Ordered Cones and Approximation
This book presents a unified approach to Korovkin-type approximation theorems. It includes classical material on the approximation of real-valuedfunctions as well as recent and new results on set-valued functions and stochastic processes, and on weighted
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1517
Klaus Keimel
Walter Roth
Ordered Cones and Approximation
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest
Authors Klaus Keimel Fachbereich Mathematik Technische Hochschule Darmstadt SchloBgartenstr. 7 W-6100 Darmstadt, Germany Walter Roth Department of Mathematics University of Bahrain P. O. Box 32038 Isa Town State of Bahrain
Mathematics Subject Classification (1991): 41-02, 41A36, 41A65, 46A22, 47H04
ISBN 3-540-55445-9 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55445-9 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera ready by author/editor Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr, 46/3140-543210 - Printed on acid-free paper
Table of Contents Introduction Chapter I:
1
Locally Convex Cones
1.
Cones and preordered cones
2.
Locally convex cones . . . . . .
11
3.
Local and global preorder. Closure .
4. 5.
Cancellation . Locally convex cones via convex quasiuniform structures
15 18
Chapter II:
Uniformly Cone
8
21
Continuous Operators and
the
Dual 25
1. 2. 3.
Uniformly continuous operators . . . . Linear functionals and separation theorems Duality theory for locally convex cones
4. 5.
Extreme points and faces. Uniformly directed cones.
36 42 44
6.
Directional operators
49
28
Chapter III: Subcones 1.
Superharmonic and subharmonic elements. The Sup-Inf-Theorem
2. 3.
.
Uniformly directed subcones . . . . . . . . . . . . . Super- and subharmonicity with uniformly directed subcones
55 60
62
Chapter IV: Approximation 1.
The Convergence Theorem .
68
2.
Some classical applications .
74
VI
Chapter V:
Nachbin Cones
1. Weighted cones of continuous cone-valued functions .
82
2.
A criterion for super- and subharmonicity
3.
Set-valued functions. . . . . . . .
89 97
Chapter VI: Quantitative Estimates 1.
Sequence cones . . . . . . . . . . . . . . . .
2. 3. 4.
Order of convergence for Korovkin type approximation . Smoothness of cone-valued functions . A criterion for the order of convergence
106 109 110 114
References
129
Index . . .
133
Introduction Korovkin type approximation theorems typically deal with certain restricted classes of continuous linear operators on locally convex vector spaces. These may be positive operators on ordered vector spaces or contractions on normed spaces as in the seminal work by Korovkin [29], [30], Shashk
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