Hausdorff Approximations - Bl. Sendov

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Résumé :
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....

Sommaire:
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.