Asymptotic Behaviour of the Iterates of Positive Linear Operators

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 𝐶[0,1]. 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 𝑋 be a compact topological space, let 𝐶(𝑋) be the linear space of all continuous real-valued functions defined on 𝑋 and endowed with the norm ‖𝑓‖∶=sup_𝑥∈𝑋|𝑓(𝑥)|, and let 𝑈∶𝐶(𝑋)→𝐶(𝑋) denote a positive linear operator, that is, 𝑈𝑓≥0 for all 𝑓≥0. The problem is to provide sufficient conditions for the convergence of the sequence of iterates (𝑈^𝑘)_𝑘∈ℕ 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, [1–22] 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 𝐶[0,1] 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:𝑋 is a compact topological space;𝐶(𝑋) is the normed linear space of all continuous real-valued functions defined on 𝑋; ‖𝑓‖∶=max_𝑥∈𝑋|𝑓(𝑥)|;𝒱 is a linear subspace of 𝐶(𝑋) including the space 𝒫₀ of constants;𝑈∶𝐶(𝑋)→C(𝑋) is a positive linear operator preserving the elements of 𝒱;𝐿∶𝐶(𝑋)→𝒱 is an interpolation operator;𝑌_𝐿 is the interpolation set of 𝐿,𝑌_𝐿=𝑦∈𝑋∣𝐿𝑓𝑦=𝑓𝑦,∀𝑓∈𝐶(𝑋)≠∅. (2.1) The existence of such an operator 𝐿 is always assured. Indeed, for fixed 𝑥₀∈𝑋, the operator 𝐿∶𝐶(𝑋)→𝒫₀, 𝐿𝑓∶=𝑓(𝑥₀) is an interpolation operator with interpolation set 𝑌_𝐿={𝑥₀}.

We also emphasize that if 𝑈∶𝐶[0,1]→𝐶[0,1] is a positive linear operator preserving the affine functions, then 𝑈𝑓 interpolates 𝑓 at the end points for all 𝑓∈𝐶[0,1]. This well-known result is a particular case of a theorem of Bauer, see [23] and [24, Proposition 1.4]. It follows that 𝑌_𝐿⊇{0,1},𝑒_𝑖∶[0,1]→ℝ are the monomial functions 𝑒_𝑖(𝑥)=𝑥^𝑖, 𝑖=0,1,…,𝐿₁∶𝐶[0,1]→𝐶[0,1] is the Lagrange interpolation operator 𝐿₁𝑓=𝑓(0)𝑒₀+(𝑓(1)−𝑓(0))𝑒₁.

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 𝜑∈𝐶(𝑋) such that 𝑈𝜑≥𝜑on𝑋,𝑈𝜑≠𝜑on𝑋⧵𝑌_𝐿, (3.1) then lim_𝑘𝑈^𝑘𝑓=𝐿𝑓,∀𝑓∈𝐶(𝑋). (3.2) Moreover, if 𝑋 is a compact metric space, then the convergence is uniform.

Proof. Let 𝑓∈𝐶(𝑋). The case when 𝐿𝑓=𝑓 is trivial. Indeed, in this case, since 𝐿𝑓∈𝒱 and 𝑈 preserves the elements of 𝒱, we have 𝑈𝑓=𝑈𝐿𝑓=𝐿𝑓, and hence 𝑈^𝑘𝑓=𝐿𝑓, 𝑘=1,2,….
If 𝐿𝑓≠𝑓, for sufficiently small 𝜀>0, the inverse image of the open set (−𝜀,𝜀) under the continuous function 𝐿𝑓−𝑓 is an open set 𝐺, 𝑌_𝐿⊆𝐺≠𝑋. It follows that 𝐿𝑓−𝑓<𝜀,on𝐺. (3.3) Since 𝑋 is compact and 𝐺 is open, it follows that 𝑋⧵𝐺⊆𝑋⧵𝑌_𝐿 is a nonempty compact subset of 𝑋, and we obtain 𝑚_𝜀∶=inf_{𝑥∈𝑋⧵𝐺}𝑈𝜑(𝑥)−𝜑(𝑥)>0. (3.4) Consequently, the following decisive inequality 𝐿𝑓−𝑓<𝜀+𝑓−𝐿𝑓𝑚_𝜀𝑈𝜑−𝜑 (3.5) is satisfied. By applying the positive operator 𝑈^𝑘 to (3.5), we get 𝐿𝑓−𝑈^𝑘𝑓<𝜀+𝑓−𝐿𝑓𝑚_𝜀𝑈^𝑘+1𝜑−𝑈^𝑘𝜑. (3.6) Since 𝑈𝜑≥𝜑, we obtain 𝜑≤𝑈^𝑘𝜑≤𝑈^𝑘+1𝜑≤𝜑,𝑘=1,2,…. (3.7) The sequence (𝑈^𝑘𝜑)_𝑘≥1 is monotone and bounded. It follows that it is pointwise convergent. Since 𝜀 was chosen arbitrarily, by using (3.6) we deduce that 𝑈^𝑘𝑓−−−−−−−→^pointwise𝐿𝑓.
In the particular case when 𝑋 is a compact metric space, since 𝑈^𝑘𝜑−−−−−−−→^pointwise𝐿𝜑∈𝐶(𝑋), (3.8) by Dini's Theorem, we obtain that 𝑈^𝑘𝜑−−−−−−−−→^uniformly𝐿𝜑. From the inequalities 𝐿𝑓−𝑈^𝑘𝑓<𝜀+𝑓−𝐿𝑓𝑚_𝜀𝑈^𝑘+1𝜑−𝑈^𝑘𝜑, (3.9) we deduce that 𝑈^𝑘𝑓−−−−−−−−→^uniformly𝐿𝑓.

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

Theorem 3.2. The limit interpolation operator 𝐿 is unique, positive, and satisfies the equalities 𝑈𝐿=𝐿𝑈=𝐿. (3.10)

Proof. The unicity and positivity of the operator 𝐿 follow from the existence of limit lim_𝑘𝑈^𝑘=𝐿. Since 𝑈 preserves the elements of 𝒱, we obtain that 𝑈𝐿=𝐿. Taking into account the relations ∅⊊𝑌_𝐿⊆𝑦∈𝑋∣𝐿𝑈𝑓𝑦=𝑈𝑓𝑦,∀𝑓∈𝐶(𝑋), (3.11) we can repeat the proof of Theorem 3.1 by starting with 𝑈𝑓 instead of 𝑓. We deduce that lim_𝑘𝑈^𝑘𝑓=𝐿𝑈𝑓, and hence 𝐿𝑈=𝐿.

An immediate corollary of Theorem 3.1 is the following.

Corollary 3.3. If 𝑈∶𝐶(𝑋)→𝐶(𝑋) is a positive linear operator possessing an interpolation point 𝑥₀∈𝑋 and there exists 𝜑∈𝐶(𝑋) such that 𝑈𝜑≶𝜑 on 𝑋⧵{𝑥₀}, then lim_𝑘𝑈^𝑘𝑓=𝑓𝑥₀,∀𝑓∈𝐶(𝑋). (3.12)

Proof. In Theorem 3.1 we take 𝐿𝑓=𝑓(𝑥₀).

Remark 3.4. The existence of the function 𝜑 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 𝑈∶𝐶[0,1]→𝐶[0,1] defined by 𝑈𝑓(𝑥)=𝑥𝑓(0)+(1−𝑥)𝑓(1) (3.13) preserves the space of constants 𝒫₀, and the operator 𝐿∶𝐶[0,1]→𝒫₀, 𝐿𝑓(𝑥)=𝑓(0), is interpolator on 𝑌_𝐿={0}. However, there exists no continuous function 𝜑∈𝐶[0,1] such that 𝑈𝜑(𝑥)>𝜑(𝑥), for all 𝑥∈[0,1]⧵{0}(see(𝑈𝜑(0)−𝜑(0))(𝑈𝜑(1)−𝜑(1))=−(𝜑(1)−𝜑(0))²≤0) and the sequence (𝑈^𝑛𝑒₁)_𝑛≥1 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 𝐶[0,1] Preserving Linear Functions

In the case of the particularisations, 𝑋=[0,1], 𝒱 is the space of all linear functions in 𝐶[0,1], 𝐿 is the Lagrange interpolation operator of degree one associated to 𝑓 at the endpoints 0 and 1, and 𝑌_𝐿={0,1}, by Theorem 3.1, we have that the following corollary holds.

Corollary 4.1. Let 𝑈∶𝐶[0,1]→𝐶[0,1] be a positive linear operator preserving the linear functions. If there exists 𝜑∈𝐶[0,1] such that 𝑈𝜑≶𝜑 on (0,1), then the sequence of the iterates of 𝑈 converges uniformly to the Lagrange operator 𝐿₁.

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 𝐌𝐊𝐙_𝑛∶𝐶[0,1]→𝐶[0,1], 𝑛∈ℕ, in the modified version of Cheney and Sharma, see [26], are defined by𝐌𝐊𝐙_𝑛𝑓(𝑥)=^∞_𝑘=0𝑘()𝑛+𝑘1−𝑥^𝑛+1𝑥^𝑘𝑓𝑘[),𝑓(1)𝑛+𝑘,𝑥∈0,1,𝑥=1. (4.1) Moreover, from [27, Equation (2.4)], see also [28], we have that𝐌𝐊𝐙_𝑛𝑒₂−𝑒₂≥(𝑛+1)⁻¹𝑒₁1−𝑒₁²>0on(0,1),𝑛≥1. (4.2) For 𝜑=𝑒₂, 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 𝐿₁.

4.3. The May Positive Linear Operators

The May operators, see [29], are defined by𝐌_𝑛𝑓(𝑥)∶=¹₀𝑓(𝑡)𝜌_𝑛(𝑥,𝑡)d𝑡,𝑛∈ℕ, (4.3) where 𝜌_𝑛 denotes a kernel function. They satisfy𝐌_𝑛𝑒₂−𝑒₂=𝜆𝑛⁻¹𝑒₁−𝑒₂>0on(0,1),𝑛≥1, (4.4) for some 𝜆>0 and preserve linear functions. For 𝜑=𝑒₂, as a consequence of Corollary 3.3, the following corollary holds true.

Corollary 4.3. The sequence (𝐌^𝑘_𝑛)_k∈ℕ of the iterates of the May operators (4.3) converges uniformly to 𝐿₁.

4.4. The Bernstein Operator on a Simplex

Consider the simplex 𝑆^𝑝 in ℝ^𝑝, 𝑝≥1, given by𝑆^𝑝=𝑥∶=𝑥₁,…,𝑥_𝑝∣𝑥_𝑖≥0,|𝑥|∶=𝑥₁+…+𝑥_𝑝≤1. (4.5) The 𝑝+1 vertices of the simplex 𝑆^𝑝 are the points 𝑣_𝑖∈ℝ^𝑝, where𝑣₀=(),𝑣0,0,…,0₁=(),⋮𝑣1,0,…,0_𝑝=(0,0,…,1). (4.6) Withℳ=𝑚∶=𝑚₁,…,𝑚_𝑝∣𝑚_𝑖∈{0,1,…,𝑛},|𝑚|≤𝑛, (4.7) the Bernstein approximation operator 𝐁_𝑛∶𝐶(𝑆^𝑝)→𝐶(𝑆^𝑝) is defined by 𝐁_𝑛𝑓(𝑥)=_𝑚∈ℳ𝑛!𝑚₁!⋯𝑚_𝑝!(|𝑚!𝑥𝑛−|)^1𝑚1⋯𝑥_𝑝^𝑚𝑝(1−|𝑥|)^{𝑛−|𝑚|}𝑓𝑚𝑛. (4.8) The operator 𝐁_𝑛 preserves the subspace of linear functions𝒱=𝑓∣𝑓𝑥₁,…,𝑥_𝑝=𝑎₀+𝑎₁𝑥₁+⋯+𝑎_𝑝𝑥_𝑝,𝑎₀,𝑎₁,…,𝑎_𝑝∈ℝ. (4.9) The Lagrange interpolation operator 𝐿∶𝐶(𝑆^𝑝)→𝒱 is defined by𝐿𝑓(𝑥)=1−^𝑝_𝑘=1𝑥_𝑘𝑓𝑣₀+^𝑝_𝑘=1𝑥_𝑘𝑓𝑣_𝑘 (4.10) and interpolates all functions in 𝐶(𝑆^𝑝) on the set𝑌_𝐿=𝑣₀,…,𝑣_𝑝. (4.11) For 𝜑(𝑥)=𝑥²₁+…+𝑥²_𝑝, we have𝐁_𝑛𝜑(𝑥)(𝑥)=1−𝜑𝑛𝑥₁1−𝑥₁+⋯+𝑥_𝑝1−𝑥_𝑝>0,∀𝑥∈𝑆^𝑝−𝑌_𝐿, (4.12) 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 𝑆^𝑝 (4.8) converges uniformly to the Lagrange interpolation operator (4.10).

4.5. Positive Operators on 𝐶[𝑎,𝑏] Preserving 𝑒₀ and 𝑒₂

In [15] Agratini introduced a sequence of positive linear operators Λ_𝑛∶𝐶[𝑎,𝑏]→𝐶[𝑎,𝑏] preserving 𝑒₀ and 𝑒₂. In the case of the particularisations, 𝐷=[𝑎,𝑏], 𝑌_𝐿={𝑎,𝑏}, 𝒱=span{𝑒₀,𝑒₂}, 𝜑=𝑒₄, and 𝐿 is the interpolation operator:(𝑥)=1𝐿𝑓𝑏²−𝑎²𝑓(𝑎)𝑏²−𝑓(𝑏)𝑎²+𝑓(𝑏)−𝑓(𝑎)𝑥², (4.13) as a corollary of Theorem 3.1, we obtain a result of Agratini [15, Theorem  3.1].

4.6. Bernstein-Type Operators Preserving 𝑒₀ and 𝑒_𝑗

Let 𝑛,𝑗∈ℕ, 𝑛>𝑗>1. Aldaz et al. [30, Proposition 11] had recently considered the Bernstein-type operators ℬ_𝑗,𝑛∶𝐶[0,1]→𝐶[0,1],ℬ_𝑗,𝑛𝑓(𝑥)=^𝑛_𝑘=0𝑛𝑘𝑓𝑘(𝑘−1)⋯𝑘−𝑗+1𝑛(𝑛−1)⋯𝑛−𝑗+1^1/𝑗𝑥^𝑘(1−𝑥)^𝑛−𝑘. (4.14) The operators ℬ_𝑗,𝑛 satisfyℬ_𝑗,𝑛𝑒₀=𝑒₀,ℬ_𝑗,𝑛𝑒_𝑗=𝑒_𝑗. (4.15)

Considering the particularisations, 𝐷=[0,1], 𝑌_𝐿={0,1}, 𝒱=span{𝑒₀,𝑒_𝑗}, 𝜑=𝑒_2𝑗, and 𝐿_𝑗is the interpolation operator: 𝐿_𝑗𝑓=𝑓(0)𝑒₀+𝑓(1)−𝑓(0)𝑒_𝑗, (4.16) as a corollary of Theorem 3.1, we obtain the following corollary.

Corollary 4.5. The sequence (ℬ^𝑘_𝑗,𝑛)_𝑘∈ℕ of the iterates of the Bernstein-type operators converges uniformly to the operator 𝐿_𝑗 in (4.16).

4.7. The Cesàro Operator on 𝐶[0,1]

In the case when 𝑋=[0,1], 𝒱 is the space 𝒫₀ of constants, 𝐿𝑓=𝑓(0), 𝑌_𝐿={0}; 𝜑=𝑒₁, by Theorem 3.1, we generalize the following recent result of Galaz Fontes and Solís.

Corollary 4.6 (see [16, Theorem 3]). Let 𝜌∈𝐶[0,1] be positive on (0,1) such that ∫¹₀𝜌(𝑡)d𝑡=1, and let 𝒞∶C[0,1]→𝐶[0,1] be the Cesàro mean operator, 𝓒𝑓(𝑥)=¹₀𝑓(𝑡𝑥)𝜌(𝑡)d𝑡,𝑥∈[0,1]. (4.17) Then, 𝓒^𝑘𝑓−−−−−−−−→^uniformly𝑓(0),∀𝑓∈𝐶[0,1]. (4.18)

4.8. The Bernstein-Stancu Operators

Let 𝑏>0. The Bernstein-Stancu operators 𝐒_𝑛,0,𝑏∶𝐶[0,1]→𝐶[0,1] (see, e.g., [31]),𝐒_𝑛,0,𝑏𝑓(𝑥)=^𝑛_𝑘=0𝑛𝑛(1−𝑥)^𝑛−𝑘𝑥^𝑘𝑓𝑘𝑛+𝑏, (4.19) satisfy the following:𝐒_𝑛,0,𝑏𝑒₀=𝑒₀,𝐒_𝑛,0,𝑏𝑒₁=𝑛𝑒𝑛+𝑏¹. (4.20) For 𝑋=[0,1], 𝑥₀=0 and 𝜑=𝑒₁, 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 𝑓(0).

4.9. The Cheney-Sharma Operator

Let 𝑡_𝑛≥0, 𝑛∈ℕ and let 𝐂𝐒_𝑛 be the 𝑛th Bernstein-Cheney-Sharma operator on 𝐶[0,1], defined by𝐂𝐒_𝑛𝑓(𝑥)=1+𝑛𝑡_𝑛^𝑛−𝑛_𝑘=0𝑛𝑘𝑥𝑥+𝑘𝑡_𝑛^𝑘−11−𝑥+(𝑛−𝑘)𝑡_𝑛^𝑛−𝑘𝑓𝑘𝑛. (4.21) It is known that 0≤𝐂𝐒_𝑛𝑒₁≤𝑒₁/(1+𝑡_𝑛) (see, e.g., [32] and [3, (5.3.7)]. For 𝑡_𝑛>0 one has that𝐂S_𝑛𝑒₀=𝑒₀,0≤𝐂S_𝑛𝑒₁≤11+𝑡_𝑛𝑒₁. (4.22)

Taking 𝑋=[0,1], 𝑥₀=0, and 𝜑=𝑒₁, 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 f(0).

4.10. The Schurer Operator

For 𝑝,𝑛∈ℕ, 𝑛≥𝑝, the Bernstein-Schurer-type operator [3, (5.3.1)], _𝑛∶𝐶[0,1]→𝐶[0,1] is defined byℬ_𝑛𝑓(𝑥)=^𝑛−𝑝_𝑘=0𝑓𝑘𝑛𝑘𝑥𝑛−𝑝_𝑘(1−𝑥)^{𝑛−𝑝−𝑘}. (4.23) One can prove that this operator satisfiesℬ_𝑛𝑒₀=𝑒₀,ℬ_𝑛𝑒₁=𝑛−𝑝𝑛𝑒₁. (4.24) In the case, when 𝑋=[0,1], 𝑥₀=0, and 𝜑=𝑒₁, 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 𝑓(0).

4.11. Piecewise Bernstein Operators

Let 𝑓∈𝐶[𝑎,𝑏]. Consider the Bernstein operator 𝐵_{𝑛,𝑎,𝑏}∶𝐶[𝑎,𝑏]→𝐶[𝑎,𝑏]:𝐵_{𝑛,𝑎,𝑏}𝑓(𝑥)=1(𝑏−𝑎)^𝑛𝑛_𝑘=0𝑛𝑘(𝑏−𝑥)^𝑛−𝑘(𝑥−𝑎)^𝑘𝑓𝑘()𝑎+𝑏−𝑎𝑛,𝑛=0,1,…. (4.25)

In the case of the particularisations, 𝑋=[𝑎,𝑐], 𝒱 is the linear space of polygonal lines with vertices possessing abscissae at 𝑎,𝑏, and 𝑐, 𝐿𝑓 is the polygonal line with vertices at (𝑎,𝑓(𝑎)), (𝑏,𝑓(𝑏)), (𝑐,𝑓(𝑐)), 𝑌_𝐿={𝑎,𝑏,𝑐}, and 𝜑(𝑥)=𝑥², by Theorem 3.1, we have that the following corollary holds.

Corollary 4.10. For 𝑎<𝑏<𝑐, consider the composite Bernstein operator 𝐵∶𝐶[𝑎,𝑐]→𝐶[𝑎,𝑐], (𝑥)=𝐵𝐵𝑓^{𝑛,𝑎,𝑏}𝑓(𝑥)[],𝐵,𝑥∈𝑎,𝑏_{𝑛,𝑏,𝑐}𝑓(𝑥),𝑥∈[𝑏,𝑐]. (4.26) Then, 𝐵^𝑘𝑓^uniformly𝐿[],𝐿[]−−−−−−−−→[]𝑎,𝑏;𝑓,on𝑎,𝑏𝑏,𝑐;𝑓,on𝑏,𝑐∀𝑓∈𝐶0,1. (4.27)

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.