82 research outputs found
An invitation to the study of evolution equations by means of positive linear operators,
No full textThis paper provides a detailed survey on a series of methods and results which are centered around the problem of the constructive approximation of the positive semigroups generated by degenerate second order differential operators by means of iterates of positive linear operators. This problem, first states in [7], has been tackled n the last fifteen years from different angles and in different settings such us spaces of bounded and unbounded continuous functions on real intervals as well as on multidimensional convex compact subsets. For the sake of simplicity the presentation is restricted in the framework of compact intervals. The exposition includes several applications as well ad several references for further possible deepening and developments
Approximation of some Feller semigroups associated with a modification of Szasz-Mirakjan-Kantorovich operators
No full textIn this paper, we deepen the study of a sequence of positive linear operators, first introduced in
\cite{altomarecappellettileonessa1}, that generalize the classical
Sz\'{a}sz-Mirakjan-Kantorovich operators. In particular, we
present some qualitative properties and an asymptotic formula for
such a sequence. Moreover, we prove that, under suitable
assumptions, the Feller semigroups generated by the second order
differential operator ( on suitable domains of continuous or
integrable functions may be approximated by means of iterates of
the 's
On a sequence of positive linear operators associated with a continuous selection of Borel measures
No full textn this paper we introduce and study a new sequence of positive linear operators acting on the space of Lebesgue-integrable functions on the unit interval. These operators are defined by means of continuous selections of Borel measures and generalize the Kantorovich operators. We investigate their approximation properties by presenting several estimates of the rate of convergence by means of suitable moduli of smoothness. Some shape preserving properties are also shown
On Bernstein-Schnabl operators on the unit interval
No full textBernstein-Schnabl operators were first introduced by
R. Schnabl in 1968 in the context
of sets of probability Radon measures on compact Hausdorff spaces.
Subsequently Grossman proposed a general method
of constructing Bernstein-Schnabl operators on an arbitrary convex
compact subset of a locally convex space and he showed that they
are an approximation process for continuous functions.
A particular class of these operators has been also studied by the
F. Altomare and,
subsequently, by several other authors. Their construction
essentially involves positive projections and they satisfy many additional
properties useful for the study of evolution problems.
In this paper we deep the study of the Bernstein-Schnabl
operators associated with a general continuous selection of
probability Borel measures on the interval [0,1], which not
necessarily arise from a positive projection. These operators seem
to have some interest because they furnish new general
approximation processes for continuous functions and they also
approximate the solutions of the initial-boundary problems
associated with a class of degenerate diffusion equations.
In the first section we recall their definition and discuss some
examples of them. After that, we investigate their approximation
properties and show several estimates of the rate of convergence
by means of suitable moduli of smoothness.
Shape preserving properties are discussed in Section 2.
In particular, we investigate some conditions under which these
operators preserve the convexity.
In the third section we show that suitable iterates of
Bernstein-Schnabl operators converge to a Markov semigroup on
C([0,1]) whose generator is a degenerate differential
operator of the form Au(x):=\alpha(x) u''(x) (x \in [0,1]
defined on a suitable subspace of smoot functions satisfying the so-called Wentcel boundary conditions.
By means of Bernstein-Schnabl operators we establish some
qualitative properties of this semigroup and, in particular, its
asymptotic behaviour.
In the same section we also study the generation properties of
general differential operators and
determine suitable continuous selections of Borel measures such
that the iterates of the corresponding Bernstein-Schnabl operators
converge to the given Markov semigroup
A note on Weibull operators and their modifications
No full textIn this paper we study, in the context of weighted function spaces of continuous
functions, a class of integral operators of probabilistic type, which are constructed by
means of the Weibull distribution, and a suitable modification that preserves linear
functions.We study the approximation properties of those two sequences of operators
and we present estimates of the rate of convergence, as well as some asymptotic
formulae. In particular, we prove that the proposed modification has a better rate of
convergence. Finally, we show how the modified Weibull operators can be fruitfully
employed to approximate suitable positive C_0-semigroups
A generalization of Bernstein-Durrmeyer operators on hypercubes by means of an arbitrary measure
No full textIn this paper we introduce and study a sequence of Bernstein-Durrmeyer type operators , acting on spaces of continuous or integrable functions on the multi-dimensional hypercube of (), defined by means of an arbitrary measure . We investigate their approximation properties both in the space of all continuous functions and in -spaces with respect to , also furnishing some esitmates of the rate of convergence. Further, we prove an asymptotic formula for the 's. The paper ends with a concrete example
Continuous selections of Borel measures, positive operators and degenerate evolution problems
No full textIn this paper we continue the study of a sequence of positive lin- ear 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
Generalized Kantorovich Operators on Convex Compact Subsets and Their Application to Evolution Problems
No full textIn this short survey paper, we review some of recent results contained in Altomare et al. (Banach J Math Anal 11:591–614, 2017; J Math Anal Appl 458:153–173, 2018) and concerning with the generalized Kantorovich operators C n defined on convex compact subsets of R d (d ≥ 1). Such operators constitute a positive approximation process for continuous functions and, in some cases, for integrable functions. Moreover, an asymptotic formula for such approximating operators leads to a differential operator which pregenerates a Markov semigroup on C(K) for which we obtain an approximation formula, in terms of suitable powers of C n , useful to infer some preservation properties of it and, as a consequence, of solutions to evolution problems associated with the generators
A Sequence of Kantorovich-Type Operators on Mobile Intervals
No full textIn this paper, we introduce and study a new sequence of positive linear operators, acting on both
spaces of continuous functions as well as spaces of integrable functions on [0; 1]. We state some qualitative properties
of this sequence and we prove that it is an approximation process both in C([0; 1]) and in Lp([0; 1]), also providing
some estimates of the rate of convergence. Moreover, we determine an asymptotic formula and, as an application,
we prove that certain iterates of the operators converge, both in C([0; 1]) and, in some cases, in Lp([0; 1]), to a limit
semigroup. Finally, we show that our operators, under suitable hypotheses, perform better than other existing ones in
the literature
- …
