Approximation of set-valued functions: adaptation of classical approximation operators
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
London
Imperial College Press
2014
|
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (pages 145-150) and index |
| Physical Description: | XIII, 153 S. Ill., graph. Darst. |
| ISBN: | 9781783263028 1783263024 |
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| 245 | 1 | 0 | |a Approximation of set-valued functions |b adaptation of classical approximation operators |c Nira Dyn ; Elza Farkhi ; Alona Mokhov |
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Record in the Search Index
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| adam_text | 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 a 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 previous results on this topic are mainly
confined to the convex case. The approach taken in this book is to adapt
classical approximation operators and to provide error estimates in terms
of the regularity properties of the approximated set-valued functions.
Specialized results are given for functions with 1 D sets as values.
Contents^
Preface
v
Notations
x
I Scientific Background 1
1. On Functions with Values in Metric Spaces 3
FI Basic Notions........................................... 3
F2 Basic Approximation Methods............................. 6
F3 Classical Approximation Operators....................... 7
F3.1 Positive operators................................ 8
F3.2 Interpolation operators.......................... 12
F3.3 Spline subdivision schemes....................... 13
F4 Bibliographical Notes.................................. 15
2· On Sets 27
2-1 Sets and Operations Between Sets...................... 17
2.1.1 Definitions and notation ........................ 17
2.1.2 Minkowski linear combination..................... 18
2.1.3 Metric average................................... 19
2.1.4 Metric linear combination ....................... 21
2-2 Pararnetrizations of Sets............................. 23
2*2.1 Induced metrics and operations . .
2.2.2 Convex sets by support functions .
2-2.3 Parametrization of sets in 1 . . . .
2.2.4 Star-shaped sets by radial functions
vii
viii Approximation of Set- Valued Functions
2.2.5 General sets by signed distance functions .... 28
2.3 Bibliographical Notes.................................. 29
3. On Set-Valued Functions (SVFs) 31
3.1 Definitions and Examples............................... 31
3.2 Representations of SVFs................................ 32
3.3 Regularity Based on Representations.................... 35
3.4 Bibliographical Notes.................................. 37
II Approximation of SVFs with Images in Mn 39
4. Methods Based on Canonical Representations 41
4.1 Induced Operators...................................... 41
4.2 Approximation Results ................................. 43
4.3 Application to SVFs with Convex Images................. 45
4.4 Examples and Conclusions .............................. 48
4.5 Bibliographical Notes.................................. 51
5. Methods Based on Minkowski Convex Combinations 53
5.1 Spline Subdivision Schemes for Convex Sets............. 54
5.2 Non-Convexity Measures of a Compact Set................ 57
5.3 Convexification of Sequences of Sample-Based
Positive Operators..................................... 59
5.4 Convexification by Spline Subdivision Schemes.......... 61
5.5 Bibliographical Notes.................................. 63
6. Methods Based on the Metric Average 65
6.1 Schoenberg Spline Operators............................ 66
6.2 Spline Subdivision Schemes............................. 71
6.3 Bernstein Polynomial Operators......................... 76
6.4 Bibliographical Notes.................................. 82
7. Methods Based on Metric Linear Combinations 85
7.1 Metric Piecewise Linear Interpolation.................. 86
7.2 Error Analysis......................................... 91
7.3 Multifunctions with Convex Images...................... 94
7.4 Specific Metric Operators.............................. 95
7.4.1 Metric Bernstein operators...................... 95
Contents ix
7.4.2 Metric Schoenberg operators................... 96
7.4.3 Metric polynomial interpolation............... 97
7.5 Bibliographical Notes................................ 99
8. Methods Based on Metric Selections 101
8.1 Metric Selections.................................... 101
8.2 Approximation Results ............................... 104
8.3 Bibliographical Notes................................ 106
III Approximation of SVFs with Images in R 107
9. SVFs with Images in 1 109
9.1 Preliminaries on the Graphs of SVFs.................. 110
9.2 Continuity of the Boundaries of a CBV
Multifunction ....................................... 112
9.3 Regularity Properties of the Boundaries................ 116
10. Multi-Segmental and Topological Representations 121
10.1 Multi-Segmental Representations (MSRs)................. 121
10.2 Topological MSRs....................................... 126
10.2.1 Existence of a topological MSR.................. 127
10.2.2 Conditions for uniqueness of a TMSR............. 130
10.3 Representation by Topological Selections .............. 134
10.4 Regularity of SVFs Based on MSRs....................... 135
11. Methods Based on Topological Representation 137
11.1 Positive Linear Operators Based on TMSRs............... 137
11.1.1 Bernstein polynomial operators ................. 139
11.1.2 Schoenberg operators............................ 141
11.2 General Operators Based on Topological Selections . . . 142
11.3 Bibliographical Notes to Part III...................... 144
Bibliography 145
Index
151
|
| any_adam_object | 1 |
| author | Dyn, Nira Farkhi, Elza Mokhov, Alona |
| author_GND | (DE-588)1065934696 (DE-588)1065934564 (DE-588)1065934475 |
| author_facet | Dyn, Nira Farkhi, Elza Mokhov, Alona |
| author_role | aut aut aut |
| author_sort | Dyn, Nira |
| author_variant | n d nd e f ef a m am |
| building | Verbundindex |
| bvnumber | BV042527496 |
| callnumber-first | Q - Science |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 905 |
| ctrlnum | (OCoLC)911245736 (DE-599)BVBBV042527496 |
| dewey-full | 515/.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.8 |
| dewey-search | 515/.8 |
| dewey-sort | 3515 18 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV042527496 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:24:10Z |
| institution | BVB |
| isbn | 9781783263028 1783263024 |
| language | English |
| lccn | 014023451 |
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| physical | XIII, 153 S. Ill., graph. Darst. |
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| spelling | Dyn, Nira Verfasser (DE-588)1065934696 aut Approximation of set-valued functions adaptation of classical approximation operators Nira Dyn ; Elza Farkhi ; Alona Mokhov London Imperial College Press 2014 XIII, 153 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (pages 145-150) and index Approximation theory Linear operators Function spaces Approximation (DE-588)4002498-2 gnd rswk-swf Mengenwertige Funktion (DE-588)4342651-7 gnd rswk-swf Mengenwertige Funktion (DE-588)4342651-7 s Approximation (DE-588)4002498-2 s DE-604 Farkhi, Elza Verfasser (DE-588)1065934564 aut Mokhov, Alona Verfasser (DE-588)1065934475 aut Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027961794&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027961794&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dyn, Nira Farkhi, Elza Mokhov, Alona Approximation of set-valued functions adaptation of classical approximation operators Approximation theory Linear operators Function spaces Approximation (DE-588)4002498-2 gnd Mengenwertige Funktion (DE-588)4342651-7 gnd |
| subject_GND | (DE-588)4002498-2 (DE-588)4342651-7 |
| title | Approximation of set-valued functions adaptation of classical approximation operators |
| title_auth | Approximation of set-valued functions adaptation of classical approximation operators |
| title_exact_search | Approximation of set-valued functions adaptation of classical approximation operators |
| title_full | Approximation of set-valued functions adaptation of classical approximation operators Nira Dyn ; Elza Farkhi ; Alona Mokhov |
| title_fullStr | Approximation of set-valued functions adaptation of classical approximation operators Nira Dyn ; Elza Farkhi ; Alona Mokhov |
| title_full_unstemmed | Approximation of set-valued functions adaptation of classical approximation operators Nira Dyn ; Elza Farkhi ; Alona Mokhov |
| title_short | Approximation of set-valued functions |
| title_sort | approximation of set valued functions adaptation of classical approximation operators |
| title_sub | adaptation of classical approximation operators |
| topic | Approximation theory Linear operators Function spaces Approximation (DE-588)4002498-2 gnd Mengenwertige Funktion (DE-588)4342651-7 gnd |
| topic_facet | Approximation theory Linear operators Function spaces Approximation Mengenwertige Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027961794&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027961794&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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