Possibility of Approximation |
Basic notions / 1: |
Linear operators / 2: |
Approximation theorems / 3: |
The theorem of Stone / 4: |
Notes / 5: |
Polynomials of Best Approximation |
Existence of polynomials of best approximation |
Characterization of polynomials of best approximation |
Applications of convexity |
Chebyshev systems |
Uniqueness of polynomials of best approximation |
Chebyshev's theorem / 6: |
Chebyshev polynomials / 7: |
Approximation of some complex functions / 8: |
Properties of Polynomials and Moduli of Continuity / 9: |
Interpolation |
Inequalities of Bernstein |
The inequality of Markov |
Growth of polynomials in the complex plane |
Moduli of continuity |
Moduli of smoothness |
Classes of functions |
The Degree of Approximation by Trigonometric Polynomials |
Generalities |
The theorem of Jackson |
The degree of approximation of differentiable functions |
Inverse theorems |
Differentiable functions |
The Degree of Approximation by Algebraic Polynomials |
Preliminaries |
The approximation theorems |
Inequalities for the derivatives of polynomials |
Approximation of analytic functions |
Approximation by Rational Functions |
Functions of Several Variables |
Degree of rational approximation |
Periodic functions of several variables |
Approximation by algebraic polynomials |
Approximation by Linear Polynomial Operators |
Sums of de la Vallee-Poussin |
Positive operators |
The principle of uniform boundedness |
Operators that preserve trigonometric polynomials |
Trigonometric saturation classes |
The saturation class of the Bernstein polynomials |
Approximation of Classes of Functions |
Introduction |
Approximation in the space $L^1$ |
The degree of approximation of the classes $W^*_p$ |
Distance matrices |
Approximation of the classes $\Lambda_{\omega}$ |
Arbitrary moduli of continuity; Approximation by operators |
Analytic functions |
Widths |
Definitions and basic properties |
Sets of continuous and differentiable functions |
Widths of balls |
Applications of theorem 2 |
Differential operators |
Widths of the sets $\mathfrak{R}_l$ |
Entropy |
Entropy and capacity |
Entropy of classes of analytic functions |
More general sets of analytic functions |
Relations between entropy and widths |
Representation of Functions of Several Variables by Functions of One Variable |
The Theorem of Kolmogorov |
The fundamental lemma |
The completion of the proof |
Functions not representable by superpositions |
Bibliography |
Index |
Possibility of Approximation |
Basic notions / 1: |
Linear operators / 2: |