This paper is concerned with asymptotic formulae for sequences of positive linear operators acting on weighted spaces of continuous functions defined on a real interval. The main result provides a characterization of those continuous functions for which a pointwise asymptotic formula holds true. The method is essentially based on a characterization of the domains of generators of C0-semigroups in terms of asymptotic formulae. Finally, several applications concerning, among others, Kantorovich operators, exponential operators, Gauss-Weierstrass operators and

In this paper we study a class of elliptic second-order differential operators on finite dimensional convex compact sets whose principal part degenerates on a subset of the boundary of the domain. We show that the closures of these operators generate Feller semigroups. Moreover, we approximate these semigroups by iterates of suitable positive linear operators which we also introduce and study in this paper for the first time, and which we refer to as modified Bernstein-Schnabl operators. As a consequence of this approximation we investigate some regularity properties preserved by the semigroup. Finally, we consider the special case of the finite dimensional simplex and the well-known Wright-Fisher diffusion model of gene frequency used in population genetics.

Deepening the study of a new approximation sequence of positive linear operators we introduced and studied in [10], in this paper we disclose its relationship with the Markov semigroup (pre)generation problem for a class of degenerate second-order elliptic dierential operators which naturally arise through an asymptotic formula, as well as with the approximation of the relevant Markov semigroups in terms of the approximating operators themselves. The analysis is carried out in the context of the space C (K) of all continuous functions dened on an arbitrary compact convex subset K of Rd, d 1, having non-empty interior and a not necessarily smooth boundary, as well as, in some particular cases, in Lp(K) spaces, 1 <p < +\infty. The approximation formula also allows to infer some preservation properties of the semigroup such as the preservation of the Lipschitz-continuity as well as of the convexity. We nally apply the main results to some noteworthy particular settings such as balls and ellipsoids, the unit interval and multidimensonal hypercubes and simplices. In these settings the relevant dierential operators fall into the class of Fleming-Viot operators.

Following a well-established tradition initiated on 1989, during the week of September 24-30, 2009, the Sixth International Conference on Functional Analysis and Approximation Theory was held at Acquafredda di Maratea (Potenza). Earlier meetings were held in 1989, 1992, 1996, 2000 and 2004 at the same place. The Conference was organized by the Center for Studies on Functional Analysis and Approximation Theory of the University of Basilicata (Potenza), directed by Prof. Giuseppe Mastroianni, with the collaboration of the Research Group “Real Analysis and Methods of Functional Analysis for Differential Problems and Approximation Problems” of the University of Bari, directed by Prof. Francesco Altomare. Besides Francesco Altomare and Giuseppe Mastroianni, the organizing committee consisted also of Antonio Attalienti, Michele Campiti, Mirella Cappelletti Montano, Lorenzo D’Ambrosio, Maria Carmela De Bonis, Sabrina Diomede, Vita Leonessa, Donatella Occorsio and Maria Grazia Russo. The meeting was devoted to some significant aspects of contemporary mathematical research on Functional Analysis, Operator Theory and Approximation Theory including the applications of these fields in other areas such as partial differential equations, integral equations and numerical analysis. A special session was dedicated to Professor Giuseppe Mastroianni on the occasion of his 70th birthday, as a fair token of appreciation for his contribution to mathematics and to the organization of all the Maratea conferences (1989 – 2009). The Conference was attended by over 150 mathematicians coming from more than 30 countries. The scientific program consisted of 14 plenary lectures and about 120 communications. Several thematic sessions were arranged. Among the main topics treated therein we mention: - Banach spaces, approximation in normed spaces. - Fixed points theorems and best approximation. - Banach lattices, - Banach algebras, topological algebras and operator algebras. - Function spaces. - Linear and nonlinear operators on Banach spaces, positive operators, spectral theory - Semigroups of operators, Feller semigroups and evolution equations. - Ordinary and partial differential equations. - Real functions and inequalities, complex approximation. - Approximation by positive operators, approximation by nonlinear expressions, rate of convergence. - Polynomial approximation, rational approximation. - Interpolation and quadrature formulas. - Orthogonal polynomials, special functions. - Integral equations. - Integral operators, integral transforms and Fourier analysis. - Integral inequalities. - Numerical analysis. In this volume we collect some invited lectures together with a selection of the papers corresponding to the research talks, which have been carefully refereed. Our thanks go to the referees for their collaboration as well as for their accurate work. We gratefully acknowledge the financial support of the Institutions listed below: • the University of Basilicata, • the University of Bari, • the Faculty of Sciences of the University of Basilicata, • the Department of Mathematics and Computer Science of the University of Basilicata (Potenza), • the Department of Mathematics of the University of Bari, • the Department of Economic Sciences and Mathematical Methods of the University of Bari, • the Department of Mathematics of the University of Salento, • the National Group for Mathematical Analysis, Probability and their Applications (IndAM - G.N.A.M.P.A.), • The Research Group “Partial Differential Equations and Mathematical Finance”, University of Bari, • the Basilicata Tourism Board. We are particularly indebted to Sabina Milella and Graziana Musceo as well as to several other Ph.D and Post-Doc students who assisted and helped us in organizing the Conference secretariat ser

In this paper we deepen the study of a sequence of positive linear operators acting on $L^1([0,1]^N)$, $N \geq 1$, that have been introduced in \cite{AltomareCappellettiLeonessa} and that generalize the multidimensional Kantorovich operators (see \cite{zhou}). We show that particular iterates of these operators converge on $\mathscr{C}([0,1]^N)$ to a Markov semigroup and on $L^p([0,1]^N)$, $1 \leq p &lt;+\infty$, to a positive $C_0$-semigroup (that is an extension of the previous one). The generators of these $C_0$-semigroups are determined in a core of their domains, where they coincide with an elliptic second-order differential operator whose principal part degenerates on the vertices of the hypercube $[0,1]^N$.

The main aim of this survey paper is to give a detailed self-contained introduction to the field as well as a secure entry into a theory that provides useful tools for understanding and unifying several aspects pertaining, among others, to real and functional analysis and which leads to several applications in constructive approximation theory and numerical analysis. This paper, however, not only presents a survey on Korovkin-type theorems but also contains several new results and applications. Moreover, the organization of the subject follows a simple and direct approach which quickly leads both to the main results of the theory and to some new ones. In Sections 3 and 4, we discuss the first and the second theorem of Korovkin. We obtain both of them from a simple unifying result which we state in the setting of metric spaces (see Theorem 3.2). This general result also implies the multidimensional extension of Korovkin's theorem due to Volkov (see Theorem 4.1). Moreover, a slight extension of it into the framework of locally compact metric spaces allows to extend the Korovkin's theorems to arbitrary real intervals or, more generally, to locally compact subsets of Rd. Throughout the two sections, we present some applications concerning several classical approximation processes ranging from Bernstein operators on the unit interval or on the canonical hypercube and the multidimensional simplex, to Kantorovich operators, from Fejér operators to Abel-Poisson operators, from Sz\'{a}sz-Mirakjan operators to Gauss-Weierstrass operators. We also prove that the first and the second theorems of Korovkin are actually equivalent to the algebraic and the trigonometric version, respectively, of the classical Weierstrass approximation theorem. Starting from Section 5, we enter into the heart of the theory by developing some of the main results in the framework of the space C0(X) of all real-valued continuous functions vanishing at infinity on a locally compact space X and, in particular, in the space C(X) of all real-valued continuous functions on a compact space X. We choose these continuous function spaces because they play a central role in the whole theory and are the most useful for applications. Moreover, by means of them it is also possible to easily obtain some Korovkin-type theorems in weighted continuous function spaces and in Lp-spaces. These last aspects are treated at the end of Section 6 and in Section 8. We point out that we discuss Korovkin-type theorems not only with respect to the identity operator but also with respect to a positive linear operator on C0(X) opening the door to a variety of problems some of which are still unsolved. In particular, in Section 10, we present some results concerning positive projections on C(X), X compact, as well as their applications to the approximation of the solutions of Dirichlet problems and of other similar problems. In Sections 6 and 7, we present several results and applications concerning Korovkin sets for the identity operator. In particular, we show that, if M is a subset of C0 (X) that separates the points of X and if f0 in C0(X) is strictly positive, then f0 ,f 0 M, f0 M2 is a Korovkin set in C0(X). This result is very useful because it furnishes a simple way to construct Korovkin sets, but in addition, as we show in Section 9, it turns out that it is equivalent to the Stone generalization to C0 (X)-spaces of the Weierstrass theorem. We also mention that, at the end of Sections 7 and 10, we present some applications concerning Bernstein-Schnabl operators associated with a positive linear operator and, in particular, with a positive projection. These operators are useful for the approximation of not just continuous functions but also (and this was the real reason for the increasing interest in them) positive semigroups and hence the solut

We are mainly concerned with the asymptotic behaviour of both discrete and continuous semigroups of Markov operators acting on the space C(X) of all continuous functions on a compact metric space X. We establish a simple criterion under which such semigroups admit a unique invariant probability measure $\mu$ on X that determines their limit behaviour on C(X) and on L^p(X,\mu). The criterion involves the behaviour of the semigroups on Lipschitz continuous functions and on the relevant Lipschitz seminorms. Finally, we discuss some applications concerning the Kantorovich operators on the hypercube and the Bernstein-Durrmeyer operator with Jacobi weights on [0,1]. As a consequence we determine the limit of the iterates of these operators as well as of their corresponding Markov semigroups whose generators fall in the class of Fleming-Viot differential operators arising in population genetics.

During the last twenty years important progresses have been made, from the point of view of constructive approximation theory, in the study of initial-boundary value differential problems of parabolic type governed by positive 0-semigroups of operators. The main aim of this approach is to construct suitable positive approximation processes whose iterates strongly converge to the semigroups which, as it is well-known, in principle furnish the solutions to the relevant initial-boundary value differential problems. By means of such kind of approximation it is then possible to investigate, among other things, preservation properties and the asymptotic behavior of the semigroups, i.e., spatial regularity properties and asymptotic behavior of the solutions to the differential problems. This series of researches, for a survey on which we refer to [25] and the references therein, has its roots in several studies developed between 1989 and 1994 which are documented in Chapter 6 of the monograph [18]. These studies were concerned with special classes of second-order elliptic differential operators acting on spaces of smooth functions on finite dimensional compact convex subsets, which are generated by a positive projection. The projections themselves offer the tools to construct an approximation process whose iterates converge to the relevant semigroup, making possible the development of a qualitative analysis as above. This theory disclosed several interesting applications by stressing the relationship among positive semigroups, initial-boundary value problems, Markov processes and approximation theory, and by offering, among other things, a unifying approach to the study of diverse differential problems. Nevertheless, over the subsequent years, it has naturally arisen the need to extend the theory by developing a parallel one for positive operators rather than for positive projections and by trying to include in the same project of investigations more general differential operators having a first-order term. The aim of this research monograph is to accomplish such an attempt by considering complete second-order (degenerate) elliptic differential operators whose leading coefficients are generated by a positive operator, by means of which it is possible to construct suitable approximation processes which approximate the relevant semigroups. Some aspects of the theory are treated also in infinite dimensional settings. viii Preface The above described more general framework guarantees to notably enlarge the class of differential operators by including, in particular, those which can be obtained by usual operations with positive operators such as convex combinations, compositions, tensor products and so on. Moreover, this non-trivial generalization discloses new challenging problems as well. However, the approximation processes which we construct in terms of the given positive operator seem to have an interest in their own also for the approximation of continuous functions. For these reasons, a special emphasis is placed upon various aspects of this theme as well.

Of concern are multiplicative perturbations of the Laplacian acting on weighted spaces of continuous functions on R^N. It is proved that such differential operators, defined on their maximal domains, are pre-generators of positive quasicontractive C_0-semigroups of operators that fulfil the Feller property. Accordingly, these semigroups are associated with a suitable probability transition function and hence with a Markov process on R^N. An approximation formula for these semigroups is also stated in terms of iterates of integral operators that generalize the classical Gauss-Weierstrass operators. Some applications of such approximation formula are finally shown concerning both the semigroups and the associated Markov processes.

In this paper we introduce and study two new sequences of positive linear operators acting on the space of all Lebesgue integrable functions defined, respectively, on the $N$-dimensional hypercube and on the $N$-dimensional simplex ($N \geq 1$). These operators represent a natural generalization to the multidimensional setting of the ones introduced in \cite{AltomareLeonessaCn1} and, in a particular case, they turn into the multidimensional Kantorovich operators on these frameworks. We study the approximation properties of such operators with respect both to the sup-norm and to the $L^p$-norm and we give some estimates of their rate of convergence by means of certain moduli of smoothness.

This paper is mainly concerned with the study of the generators of those positive C0-semigroups on weighted continuous function spaces that leave invariant a given closed sublattice of bounded continuous functions and whose relevant restrictions are Feller semigroups. Additive and multiplicative perturbation results for this class of generators are also established. Finally, some applications concerning multiplicative perturbations of the Laplacian on Rn, n ≥ 1, and degenerate second-order differential operators on unbounded real intervals are showed.

In this paper we introduce and study a sequence of positive linear operators acting on suitable spaces of measurable functions on [0, +\infty[, including L^p([0, +\infty[) spaces, 1 \leq p <+\infty, and continuous function spaces with polynomial weights. These operators generalize the Sz\'{a}sz-Mirakjan-Kantorovich operators and they allow to approximate (or to reconstruct) suitable measurable functions by knowing their mean values on a sequence of subintervals of [0,+\infty[ that do not constitute a subdivision of it. We also give some estimates of the rates of convergence by means of suitable moduli of smoothness.

In this paper we introduce and study a new class of elliptic second-order differential operators on a convex compact subset K of R^d, which are associated with a Markov operator T on C(K) and which degenerate on a suitable subset of K containing its extreme points. Among other things, we show that the closures of these operators generate Markov semigroups. Moreover, we prove that these semigroups can be approximated by means of iterates of suitable positive linear operators, which are referred to as the Bernstein-Schnabl operators associted with T. As a consequence we show that the semigroups preserve polynomials of a given degree as well as Holder continuity which gives rise some spatial regularity properties of the solutions of the relevant evolution equations.

The paper is concerned with a special class of positive linear operators acting on the space C(K) of all continuous functions defined on a convex compact subset K of R^d, having non-empty interior. Actually, this class consists of all positive linear operators T on C(K) which leave invariant the polynomials of degree at most $1$ and which, in addition, map polynomials into polynomials of the same degree. Among other things, we discuss the existence of such operators in the special case where K is strictly convex by also characterizing them within the class of positive projections. In particular we show that such operators exist if and only if the boundary of K is an ellipsoid. Furthermore, a characterization of balls of R^d in terms of a special class of them is furnished. Additional results and illustrative examples are presented as well.

We establish a simple criterion concerning the convergence of nets (or generalized sequences) of positive linear operators on $C(X)$, $X$ compact, toward a positive linear operator. As a consequence we discuss several criteria about the asymptotic behaviour of the iterates of Markov operators. Finally, we present some applications concerning the asymptotic behaviour of the iterates of Bernstein-Schnabl operators on a convex compact subset (not necessarily associated with a positive projection) and the iterates of a special linear operator generalizing the so-called Ces\`{a}ro operator on $C([0,1])$

In this paper we study a class of degenerate second-order elliptic differential operators, often referred to as Fleming-Viot type operators, in the framework of function spaces defined on the d-dimensional hypercube Qd of Rd, d ≥ 1. By making mainly use of techniques arising from approximation theory, we show that their closures generate positive semigroups both in the space of all continuous functions and in weighted Lp-spaces. In addition, we show that the semigroups are approximated by iterates of certain polynomial type positive linear operators, which we introduce and study in this paper and which generalize the Bernstein-Durrmeyer operators with Jacobi weights on [0, 1]. As a consequence, after determining the unique invariant measure for the approximating operators and for the semigroups, we establish some of their regularity properties along with their asymptotic behaviours.