*(English)*Zbl 0924.41001

In 1953 P. P. Korovkin discovered a criterion in order to decide whether a given sequence $\left({L}_{n}\right)$ of positive linear operators on the space $C[0,1]$ is an approximation process, i.e., ${L}_{n}f\to f$ uniformly on [0,1] for every $f\in C[0,1]$: in fact, it is sufficient to verify that ${L}_{n}f\to f$ only for $f\in \{1,x,{x}^{2}\}$.

This result has been extended to other function spaces and to abstract spaces. Now the Korovkin approximation theory (KAT) has fruitful connections with functional analysis (notably with Choquet’s theory and Banach algebras theory), harmonic analysis, measure and probability theory, partial differential equations; an important feature of this monograph is the systematic presentation of all these connections.

The book is addressed to specialists in the above mentioned fields; a large part of it can also serve as a textbook for a graduate-level course. The authors have contributed significantly to the development of this theory. They give a survey on classical as well as recent developments in the field. Most of the results appear in book form for the first time.

In order to make the exposition self-contained, the prerequisites (from topology and analysis, measure and probability theory, locally convex spaces and Choquet’s integral representation theory, semigroups of operators) are collected in Chapter 1. Chapters 2, 3 and 4 are devoted to the main aspects of KAT in ${C}_{0}\left(X\right)$ $(X$ locally compact) and $C\left(X\right)$ $(X$ compact). The fundamental problem consists in studying, for a given positive linear operator $T:{C}_{0}\left(X\right)\to {C}_{0}\left(Y\right)$, those subspaces $H$ of ${C}_{0}\left(X\right)$ (if any) which have the property that every equicontinuous net of positive linear operators (or positive contractions) from ${C}_{0}\left(X\right)$ into ${C}_{0}\left(Y\right)$ converges strongly to $T$ whenever it converges to $T$ on $H$. (Such subspaces are called Korovkin subspaces for $T$). The classical case when $T$ is the identity operator is discussed in Chapter 4, where the authors present also the strong interplay between KAT and Choquet’s integral representation theory, as well as Stone-Weierstrass-type theorems. Furthermore, the existence of finite dimensional Korovkin subspaces is carefully analysed. The Korovkin subspaces for arbitrary positive linear operators, positive projections and finitely defined operators are characterized in Chapter 3. Here we find also applications to potential theory, Bauer simplices and Chebyshev systems. The results contained in Chapters 3 and 4 are based on the general Korovkin-type theorems for “bounded positive Radon measures, presented in Chapter 2 in connection with the theory of Choquet boundaries. Chapter 5 contains applications to the approximation of continuous functions defined on real intervals, by means of various kinds of operators. The rates of convergence are described by using classical moduli of continuity.

Chapter 6 contains a detailed analysis of some sequences $\left({B}_{n}\right)$ of positive linear operators that have been studied recently; they connect the theory of ${C}_{0}$-semigroups of operators, partial differential equations and Markov processes. The operators ${B}_{n}$ are constructed by means of a positive projection $T$ acting on the space of continuous functions on a convex compact set $K$. This general construction has been proposed by F. Altomare [Ann. Sc. Norm. Sup. Pisa, Cl. Sci., IV. Ser. 16, No. 2, 259-279 (1989; Zbl 0706.47022]. By specializing $K$ and $T$ some well-known approximation processes can be obtained; new approximation processes (e.g., in the context of the infinite-dimensional Bauer simplices) are described and investigated.

The approximation properties of the operators ${B}_{n}$, their monotonicity properties and the preservation of some global smoothness properties are carefully analysed. Then a Feller semigroup is constructed in terms of powers of ${B}_{n}$. The infinitesimal generator of this semigroup is explicitly determined in a core of its domain; in the finite dimensional case it is an elliptic second-order differential operator which degenerates on the Choquet boundary of the range of $T$.

This theory is used to derive a representation and some qualitative properties of the solutions of the Cauchy problems which correspond to the involved diffusion equations. The transition function and the asymptotic behavior of the Markov processes governed by these diffusion equations are also described.

After the publication of the book, the theory presented in Chapter 6 has been further developed; see , e.g., [*M. Campiti, G. Metafune*, J. Approximation Theory 87, No. 3, 243-269 (1996; Zbl 0865.41027) and ibid., 270-290 (1996; Zbl 0874.41010)], [*F. Altomare, I. Carbone*, J. Math. Anal. Appl. 213, No. 1, 308-333 (1997; Zbl 0894.35044)], [*F. Altomare, A. Attalienti*, Math. Z. 225, No. 2, 211-229 (1997; Zbl 0871.41016)], [*A. Favini, J. A. Goldstein, S. Romanelli*, in: Stochastic processes and functional analysis, *J. A. Goldstein*, Marcel Dekker, New York, 1997, 85-100 (1997; Zbl 0889.35039)], [*M. Romito*, Mh. Math. (to appear)], [*F. Altomare* and the reviewer, Atti Semin. Mat. Fis. Univ. Modena 46, Suppl. 13-38 (1998; Zbl 0917.35042)].

In two appendices, written by M. Pannenberg and F. Beckhoff respectively, the main developments of KAT in the setting of Banach algebras are outlined.

The book contains also a subject classification, a symbol index and a subject index, all of them very useful. 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.

##### MSC:

41-02 | Research monographs (approximations and expansions) |

41A36 | Approximation by positive operators |

41A25 | Rate of convergence, degree of approximation |

41A35 | Approximation by operators (in particular, by integral operators) |

41A63 | Multidimensional approximation problems |