Korovkin-type Approximation Theory and its Applications

The power of the original result by Korovkin impressed many mathematicians and hence a considerable amount of research extended this theorem to the setting of different function spaces or more general abstract spaces such as Banach lattices, Banach algebras, Banach spaces and so on. At the same time, strong and fruitful connections of this theory have also been revealed not only with classical approximation theory, but also with other fields such as functional analysis (abstract Choquet boundaries and convexity theory, uniqueness of extensions of positive linear forms, convergence of sequences of positive linear operators in Banach lattices, structure theory of Banach lattices, convergence of sequences of linear operators in Banach algebras and in C*-algebras, structure theory of Banach algebras, approximation problems in function algebras), harmonic analysis (convergence of sequences of convolution operators on function spaces and function algebras on (locally) compact topological groups, structure theory of topological groups), measure theory and probability theory (weak convergence of sequences of positive Radon measures and positive approximation processes constructed by probabilistic methods), and partial differential equations (approximation of solutions of Dirichlet problems and of diffusion equations). This work, in fact, delineated a new theory called Korovkin-type approximation theory. The reader will find a quite complete picture of what has been achieved in the field, a modern and comprehensive exposition of the main aspects of Korovkin-type approximation theory in spaces of continuous real functions together with its main applications. The function spaces we have chosen to treat play a central role in the whole theory and are the most useful for the applications in the various univariate, multivariate and infinite dimensional settings. The book is mainly intended as a reference text for research workers in the field; a large part of it can also serve as a textbook for a graduate level course. The organization of the material does not follow the historical development of the subject and allows us to present the most important part of the theory in a concise way. Chapters 2, 3 and 4 are devoted to the main aspects of Korovkin-type approximation theory in C_0(X) and C(X)-spaces. In Chapter IV we also point out the strong interplay between KAT and Choquet's integral representation theory, as well as Stone-Weierstrass-type theorems. Chapters 5 and 6 are mainly concerned with applications to: Approximation of continuous functions by means of positive linear operators, Approximation and representation of the solutions of particular partial differential equations of diffusion type, by means of powers of positive linear operators, More precisely, in Chapter 5 we give the first and best-known applications of Korovkin-type approximation theory. We describe different kinds of positive approximation processes. Particular care is devoted to probabilistic-type operators, discrete-type operators, convolution operators for periodic functions and summation methods. In the final Chapter 6 we present a detailed analysis of some further sequences of positive linear operators that have been studied recently. These operators play an important role in some fine aspects of approximation theory. They connect the theory of C_0-semigroups of operators, partial differential equations and Markov processes. The main examples we consider are the Bernstein-Schnabl operators, the Stancu-Schnabl operators and the Lototsky-Schnabl operators. Subsequently we show how these operators are strongly connected with initial and (Ventcel-type) boundary value problems in the theory of partial differential equations. Although the aim of the book is to survey both classical and recent results in the field, the reader will find a certain amount of new material. In any case, the majority of the results presented here appears in a book for the first time.