Approximation of set-valued functions : adaptation of classical approximation operators / Nira Dyn, Tel Aviv University, Israel, Elza Farkhi, Tel Aviv University, Israel, Alona Mokhov, Tel Aviv University, Israel
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
Summary
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 also comes from the newer field of geometric modeling, in particular from the problem of reconstruction of 3D objects from 2D cross-sections. This is reflected in the focus of this book, which is the approximation of set-valued functions with general (not necessarily convex) sets as values, while previo
