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'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.

1 Elements of segment analysis.- § 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.- § 1.2. Segment sequences.- 1.2.1. Segment limits.- 1.2.2. Theorems on segment limits.- § 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.- § 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.- 3.3.1. The Fejer operator.- 3.3.2 The Jackson operator.- 3.3.3. The generalized Jackson operator.- 3.3.4 The Vallée-Poussin operator.- § 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.- § 3.5. Approximation of functions by summation formulas on a finite closed interval.- 3.5.1. Bernstein polynomials.- 3.5.2. Fejer inteipolational polynomials.- § 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.- § 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.- 4.1.1. 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.- § 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.- § 4.3. Best approximation by spline functions.- 4.3.1. Spline functions with equidistant knots.- 4.3.2. Spline functions with free knots.- § 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.- 5.2.1. The trigonometric case.- 5.2.2. The algebraic case.- § 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.- 6.3.1. Passages in labyrinths.- § 6.4. ?-entropy and ?-capacity of bounded sets of connected compact sets.- § 6.5. Widths.- 6.5.1. Widths of the set of bounded real functions.- 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.- 8.1.1. Existence and uniqueness of the polynomial of best onesided approximation.- § 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.