Korovkin-type Approximation Theory and Its Applications

✍ Scribed by Francesco Altomare; Michele Campiti

• Publisher: De Gruyter
• Year: 1994
• Tongue: English
• Leaves: 640
• Series: De Gruyter Studies in Mathematics; 17
• Edition: Reprint 2011
• Category: Library

✦ Synopsis

"The book is an excellent reference text and may be used as a textbook for a graduate level course. [...]The authors have to be congratulated for their colossal effort in systematizing their own original results with the most important other results in KAT in a book form easy to use for references." Mathematical Reviews

"The historical notes and the references are very detailed. Well written and well produced, this comprehensive research monograph is a timely and welcome addition to the mathematical literature." Zentralblatt für Mathematik

✦ Table of Contents

Introduction
Interdependence of sections
Notation
Chapter 1. Preliminaries
1.1 Topology and analysis
1.2 Radon measures
* 1.3 Some basic principles of probability theory
1.4 Selected topics on locally convex spaces
* 1.5 Integral representation theory for convex compact sets
* 1.6 Co-Semigroups of operators and abstract Cauchy problems
Chapter 2. Korovkin-type theorems for bounded positive Radon measures
2.1 Determining subspaces for bounded positive Radon measures
2.2 Determining subspaces for discrete Radon measures
2.3 Determining subspaces and Chebyshev systems
2.4 Convergence subspaces associated with discrete Radon measures
2.5 Determining subspaces for Dirac measures
2.6 Choquet boundaries
Chapter 3. Korovkin-type theorems for positive linear operators
3.1 Korovkin closures and Korovkin subspaces for positive linear operators
3.2 Special properties of Korovkin closures
3.3 Korovkin subspaces for positive projections
3.4 Korovkin subspaces for finitely defined operators
Chapter 4. Korovkin-type theorems for the identity operator
4.1 Korovkin closures and Korovkin subspaces for the identity operator
4.2 Strict Korovkin subsets. Korovkin’s theorems
4.3 Korovkin closures and state spaces. Spaces of parabola-like functions
4.4 Korovkin closures and Stone-Weierstrass theorems
4.5 Finite Korovkin sets
Chapter 5. Applications to positive approximation processes on real intervals
5.1 Moduli of continuity and degree of approximation by positive linear operators
5.2 Probabilistic methods and positive approximation processes
5.3 Discrete-type approximation processes
5.4 Convolution operators and summation processes
Chapter 6. Applications to positive approximation processes on convex compact sets
6.1 Positive approximation processes associated with positive projections
6.2 Positive projections and their associated Feller semigroups
6.3 Miscellaneous examples and degenerate diffusion equations on convex compact subsets of ℝP
Appendices
A. Korovkin-type approximation theory on commutative Banach algebras
A.1 Universal Korovkin-type approximation theory on commutative Banach algebras
A.2 Commutative group algebras
A.3 Finitely generated commutative Banach algebras and polydisk algebras
A.4 Generalized analytic functions and algebras generated by inner functions
A.5 Extreme spectral states and the Gleason-Kahane-Zelazko property
B. Korovkin-type approximation theory on C*-algebras
B.1 Approximation by positive linear functionals
B.2 Approximation by positive linear operators
C. A list of determining sets and Korovkin sets
C.1 Determining sets in C0(X) (X locally compact Hausdorff space)
C.2 Determining sets in C(X) (X compact)
C.3 Korovkin sets in C0(X) (X locally compact Hausdorff space)
C.4 Korovkin sets in C(X) (X compact Hausdorff space)
C.5 Korovkin sets in Lp(Χ,μ)-spaces
D. A subject classification of Korovkin-type approximation theory with a subject index
D.1 Subject classification (SC)
D.2 Subject index
Bibliography
Symbol index
Subject index