Korovkin-Type Approximation Theory and Its Applications

Description

Since their discovery the simplicity and, at the same time, the power of the classical theorems of Korovkin impressed several mathematicians. During the last thirty years a considerable amount of research extended these theorems to the setting of different function spaces or more general abstract spaces such as Banach lattices, Banach algebras, Banach spaces and so on. This work, in fact, delineated a new theory that we may now call Korovkin- type approximation theory. At the same time, strong and fruitful connections of this theory have been revealed not only with classical approximation theory, but also with other fields such as functional analysis, harmonic analysis, measure theory, probability the- ory and partial differential equations. This has been accomplished by a large number of mathematicians ranging from specialists in approximation theory to functional analysts. A selected part of the theory is already documented in the monographs of Donner [1982J and Keimel and Roth [1992]. With this book we hope to contribute further to the subject by presenting a modern and comprehensive exposition of the main aspects of the theory in spaces of continuous functions (vanishing at infinity, respectively) defined on a compact space (a locally compact space, respectively), together with its main applications. We have chosen to treat these function spaces since they playa central role in the whole theory and are the most useful for the applications. Besides surveying both classical and recent results in the field, the book also contains a certain amount of new material. In any case, the majority of the results appears in a book for the first time.