Hausdorff Approximations: (50 Mathematics and its Applications)

'Et moi, ..., si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point a1Ie.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Table of Contents:
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.