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Chapter 1．Problems of Polynomial Approximation1

1．Examples of Polynomials of Best Approximation1

2．Distribution of Alternation Points of Polynomials of Best Approximation4

3．Distribution of Zeros of Polynomials of Best Approximation11

4．Error of Approximation20

5．Approximation on(-∞,∞)by Linear Combinations of Functions(x-c)-123

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

7．Spaces of Approximation Theory33

8．Problems and Notes37

Chapter 2．Approximation Problems with Constraints39

1．Introduction39

2．Growth Restrictions for the Coefficients39

3．Monotone Approximation43

4．Polynomials with Integral Coefficients49

5．Determination of the Characteristic Sets59

6．Markov-Type Inequalities64

7．The Inequality of Remez73

8．One-sided Approximation by Polynomials76

9．Problems81

10．Notes82

Chapter 3．Incomplete Polynomials85

1．Incomplete Polynomials85

2．Incomplete Chebyshev Polynomials89

3．Incomplete Trigonometric Polynomials92

4．Sequences of Polynomials with Many Real Zeros98

5．Problems104

6．Notes104

Chapter 4．Weighted Polynomials105

1．Essential Sets of Weighted Polynomials105

2．Weighted Chebyshev Polynomials109

3．The Equilibrium Measure117

4．Determination of Minimal Essential Sets125

5．Weierstrass Theorems and Oscillations131

6．Weierstrass Theorem for Freud Weights134

7．Problems141

8．Notes141

Chapter 5．Wavelets and Orthogonal Expansions145

1．Multiresolutions and Wavelets145

2．Scaling Functions with a Monotone Majorant151

3．Periodization156

4．Polynomial Schauder Bases160

5．Orthonormal Polynomial Bases164

6．Problems and Notes172

Chapter 6．Splines175

1．General Facts175

2．Splines of Best Approximation181

3．Periodic Splines189

4．Convergence of Some Spline Operators196

5．Notes202

Chapter 7．Rational Approximation205

1．Introduction205

2．Best Rational Approximation210

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 Functions231

7．Theorems of Popov235

8．Properties of the Operator of Best Rational Approximation in C and Lp242

9．Appro ？imation by Rational Functions with Arbitrary Powers248

10．Problems251

11．Notes252

Chapter 8．Stahl's Theorem255

1．Introduction and Main Result255

2．A Dirichlet Problem on[1/2,1/ρn]256

3．The Second Approach to the Dirichlet Problem263

4．Proof of Theorem 1.1271

5．Notes276

Chapter 9．Padé Approximation277

1．The Padé Table277

2．Convergence of the Rows of the Padé Table282

3．The Nuttall-Pommerenke Theorem290

4．Problems296

5．Notes296

Chapter 10．Hardy Space Methods in Rational Approximation299

1．Bernstein-Type Inequalities for Rational Functions300

2．Uniform Rational Approximation in Hardy Spaces308

3．Approximation by Simple Functions314

4．The Jackson-Rusak Operator;Rational Approximation of Sums of Simple Functions320

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．Problems341

8．Notes341

Chapter 11．Müntz Polynomials345

1．Definitions and Simple Properties345

2．Müntz-Jackson Theorems347

3．An Inverse Müntz-Jackson Theorem353

4．The Index of Approximation360

5．Markov-Type Inequality for Müntz Polynomials362

6．Problems365

7．Notes366

Chapter 12．Nonlinear Approximation369

1．Definitions and Simple Properties369

2．Varisolvent Families371

3．Exponential Sums376

4．Lower Bounds for Errors of Nonlinear Approximation383

5．Continuous Selections from Metric Projections386

6．Approximation in Banach Spaces:Suns and Chebyshev Sets390

7．Problems395

8．Notes396

Chapter 13．Widths Ⅰ399

1．Definitions and Basic Properties399

2．Relations Between Different Widths407

3．Widths of Cubes and Octahedra410

4．Widths in Hilbert Spaces412

5．Applications of Borsuk's Theorem418

6．Variational Problems and Spectral Functions423

7．Results of Buslaev and Tikhomirov432

8．Classes of Differentiable Functions on an Interval441

9．Classes of Analytic Functions443

10．Problems445

11．Notes447

Chapter 14．Widths Ⅱ:Weak Asymptotics for Widths of Lipschitz Balls,Random Approximants449

1．Introduction449

2．Discretization451

3．Weak Equivalences for Widths.Elementary Methods453

4．Distribution of Scalar Products of Unit Vectors461

5．Kashin's Theorems465

6．Gaussian Measures469

7．Linear Widths of Finite Dimensional Balls472

8．Linear Widths of the Lipschitz Classes478

9．Problems481

10．Notes481

Chapter 15．Entropy485

1．Entropy and Capacity485

2．Elementary Estimates489

3．Linear Approximation and Entropy492

4．Relations Between Entropy and Widths497

5．Entropy of Classes of Analytic Functions502

6．The Birman-Solomyak Theorem506

7．Entropy Numbers of Operators509

8．Notes514

Chapter 16．Convergence of Sequences of Operators517

1．Introduction517

2．Simple Necessary and Sufficient Conditions518

3．Geometric Properties of Dominating Sets523

4．Strict Dominating Systems;Minimal Systems;Examples528

5．Shadows of Sets of Continuous Functions536

6．Shadows in Banach Function Spaces541

7．Positive Contractions545

8．Contractions547

9．Notes551

Chapter 17．Representation of Functions by Superpositions553

1．The Theorems of Kolmogorov553

2．Proof of the Theorems555

3．Functions Not Representable by Superpositions559

4．Linear Superpositions562

5．Notes564

Appendix 1．Theorems of Borsuk and of Brunn-Minkowski567

1．Borsuk's Theorem567

2．The Brunn-Minkowski Inequality572

Appendix 2．Estimates of Some Elliptic Integrals575

Appendix 3．Hardy Spaces and Blaschke Products581

1．Hardy Spaces581

2．Conjugate Functions and Cauchy Integrals584

3．Atomic Decompositions in Hardy Spaces587

4．Blaschke Products591

Appendix 4．Potential Theory and Logarithmic Capacity595

1．Logarithmic Potentials595

2．Equilibrium Distribution and Logarithmic Capacity603

3．The Dirichlet Problem and Green's Function614

4．Balayage Methods615

Bibliography621

Author Index641

Subject Index647