Applications of Borsuk's Theorem 418

6．Variational Problems and Spectral Functions 423

7．Results of Buslaev and Tikhomirov 432

8．Classes of Differentiable Functions on an Interval 441

9．Classes of Analytic Functions 443

10．Problems 445

11．Notes 447

Chapter 14．Widths Ⅱ:Weak Asymptotics for Widths of Lipschitz Balls,Random Approximants 449

1．Introduction 449

2．Discretization 451

3．Weak Equivalences for Widths.Elementary Methods 453

4．Distribution of Scalar Products of Unit Vectors 461

5．Kashin's Theorems 465

6．Gaussian Measures 469

7．Linear Widths of Finite Dimensional Balls 472

8．Linear Widths of the Lipschitz Classes 478

9．Problems 481

10．Notes 481

Chapter 15．Entropy 485

1．Entropy and Capacity 485

2．Elementary Estimates 489

3．Linear Approximation and Entropy 492

4．Relations Between Entropy and Widths 497

5．Entropy of Classes of Analytic Functions 502

6．The Birman-Solomyak Theorem 506

7．Entropy Numbers of Operators 509

8．Notes 514

Chapter 16．Convergence of Sequences of Operators 517

1．Introduction 517

2．Simple Necessary and Sufficient Conditions 518

3．Geometric Properties of Dominating Sets 523

4．Strict Dominating Systems;Minimal Systems;Examples 528

5．Shadows of Sets of Continuous Functions 536

6．Shadows in Banach Function Spaces 541

7．Positive Contractions 545

8．Contractions 547

9．Notes 551

Chapter 17．Representation of Functions by Superpositions 553

1．The Theorems of Kolmogorov 553

2．Proof of the Theorems 555

3．Functions Not Representable by Superpositions 559

4．Linear Superpositions 562

5．Notes 564

Appendix 1．Theorems of Borsuk and of Brunn-Minkowski 567

1．Borsuk's Theorem 567

2．The Brunn-Minkowski Inequality 572

Appendix 2．Estimates of Some Elliptic Integrals 575

Appendix 3．Hardy Spaces and Blaschke Products 581

1．Hardy Spaces 581

2．Conjugate Functions and Cauchy Integrals 584

3．Atomic Decompositions in Hardy Spaces 587

4．Blaschke Products 591

Appendix 4．Potential Theory and Logarithmic Capacity 595

1．Logarithmic Potentials 595

2．Equilibrium Distribution and Logarithmic Capacity 603

3．The Dirichlet Problem and Green's Function 614

4．Balayage Methods 615

Bibliography 621

Author Index 641

Subject Index 647