Hausdorff Approximations

1 Elements of segment analysis.- x 1.1. Segment arithmetic.- 1.1.1. Partial orderings.- 1.1.2. Lattice operations.- 1.1.3. Arithmetic operations.- 1.1.3.1. Addition and subtraction.- 1.1.3.2. Multiplication and division.- 1.1.4. Distance and norm.- x 1.2. Segment sequences.- 1.2.1. Segment limits.- 1.2.2. Theorems on segment limits.- x 1.3. Segment functions.- 1.3.1. The segment limit of a segment function.- 1.3.2. Segment derivatives.- 1.3.3. Segment continuity.- 1.3.4. H-continuity.- 2 Hausdorff distance.- x 2.1. Hausdorff distance between subsets of a metric space.- x 2.2. The metric space F?.- x 2.3. H-distancein A? and its properties.- x 2.4. Relationships between uniform distance and the Hausdorff distance.- x 2.5. The modulus of H-continuity.- x 2.6. The order of the modulus of H-continuity.- x 2.7. H-continuity on a subset.- x 2.8. H-distance with weight.- 3 Linear methods of approximation.- x 3.1. Convergence of sequences of positive operators.- x 3.2. The order of approximation of functions by positive linear operators.- x 3.3. Approximation of periodic functions by positive integral operators.- 3.3.1. The Fejer operator.- 3.3.2 The Jackson operator.- 3.3.3. The generalized Jackson operator.- 3.3.4 The Vallee-Poussin operator.- x 3.4. Approximation of functions by positive integral operators on a finite closed interval.- 3.4.1. The Landau operator.- 3.4.2. The generalized Landau operator.- x 3.5. Approximation of functions by summation formulas on a finite closed interval.- 3.5.1. Bernstein polynomials.- 3.5.2. Fejer inteipolational polynomials.- x 3.6. Approximation of nonperiodic functions by integral operators on the entire real axis.- 3.6.1 The Fejer operator in the nonperiodic case.- 3.6.2. The generalized Jackson operator in the nonperiodic case.- 3.6.3. The Weierstrass operator.- x 3.7. Convergence of derivatives of linear operators.- x 3.8. A-distance.- x 3.9. Approximation by partial sums of Fourier series.- 4 Best Hausdorff approximations.- x 4.1. Best approximation by algebraic and trigonometric polynomials.- 4.1.1. Uniqueness conditions for the polynomial of best approximation.- 4.1.2. Estimates for the best approximation.- 4.1.2.1. Best approximation of the delta-function.- 4.1.2.2. Universal estimates.- 4.1.2.3. Exact asymptotic behavior of the best approximation.- 4.1.2.4. Generalizations of Jackson's theorem.- 4.1.2.5. Approximation of certain concrete functions.- 4.1.2.6. Approximation of convex functions.- 4.1.2.7. An analogue of Nikol'skii's theorem.- 4.1.2.8. Comonotone approximations.- x 4.2. Best approximation by rational functions.- 4.2.1. Universal estimates for bounded functions.- 4.2.2. Unimprovability of the universal estimate.- 4.2.3. Approximation of analytic functions with singularities on the boundary of a closed interval.- x 4.3. Best approximation by spline functions.- 4.3.1. Spline functions with equidistant knots.- 4.3.2. Spline functions with free knots.- x 4.4. Best approximation by piecewise monotone functions.- 5 Converse theorems.- x 5.1. Existence of a function with preassigned best approximations.- x 5.2. Converse theorems for the approximation by algebraic and trigonometric polynomials.- 5.2.1. The trigonometric case.- 5.2.2. The algebraic case.- x 5.3. Converse theorems for approximation by spline functions.- x 5.4. Converse theorems for approximation by rational and partially monotone functions.- x 5.5. Converse theorems for approximation by positive linear operators.- 6 ?-Entropy, ?-capacity and widths.- x 6.1. ?-entropy and ?-capacity of the set F?M.- x 6.2. The number of (p,q)-corridors.- x 6.3. Labyrinths.- 6.3.1. Passages in labyrinths.- x 6.4. ?-entropy and ?-capacity of bounded sets of connected compact sets.- x 6.5. Widths.- 6.5.1. Widths of the set of bounded real functions.- 7 Approximation of curves and compact sets in the plane.- x 7.1. Approximation by polynomial curves.- x 7.2. Characterization of best approximation in terms of metric dimension.- x 7.3. Approximation by piecewise monotone curves.- x 7.4. Other methods for the approximation of curves in the plane.- 8 Numerical methods of best Hausdorff approximation.- x 8.1. One-sided Hausdorff distance.- 8.1.1. Existence and uniqueness of the polynomial of best onesided approximation.- x 8.2. Coincidence of polynomials of best approximation with respect to one- and two-sided Hausdorff distance.- x 8.3. Numerical methods for calculating the polynomial of best one-sided approximation.- References.- Author Index.- Notation Index.