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Abstract and Applied Analysis
Volume 2011, Article ID 670509, 11 pages
http://dx.doi.org/10.1155/2011/670509
Research Article

Asymptotic Behaviour of the Iterates of Positive Linear Operators

Department of Mathematics, Technical University of Cluj-Napoca, Street Memorandumului 28, 400114 Cluj-Napoca, Romania

Received 4 December 2010; Accepted 20 March 2011

Academic Editor: Pavel Drábek

Copyright © 2011 Ioan Gavrea and Mircea Ivan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. Introduction

The outstanding results of Kelisky and Rivlin [1] and Karlin and Ziegler [2] provided new insights into the study of the limit behavior of the iterates of linear operators defined on . They have attracted a lot of attention lately, and several alternative proofs and generalizations have been given in the last fifty years (see the references).

Nevertheless, the general problem concerning the overiterates of positive linear operators remained unsolved. This can be stated as follows.

Let be a compact topological space, let be the linear space of all continuous real-valued functions defined on and endowed with the norm , and let denote a positive linear operator, that is, for all . The problem is to provide sufficient conditions for the convergence of the sequence of iterates and find its limit, which is the goal of this paper.

Various techniques from different areas such as spectral theory, probability theory, fixed point theory, and the theory of semigroups of operators, have been employed in the attempts to find a solution (see, in chronological order, [122] and the references therein).

However, although many useful contributions have been made, the problem, in its generality, remained unsolved. The limit remained unknown for a long while even for the case restricted to classical particular positive linear operators.

For the first time, a solution to the general problem of the asymptotic behavior of the iterates of positive linear operators defined on was announced by the authors of this paper at the APPCOM08 conference held in Niš, Serbia, in August 2008. Related results appeared one year later in [19].

This paper describes the employment of a number of completely new methods in solving the general problem. To the best of our knowledge, this is the most general result known up to date. As an application of the main result, the asymptotic behavior of the iterates of many classical and new positive linear operators is deduced.

2. Notations and Preliminary Results

Throughout this paper, we will use the following notations: is a compact topological space; is the normed linear space of all continuous real-valued functions defined on ; ; is a linear subspace of including the space of constants; is a positive linear operator preserving the elements of ; is an interpolation operator; is the interpolation set of , The existence of such an operator is always assured. Indeed, for fixed , the operator , is an interpolation operator with interpolation set .

We also emphasize that if is a positive linear operator preserving the affine functions, then interpolates at the end points for all . This well-known result is a particular case of a theorem of Bauer, see [23] and [24, Proposition 1.4]. It follows that , are the monomial functions , , is the Lagrange interpolation operator .

3. The Main Results

By using the notations presented in Section 2, the following theorem is the main result of the paper.

Theorem 3.1. If there exists such that then Moreover, if is a compact metric space, then the convergence is uniform.

Proof. Let . The case when is trivial. Indeed, in this case, since and preserves the elements of , we have , and hence , .
If , for sufficiently small , the inverse image of the open set under the continuous function is an open set , . It follows that Since is compact and is open, it follows that is a nonempty compact subset of , and we obtain Consequently, the following decisive inequality is satisfied. By applying the positive operator to (3.5), we get Since , we obtain The sequence is monotone and bounded. It follows that it is pointwise convergent. Since was chosen arbitrarily, by using (3.6) we deduce that .
In the particular case when is a compact metric space, since by Dini's Theorem, we obtain that . From the inequalities we deduce that .

In the following we give more information on the limit operator .

Theorem 3.2. The limit interpolation operator is unique, positive, and satisfies the equalities

Proof. The unicity and positivity of the operator follow from the existence of limit . Since preserves the elements of , we obtain that . Taking into account the relations we can repeat the proof of Theorem 3.1 by starting with instead of . We deduce that , and hence .

An immediate corollary of Theorem 3.1 is the following.

Corollary 3.3. If is a positive linear operator possessing an interpolation point and there exists such that on , then

Proof. In Theorem 3.1 we take .

Remark 3.4. The existence of the function is essential here, in the sense that, if it is not satisfied, then the statement of Theorem 3.1 might not be true. Indeed, the positive linear operator defined by preserves the space of constants , and the operator , , is interpolator on . However, there exists no continuous function such that , for all and the sequence has no limit.

4. Applications

In this section, as applications of Theorem 3.1 and Corollary 3.3, we rediscover known results and obtain new ones concerning the asymptotic behaviour of the iterates of positive linear operators.

4.1. Positive Operators on Preserving Linear Functions

In the case of the particularisations, , is the space of all linear functions in , is the Lagrange interpolation operator of degree one associated to at the endpoints 0 and 1, and , by Theorem 3.1, we have that the following corollary holds.

Corollary 4.1. Let be a positive linear operator preserving the linear functions. If there exists such that on , then the sequence of the iterates of converges uniformly to the Lagrange operator .

4.2. The Meyer-König and Zeller Operators

In 1960 Meyer-König and Zeller, see [25], introduced a sequence of positive linear operators which were studied, modified, and generalized by several authors. The classical Meyer-König and Zeller operators , , in the modified version of Cheney and Sharma, see [26], are defined by Moreover, from [27, Equation (2.4)], see also [28], we have that For , we have, as a consequence of Corollary 3.3, that the following corollary holds.

Corollary 4.2. The sequence of the iterates of the Meyer-König and Zeller operators (4.1) converges uniformly to the Lagrange interpolation operator .

4.3. The May Positive Linear Operators

The May operators, see [29], are defined by where denotes a kernel function. They satisfy for some and preserve linear functions. For , as a consequence of Corollary 3.3, the following corollary holds true.

Corollary 4.3. The sequence of the iterates of the May operators (4.3) converges uniformly to .

4.4. The Bernstein Operator on a Simplex

Consider the simplex in , , given by The vertices of the simplex are the points , where With the Bernstein approximation operator is defined by The operator preserves the subspace of linear functions The Lagrange interpolation operator is defined by and interpolates all functions in on the set For , we have and, by using Theorem 3.1, we get the following.

Corollary 4.4. The sequence of the iterates of the Bernstein operator associated with the simplex (4.8) converges uniformly to the Lagrange interpolation operator (4.10).

4.5. Positive Operators on Preserving and

In [15] Agratini introduced a sequence of positive linear operators preserving and . In the case of the particularisations, , , , , and is the interpolation operator: as a corollary of Theorem 3.1, we obtain a result of Agratini [15, Theorem  3.1].

4.6. Bernstein-Type Operators Preserving and

Let , . Aldaz et al. [30, Proposition 11] had recently considered the Bernstein-type operators , The operators satisfy

Considering the particularisations, , , , , and is the interpolation operator: as a corollary of Theorem 3.1, we obtain the following corollary.

Corollary 4.5. The sequence of the iterates of the Bernstein-type operators converges uniformly to the operator in (4.16).

4.7. The Cesàro Operator on

In the case when , is the space of constants, , ; , by Theorem 3.1, we generalize the following recent result of Galaz Fontes and Solís.

Corollary 4.6 (see [16, Theorem 3]). Let be positive on such that , and let be the Cesàro mean operator, Then,

4.8. The Bernstein-Stancu Operators

Let . The Bernstein-Stancu operators (see, e.g., [31]), satisfy the following: For , and , by Corollary 3.3, we obtain a result concerning the iterates of the Bernstein-Stancu operators.

Corollary 4.7. The sequence of the iterates of Stancu's operators (4.19) converges uniformly to .

4.9. The Cheney-Sharma Operator

Let , and let be the th Bernstein-Cheney-Sharma operator on , defined by It is known that (see, e.g., [32] and [3, ]. For one has that

Taking , , and , in Corollary 3.3, we obtain the following application.

Corollary 4.8. The sequence of the iterates of the Cheney-Sharma operators (4.21) converges uniformly to .

4.10. The Schurer Operator

For , , the Bernstein-Schurer-type operator [3, (5.3.1)], is defined by One can prove that this operator satisfies In the case, when , , and , by Corollary 3.3, we obtain the following result.

Corollary 4.9. The sequence of the iterates of the Schurer operators (4.23) converges uniformly to .

4.11. Piecewise Bernstein Operators

Let . Consider the Bernstein operator :

In the case of the particularisations, , is the linear space of polygonal lines with vertices possessing abscissae at , and , is the polygonal line with vertices at , , , , and , by Theorem 3.1, we have that the following corollary holds.

Corollary 4.10. For , consider the composite Bernstein operator , Then,

Acknowledgments

The authors thank the referee for useful comments and suggestions. This research was supported by the Romanian CNCSIS Grant ID 162/2008 and by the Project PN2-Partnership no. 11018 MoDef.

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