Iterates Of Positive Linear Operators Research Articles

Estimates related to the iterates of positive linear operators and their multidimensional analogues

Estimates related to the iterates of positive linear operators and their multidimensional analogues

Iterates of positive linear operators on Bauer simplices

We consider positive linear operators acting on C(K), where K is a metrizable Bauer simplex. For such an operator L we investigate the limit of the iterates L m , when m → ∞ . Qualitative results and rates of convergence are obtained. The general results are illustrated by examples involving classical operators.

The iterates of positive linear operators with the set of constant functions as the fixed point set

Let Ω ⊂ Rp, p ∈ N∗ be a nonempty subset and B(Ω) be the Banach lattice of all bounded real functions on Ω, equipped with sup norm. Let X ⊂ B(Ω) be a linear sublattice of B(Ω) and A: X → X be a positive linear operator with constant functions as the fixed point set. In this paper, using the weakly Picard operators techniques, we study the iterates of the operator A. Some relevant examples are also given.

Power Series of Bernstein Operators and Approximation of Resolvents

According to the theory developed by F. Altomare and his school, certain C 0-semigroups can be approximated by iterates of positive linear operators. A. Albanese, M. Campiti and E. Mangino [J. Appl. Funct. Anal. 1 (2006), 343-358] proved that the resolvent (λ−A)−1 of the infinitesimal generator of such a semigroup can be also approximated, for λ > 0, by suitable iterates. What happens when $${\lambda \to 0^{+}?}$$ We give an answer in the case of the semigroup approximated by the classical Bernstein operators B n on the canonical simplex S of $${\mathbb{R}^{d}}$$ . Specifically, we show that $$-A^{-1}h = \lim\limits_{n \to \infty}\frac{1}{n}{\sum\limits^{\infty}_{k=0}}{B^{k}_{n}h}$$ for h in a certain subspace of C(S). This gives a new method to investigate the qualitative properties of the inverse of A.

The iterates of positive linear operators preserving constants

In this note we introduce a simple and efficient technique for studying the asymptotic behavior of the iterates of a large class of positive linear operators preserving constant functions.

On the iterates of positive linear operators

We have devised a new method for the study of the asymptotic behavior of the iterates of positive linear operators. This technique enlarges the class of operators for which the limit of the iterates can be computed.

Asymptotic Behaviour of the Iterates of Positive Linear Operators

We present a general result concerning the limit of the iterates of positive linear operators acting on continuous functions defined on a compact set. As applications, we deduce the asymptotic behaviour of the iterates of almost all classic and new positive linear operators.

On the iterates of positive linear operators preserving the affine functions

In this note we study the limit behavior of the iterates of a large class of positive linear operators preserving the affine functions and, as a byproduct of our result, we obtain the limit of the iterates of Meyer-König and Zeller operators.

Estimates for Iterates of Positive Linear Operators Preserving Linear Functions

We consider iterates of certain (general) positive linear operators preserving linear functions and derive quantitative upper estimates in terms of weighted and non-weighted moduli of smoothness and related K-functionals. We show that corresponding lower estimates in terms of the classical moduli are not possible, while for the Ditzian–Totik modulus the situation can be different. The results can be applied to several well-known operators; we present here the Bernstein and the genuine Bernstein–Durrmeyer operators.

Overiterated Linear Operators and Asymptotic Behaviour of Semigroups

The results provided hereby are related to the asymptotic behaviour of certain strongly continuous semigroups, which may be expressed in terms of iterates of positive linear operators, in the sense of Altomare’s theory. We present some applications to concrete cases involving continuous and discrete type operators, namely the Beta and the Stancu operators.