# Asymptotic Behaviour of the Iterates of Positive Linear Operators

Authors: Ioan Gavrea; Mircea Ivan
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 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.LetX 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∥∶=supx∈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 (Uk)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 onC[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∥:=maxx∈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;YL is the interpolation set of L,(2.1)YL={y∈X∣Lf(y)=f(y),∀f∈C(X)}≠∅.
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 YL={x0}.We also emphasize that ifU: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 YL⊇{0,1},ei:[0,1]→ℝ are the monomial functions ei(x)=xi, 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 Section2, the following theorem is the main result of the paper.Theorem 3.1.
If there existsφ∈C(X)  such that
(3.1)Uφ≥φon⁡X,Uφ≠φon⁡X∖YL,
then
(3.2)lim⁡kUkf=Lf,∀f∈C(X).
Moreover, if X is a compact metric space, then the convergence is uniform.Proof.
Letf∈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 Ukf=Lf, k=1,2,….
IfLf≠f, for sufficiently small ɛ>0, the inverse image of the open set (-ɛ,ɛ) under the continuous function Lf-f is an open set G,   YL⊆G≠X. It follows that
(3.3)|Lf-f|<ɛ,on⁡G.
Since X is compact and G  is open, it follows that  X∖G⊆X∖YL  is a nonempty compact subset of X, and we obtain
(3.4)mɛ∶=inf⁡x∈X∖G(Uφ(x)-φ(x))>0.
Consequently, the following decisive inequality
(3.5)|Lf-f|<ɛ+‖f-Lf‖mɛ(Uφ-φ)
is satisfied. By applying the positive operator Uk to (3.5), we get
(3.6)|Lf-Ukf|<ɛ+‖f-Lf‖mɛ|Uk+1φ-Ukφ|.
Since Uφ≥φ, we obtain
(3.7)φ≤Ukφ≤Uk+1φ≤‖φ‖,k=1,2,….
The sequence (Ukφ)k≥1 is monotone and bounded. It follows that it is pointwise convergent. Since ɛ was chosen arbitrarily, by using (3.6) we deduce that  Ukf→pointwiseLf.
In the particular case whenX is a compact metric space, since
(3.8)Ukφ→pointwiseLφ∈C(X),
by Dini's Theorem, we obtain that Ukφ→uniformlyLφ. From the inequalities
(3.9)‖Lf-Ukf‖<ɛ+‖f-Lf‖mɛ‖Uk+1φ-Ukφ‖,
we deduce that Ukf→uniformlyLf.In the following we give more information on the limit operatorL.Theorem 3.2.
The limit interpolation operatorL is unique, positive, and satisfies the equalities
(3.10)UL=LU=L.Proof.
The unicity and positivity of the operatorL follow from the existence of limit limkUk=L. Since U preserves the elements of 𝒱, we obtain that UL=L. Taking into account the relations
(3.11)∅⊊YL⊆{y∈X∣LUf(y)=Uf(y),∀f∈C(X)},
we can repeat the proof of Theorem 3.1 by starting with Uf instead of f. We deduce that limkUkf=LUf, and hence LU=L.An immediate corollary of Theorem3.1 is the following.Corollary 3.3.
IfU: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
(3.12)lim⁡kUkf=f(x0),∀f∈C(X).Proof.
In Theorem3.1 we take  Lf=f(x0).Remark 3.4.
The existence of the functionφ 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
(3.13)Uf(x)=xf(0)+(1-x)f(1)
preserves the space of constants 𝒫0, and the operator L:C[0,1]→𝒫0, Lf(x)=f(0), is interpolator on YL={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 (Une1)n≥1 has no limit.


## 4. Applications

In this section, as applications of Theorem3.1 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 onC[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 YL={0,1}, by Theorem 3.1, we have that the following corollary holds.Corollary 4.1.
LetU: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 MKZn:C[0,1]→C[0,1], n∈ℕ, in the modified version of Cheney and Sharma, see [26], are defined by(4.1)MKZnf(x)={∑k=0∞(n+kk)(1-x)n+1xkf(kn+k),x∈[0,1),f(1),x=1.
Moreover, from [27, Equation (2.4)], see also [28], we have that(4.2)MKZne2-e2≥(n+1)-1e1(1-e1)2>0on⁡(0,1),n≥1.
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 by(4.3)Mnf(x)∶=∫01f(t)ρn(x,t)dt,n∈N,
where ρn denotes a kernel function. They satisfy(4.4)Mne2-e2=λn-1(e1-e2)>0on⁡(0,1),n≥1,
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(Mnk)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 simplexSp in ℝp, p≥1, given by(4.5)Sp={x∶=(x1,…,xp)∣xi≥0,|x|∶=x1+…+xp≤1}.
The p+1 vertices of the simplex Sp are the points vi∈ℝp, where(4.6)v0=(0,0,…,0),v1=(1,0,…,0),⋮vp=(0,0,…,1).
With(4.7)M={m∶=(m1,…,mp)∣mi∈{0,1,…,n},|m|≤n},
the Bernstein approximation operator Bn:C(Sp)→C(Sp) is defined by  (4.8)Bnf(x)=∑m∈Mn!m1!⋯mp!(n-|m|)!x1m1⋯xpmp(1-|x|)n-|m|f(mn).
The operator Bn preserves the subspace of linear functions(4.9)V={f∣f(x1,…,xp)=a0+a1x1+⋯+apxp,a0,a1,…,ap∈R}.
The Lagrange interpolation operator L:C(Sp)→𝒱 is defined by(4.10)Lf(x)=(1-∑k=1pxk)f(v0)+∑k=1pxkf(vk)
and interpolates all functions in C(Sp) on the set(4.11)YL={v0,…,vp}.
For φ(x)=x12+…+xp2, we have(4.12)Bnφ(x)-φ(x)=1n(x1(1-x1)+⋯+xp(1-xp))>0,∀x∈Sp-YL,
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 simplexSp  (4.8) converges uniformly to the Lagrange interpolation operator (4.10).


### 4.5. Positive Operators onC[a,b] Preserving e0 and e2

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


### 4.6. Bernstein-Type Operators Preservinge0 and ej

Letn,j∈ℕ, n>j>1. Aldaz et al. [30, Proposition 11] had recently considered the Bernstein-type operators ℬj,n:C[0,1]→C[0,1],(4.14)Bj,nf(x)=∑k=0n(nk)f((k(k-1)⋯(k-j+1)n(n-1)⋯(n-j+1))1/j)xk(1-x)n-k.
The operators ℬj,n satisfy(4.15)Bj,ne0=e0,Bj,nej=ej.Considering the particularisations,D=[0,1], YL={0,1}, 𝒱=span{e0,ej}, φ=e2j, and Ljis the interpolation operator: (4.16)Ljf=f(0)e0+(f(1)-f(0))ej,
as a corollary of Theorem 3.1, we obtain the following corollary.Corollary 4.5.
The sequence(ℬj,nk)k∈ℕ of the iterates of the Bernstein-type operators  converges uniformly to the operator Lj in (4.16).


### 4.7. The Cesàro Operator onC[0,1]

In the case whenX=[0,1], 𝒱 is the space 𝒫0 of constants, Lf=f(0), YL={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]).
Letρ∈C[0,1] be positive on (0,1) such that ∫01ρ(t)dt=1, and let 𝒞:C[0,1]→C[0,1] be the Cesàro mean operator,
(4.17)Cf(x)=∫01f(tx)ρ(t)dt,x∈[0,1].
Then,
(4.18)Ckf→uniformlyf(0),∀f∈C[0,1].


### 4.8. The Bernstein-Stancu Operators

Letb>0. The Bernstein-Stancu operators Sn,0,b:C[0,1]→C[0,1] (see, e.g., [31]),(4.19)Sn,0,bf(x)=∑k=0n(nn)(1-x)n-kxkf(kn+b),
satisfy the following:(4.20)Sn,0,be0=e0,Sn,0,be1=nn+be1.
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

Lettn≥0, n∈ℕ and let CSn be the  nthBernstein-Cheney-Sharma operator on C[0,1], defined by(4.21)CSnf(x)=(1+ntn)-n∑k=0n(nk)x(x+ktn)k-1(1-x+(n-k)tn)n-kf(kn).
It is known that 0≤CSne1≤e1/(1+tn) (see, e.g., [32] and [3, (5.3.7)]. For tn>0 one has that(4.22)CSne0=e0,0≤CSne1≤11+tne1.TakingX=[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

Forp,n∈ℕ, n≥p, the Bernstein-Schurer-type operator [3, (5.3.1)], ßn:C[0,1]→C[0,1] is defined by(4.23)Bnf(x)=∑k=0n-pf(kn)(n-pk)xk(1-x)n-p-k.
One can prove that this operator satisfies(4.24)Bne0=e0,Bne1=n-pne1.
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

Letf∈C[a,b]. Consider the Bernstein operator Bn,a,b:C[a,b]→C[a,b]:(4.25)Bn,a,bf(x)=1(b-a)n∑k=0n(nk)(b-x)n-k(x-a)kf(a+k(b-a)n),n=0,1,….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)), YL={a,b,c}, and φ(x)=x2, by Theorem 3.1, we have that the following corollary holds.Corollary 4.10.
Fora<b<c, consider the composite Bernstein operator B:C[a,c]→C[a,c],
(4.26)Bf(x)={Bn,a,bf(x),x∈[a,b],Bn,b,cf(x),x∈[b,c].
Then,
(4.27)Bkf→uniformly{L[a,b;f],on⁡[a,b],L[b,c;f],on⁡[b,c]∀f∈C[0,1].


## 4.1. Positive Operators onC[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 YL={0,1}, by Theorem 3.1, we have that the following corollary holds.Corollary 4.1.
LetU: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 MKZn:C[0,1]→C[0,1], n∈ℕ, in the modified version of Cheney and Sharma, see [26], are defined by(4.1)MKZnf(x)={∑k=0∞(n+kk)(1-x)n+1xkf(kn+k),x∈[0,1),f(1),x=1.
Moreover, from [27, Equation (2.4)], see also [28], we have that(4.2)MKZne2-e2≥(n+1)-1e1(1-e1)2>0on⁡(0,1),n≥1.
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 by(4.3)Mnf(x)∶=∫01f(t)ρn(x,t)dt,n∈N,
where ρn denotes a kernel function. They satisfy(4.4)Mne2-e2=λn-1(e1-e2)>0on⁡(0,1),n≥1,
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(Mnk)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 simplexSp in ℝp, p≥1, given by(4.5)Sp={x∶=(x1,…,xp)∣xi≥0,|x|∶=x1+…+xp≤1}.
The p+1 vertices of the simplex Sp are the points vi∈ℝp, where(4.6)v0=(0,0,…,0),v1=(1,0,…,0),⋮vp=(0,0,…,1).
With(4.7)M={m∶=(m1,…,mp)∣mi∈{0,1,…,n},|m|≤n},
the Bernstein approximation operator Bn:C(Sp)→C(Sp) is defined by  (4.8)Bnf(x)=∑m∈Mn!m1!⋯mp!(n-|m|)!x1m1⋯xpmp(1-|x|)n-|m|f(mn).
The operator Bn preserves the subspace of linear functions(4.9)V={f∣f(x1,…,xp)=a0+a1x1+⋯+apxp,a0,a1,…,ap∈R}.
The Lagrange interpolation operator L:C(Sp)→𝒱 is defined by(4.10)Lf(x)=(1-∑k=1pxk)f(v0)+∑k=1pxkf(vk)
and interpolates all functions in C(Sp) on the set(4.11)YL={v0,…,vp}.
For φ(x)=x12+…+xp2, we have(4.12)Bnφ(x)-φ(x)=1n(x1(1-x1)+⋯+xp(1-xp))>0,∀x∈Sp-YL,
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 simplexSp  (4.8) converges uniformly to the Lagrange interpolation operator (4.10).


## 4.5. Positive Operators onC[a,b] Preserving e0 and e2

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


## 4.6. Bernstein-Type Operators Preservinge0 and ej

Letn,j∈ℕ, n>j>1. Aldaz et al. [30, Proposition 11] had recently considered the Bernstein-type operators ℬj,n:C[0,1]→C[0,1],(4.14)Bj,nf(x)=∑k=0n(nk)f((k(k-1)⋯(k-j+1)n(n-1)⋯(n-j+1))1/j)xk(1-x)n-k.
The operators ℬj,n satisfy(4.15)Bj,ne0=e0,Bj,nej=ej.Considering the particularisations,D=[0,1], YL={0,1}, 𝒱=span{e0,ej}, φ=e2j, and Ljis the interpolation operator: (4.16)Ljf=f(0)e0+(f(1)-f(0))ej,
as a corollary of Theorem 3.1, we obtain the following corollary.Corollary 4.5.
The sequence(ℬj,nk)k∈ℕ of the iterates of the Bernstein-type operators  converges uniformly to the operator Lj in (4.16).


## 4.7. The Cesàro Operator onC[0,1]

In the case whenX=[0,1], 𝒱 is the space 𝒫0 of constants, Lf=f(0), YL={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]).
Letρ∈C[0,1] be positive on (0,1) such that ∫01ρ(t)dt=1, and let 𝒞:C[0,1]→C[0,1] be the Cesàro mean operator,
(4.17)Cf(x)=∫01f(tx)ρ(t)dt,x∈[0,1].
Then,
(4.18)Ckf→uniformlyf(0),∀f∈C[0,1].


## 4.8. The Bernstein-Stancu Operators

Letb>0. The Bernstein-Stancu operators Sn,0,b:C[0,1]→C[0,1] (see, e.g., [31]),(4.19)Sn,0,bf(x)=∑k=0n(nn)(1-x)n-kxkf(kn+b),
satisfy the following:(4.20)Sn,0,be0=e0,Sn,0,be1=nn+be1.
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

Lettn≥0, n∈ℕ and let CSn be the  nthBernstein-Cheney-Sharma operator on C[0,1], defined by(4.21)CSnf(x)=(1+ntn)-n∑k=0n(nk)x(x+ktn)k-1(1-x+(n-k)tn)n-kf(kn).
It is known that 0≤CSne1≤e1/(1+tn) (see, e.g., [32] and [3, (5.3.7)]. For tn>0 one has that(4.22)CSne0=e0,0≤CSne1≤11+tne1.TakingX=[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

Forp,n∈ℕ, n≥p, the Bernstein-Schurer-type operator [3, (5.3.1)], ßn:C[0,1]→C[0,1] is defined by(4.23)Bnf(x)=∑k=0n-pf(kn)(n-pk)xk(1-x)n-p-k.
One can prove that this operator satisfies(4.24)Bne0=e0,Bne1=n-pne1.
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

Letf∈C[a,b]. Consider the Bernstein operator Bn,a,b:C[a,b]→C[a,b]:(4.25)Bn,a,bf(x)=1(b-a)n∑k=0n(nk)(b-x)n-k(x-a)kf(a+k(b-a)n),n=0,1,….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)), YL={a,b,c}, and φ(x)=x2, by Theorem 3.1, we have that the following corollary holds.Corollary 4.10.
Fora<b<c, consider the composite Bernstein operator B:C[a,c]→C[a,c],
(4.26)Bf(x)={Bn,a,bf(x),x∈[a,b],Bn,b,cf(x),x∈[b,c].
Then,
(4.27)Bkf→uniformly{L[a,b;f],on⁡[a,b],L[b,c;f],on⁡[b,c]∀f∈C[0,1].



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