Hausdorff Approximations

1 Elements of segment analysis.- § 1.1. Segment arithmetic.- § 1.2. Segment sequences.- § 1.3. Segment functions.- 2 Hausdorff distance.- § 2.1. Hausdorff distance between subsets of a metric space.- § 2.2. The metric space F?.- § 2.3. H-distancein A? and its properties.- § 2.4. Relationships between uniform distance and the Hausdorff distance.- § 2.5. The modulus of H-continuity.- § 2.6. The order of the modulus of H-continuity.- § 2.7. H-continuity on a subset.- § 2.8. H-distance with weight.- 3 Linear methods of approximation.- § 3.1. Convergence of sequences of positive operators.- § 3.2. The order of approximation of functions by positive linear operators.- § 3.3. Approximation of periodic functions by positive integral operators.- § 3.4. Approximation of functions by positive integral operators on a finite closed interval.- § 3.5. Approximation of functions by summation formulas on a finite closed interval.- § 3.6. Approximation of nonperiodic functions by integral operators on the entire real axis.- § 3.7. Convergence of derivatives of linear operators.- § 3.8. A-distance.- § 3.9. Approximation by partial sums of Fourier series.- 4 Best Hausdorff approximations.- § 4.1. Best approximation by algebraic and trigonometric polynomials.- § 4.2. Best approximation by rational functions.- § 4.3. Best approximation by spline functions.- § 4.4. Best approximation by piecewise monotone functions.- 5 Converse theorems.- § 5.1. Existence of a function with preassigned best approximations.- § 5.2. Converse theorems for the approximation by algebraic and trigonometric polynomials.- § 5.3. Converse theorems for approximation by spline functions.- § 5.4. Converse theorems for approximation by rational and partially monotone functions.- § 5.5.Converse theorems for approximation by positive linear operators.- 6 ?-Entropy, ?-capacity and widths.- § 6.1. ?-entropy and ?-capacity of the set F?M.- § 6.2. The number of (p,q)-corridors.- § 6.3. Labyrinths.- § 6.4. ?-entropy and ?-capacity of bounded sets of connected compact sets.- § 6.5. Widths.- 7 Approximation of curves and compact sets in the plane.- § 7.1. Approximation by polynomial curves.- § 7.2. Characterization of best approximation in terms of metric dimension.- § 7.3. Approximation by piecewise monotone curves.- § 7.4. Other methods for the approximation of curves in the plane.- 8 Numerical methods of best Hausdorff approximation.- § 8.1. One-sided Hausdorff distance.- § 8.2. Coincidence of polynomials of best approximation with respect to one- and two-sided Hausdorff distance.- § 8.3. Numerical methods for calculating the polynomial of best one-sided approximation.- References.- Author Index.- Notation Index.

1 Elements of segment analysis. - § 1. 1. Segment arithmetic. - § 1. 2. Segment sequences. - § 1. 3. Segment functions. - 2 Hausdorff distance. - § 2. 1. Hausdorff distance between subsets of a metric space. - § 2. 2. The metric space F? . - § 2. 3. H-distancein A? and its properties. - § 2. 4. Relationships between uniform distance and the Hausdorff distance. - § 2. 5. The modulus of H-continuity. - § 2. 6. The order of the modulus of H-continuity. - § 2. 7. H-continuity on a subset. - § 2. 8. H-distance with weight. - 3 Linear methods of approximation. - § 3. 1. Convergence of sequences of positive operators. - § 3. 2. The order of approximation of functions by positive linear operators. - § 3. 3. Approximation of periodic functions by positive integral operators. - § 3. 4. Approximation of functions by positive integral operators on a finite closed interval. - § 3. 5. Approximation of functions by summation formulas on a finite closed interval. - § 3. 6. Approximation of nonperiodic functions by integral operators on the entire real axis. - § 3. 7. Convergence of derivatives of linear operators. - § 3. 8. A-distance. - § 3. 9. Approximation by partial sums of Fourier series. - 4 Best Hausdorff approximations. - § 4. 1. Best approximation by algebraic and trigonometric polynomials. - § 4. 2. Best approximation by rational functions. - § 4. 3. Best approximation by spline functions. - § 4. 4. Best approximation by piecewise monotone functions. - 5 Converse theorems. - § 5. 1. Existence of a function with preassigned best approximations. - § 5. 2. Converse theorems for the approximation by algebraic and trigonometric polynomials. - § 5. 3. Converse theorems for approximation by spline functions. - § 5. 4. Converse theorems for approximation by rational and partially monotone functions. - § 5. 5. Converse theorems for approximation by positive linear operators. - 6 ? -Entropy, ? -capacity and widths. - § 6. 1. ? -entropy and ? -capacity of the set F? M. - § 6. 2. The number of (p, q)-corridors. - § 6. 3. Labyrinths. - § 6. 4. ? -entropy and ? -capacity of bounded sets of connected compact sets. - § 6. 5. Widths. - 7 Approximation of curves and compact sets in the plane. - § 7. 1. Approximation by polynomial curves. - § 7. 2. Characterization of best approximation in terms of metric dimension. - § 7. 3. Approximation by piecewise monotone curves. - § 7. 4. Other methods for the approximation of curves in the plane. - 8 Numerical methods of best Hausdorff approximation. - § 8. 1. One-sided Hausdorff distance. - § 8. 2. Coincidence of polynomials of best approximation with respect to one- and two-sided Hausdorff distance. - § 8. 3. Numerical methods for calculating the polynomial of best one-sided approximation. - References. - Author Index. - Notation Index.