Iterates of Markov operators and constructive approximation of semigroups

In this paper we survey some recent results concerning the asymptotic behaviour of the iterates of a single Markov operator or of a sequence of Markov operators. Among other things, a characterization of the convergence of the iterates of Markov operators toward a given Markov projection is discussed in terms of the involved interpolation sets. Constructive approximation problems for strongly continuous semigroups of operators in terms of iterates are also discussed. In particular we present some simple criteria concerning their asymptotic behaviour. Finally, some applications are shown concerning Bernstein-Schnabl operators on convex compact sets and Bernstein-Durrmeyer operators with Jacobi weights on the unit hypercube. A final section contains some suggestions for possible further researches

On some classes of Fleming-Viot type differential operators on the unit interval

Of concern are some classes of initial-boundary value differential problems associated with one-dimensional Fleming-Viot differential operators. Among other things, these operators occur in some models from population genetics to study the fluctuation of gene frequency under the influence of mutation and selection. The main aim of this survey paper is to discuss old and more recent results about the existence, uniqueness and continuous dependence from initial data of the solutions to these problems through the theory of the C0-semigroups of operators. Other additional aspects which will be highlighted, concern the approximation of the relevant semigroups in terms of positive linear operators. The given approximation formulae allow to infer several preservation properties of the semigroups together with their asymptotic behavior. The analysis is carried out in the context of the space C([0; 1]) as well as, in some particular cases, in Lp([0; 1]) spaces, 1< p < +1. Finally, some open problems are also discussed

Korovkin-type Theorems and Approximation by Positive Linear Operators

This survey paper contains a detailed self-contained introduction to Korovkin-type theorems and to some of their applications concerning the approximation of continuous functions as well as of L^p-functions, by means of positive linear operators. The paper 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

A generalization of Kantorovich operators for convex compact subsets

In this paper we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number and a sequence of probability Borel measures. By considering special cases of these parameters for particular convex compact subsets we obtain the classical Kantorovich operators defined in the one-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, the preservation of Lipschitz-continuity as well as of convexity are discussedComment: Research articl

On a class of positive C0C_{0}-semigroups of operators on weighted continuous function spaces

This paper is mainly concerned with the study of the generators of those positive $C_{0}$-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 $\mathbb{R}^{n}$, $n\geq1$, and degenerate second-order differential operators on unbounded real intervals are showed

Kantorovich-type modifications of certain discrete-type operators on the positive real axis

The paper is concerned with the approximation properties of a modification of Kantorovich-type of a general class of operators of discrete-type. Such a modification was introduced by Agratini in 2015; in particular, we focus on extending its approximation properties in several function spaces, including polynomial weighted spaces of any degree as well as $L^p$-spaces. Some estimates of the rate of convergence are also obtained

Continuous selections of Borel measures, positive operators and degenerate evolution problems

In this paper we continue the study of a sequence of positive linear operators which we have introduced in [9] and which are associated with a continuous selection of Borel measures on the unit interval. We show that the iterates of these operators converge to a Markov semigroup whose generator is a degenerate second-order elliptic differential operator on the unit interval. Some qualitative properties of the semigroup, or equivalently, of the solutions of the corresponding degenerate evolution problems, are also investigated

On some density theorems in regular vector lattices of continuous functions

In this paper, we establish some density theorems in the setting of particular locally convex vector lattices of ontinuous functions defined on a locally compact Hausdorff space, which we introduced and studied in $[3, 4]$ and which we named regular vector lattices. In this framework, by using properties of the subspace of the so-called generalized affine functions, we give a simple description of the closed vector sublattice, the closed Stone vector sublattice and the closed subalgebra generated by a subset of a regular vector lattice. As a consequence, we obtain some density results. Finally, a connection with the Korovkin type approximation theory is also shown