Ordered cones and approximation. (English) Zbl 0752.41033

Keimel, Klaus; Roth, Walter

Lecture Notes in Mathematics. 1517. Berlin etc.: Springer-Verlag. vi, 134 p. (1992).

The purpose of these lecture notes is to present a unified treatment of Korovkin type approximation theorems. Such results typically deal with certain restricted classes of linear operators on locally convex vector spaces. The authors found it necessary to leave the setting the vector spaces and turn to more general structures called locally convex cones. It was essential to include cones which are not embeddable in vector spaces. Chapter 1 is devoted to the study of locally convex cones. In Chapter 2, uniformly continuous operators and the dual cone are studied. Subcones are discussed in Chapter 3. The main results on approximation theory are presented in Chapter 4. Some research based on ideas of Nachbin is discussed in Chapter 5. Quantitative estimates for the closeness of the approximation are presented in Chapter 6.
The text is clearly and concisely written. A wealth of material has been presented, much of which had previously been scattered in the literature. This book should prove invaluable to the research worker on approximation theory.

Reviewer: H.R.Dowson (Glasgow)

MSC:

41A65	Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41-02	Research exposition (monographs, survey articles) pertaining to approximations and expansions

Keywords:

Korovkin type approximation theorems; locally convex cones; uniformly continuous operators; Subcones