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Approximation of set-valued functions : adaptation of classical approximation operators /
This book is aimed at the approximation of set-valued functions with compact sets in an Euclidean space as values. The interest in set-valued functions is rather new. Such functions arise in various modern areas such as control theory, dynamical systems and optimization. The authors' motivation...
| Clasificación: | Libro Electrónico |
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| Autores principales: | , , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Hackensack, NJ :
Imperial College Press,
[2014]
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Preface; Contents; Notations; I Scientific Background; 1. On Functions with Values in Metric Spaces; 1.1 Basic Notions; 1.2 Basic Approximation Methods; 1.3 Classical Approximation Operators; 1.3.1 Positive operators; 1.3.2 Interpolation operators; 1.3.3 Spline subdivision schemes; 1.4 Bibliographical Notes; 2. On Sets; 2.1 Sets and Operations Between Sets; 2.1.1 Definitions and notation; 2.1.2 Minkowski linear combination; 2.1.3 Metric average; 2.1.4 Metric linear combination; 2.2 Parametrizations of Sets; 2.2.1 Induced metrics and operations; 2.2.2 Convex sets by support functions.
- 2.2.3 Parametrization of sets in R2.2.4 Star-shaped sets by radial functions; 2.2.5 General sets by signed distance functions; 2.3 Bibliographical Notes; 3. On Set-Valued Functions (SVFs); 3.1 Definitions and Examples; 3.2 Representations of SVFs; 3.3 Regularity Based on Representations; 3.4 Bibliographical Notes; II Approximation of SVFs with Images in Rn; 4. Methods Based on Canonical Representations; 4.1 Induced Operators; 4.2 Approximation Results; 4.3 Application to SVFs with Convex Images; 4.4 Examples and Conclusions; 4.5 Bibliographical Notes.
- 5. Methods Based on Minkowski Convex Combinations5.1 Spline Subdivision Schemes for Convex Sets; 5.2 Non-Convexity Measures of a Compact Set; 5.3 Convexification of Sequences of Sample-Based Positive Operators; 5.4 Convexification by Spline Subdivision Schemes; 5.5 Bibliographical Notes; 6. Methods Based on the Metric Average; 6.1 Schoenberg Spline Operators; 6.2 Spline Subdivision Schemes; 6.3 Bernstein Polynomial Operators; 6.4 Bibliographical Notes; 7. Methods Based on Metric Linear Combinations; 7.1 Metric Piecewise Linear Interpolation; 7.2 Error Analysis.
- 7.3 Multifunctions with Convex Images7.4 Specific Metric Operators; 7.4.1 Metric Bernstein operators; 7.4.2 Metric Schoenberg operators; 7.4.3 Metric polynomial interpolation; 7.5 Bibliographical Notes; 8. Methods Based on Metric Selections; 8.1 Metric Selections; 8.2 Approximation Results; 8.3 Bibliographical Notes; III Approximation of SVFs with Images in R; 9. SVFs with Images in R; 9.1 Preliminaries on the Graphs of SVFs; 9.2 Continuity of the Boundaries of a CBV Multifunction; 9.3 Regularity Properties of the Boundaries; 10. Multi-Segmental and Topological Representations.
- 10.1 Multi-Segmental Representations (MSRs)10.2 Topological MSRs; 10.2.1 Existence of a topological MSR; 10.2.2 Conditions for uniqueness of a TMSR; 10.3 Representation by Topological Selections; 10.4 Regularity of SVFs Based on MSRs; 11. Methods Based on Topological Representation; 11.1 Positive Linear Operators Based on TMSRs; 11.1.1 Bernstein polynomial operators; 11.1.2 Schoenberg operators; 11.2 General Operators Based on Topological Selections; 11.3 Bibliographical Notes to Part III; Bibliography; Index.