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	\title{Iterates of positive linear operators on Bauer simplices}
	
	\author{M. Dancs}
	\address{Technical University of Cluj-Napoca, \\ Department of Mathematics,\\
		28, Memorandumului Street,\\
		400114 Cluj-Napoca,\\
		Romania}
	\email{dancs\_madalina@yahoo.com}
	
	\author{S. Hodi\c s}
	\address{Technical University of Cluj-Napoca, \\ Department of Mathematics,\\
		28, Memorandumului Street,\\
		400114 Cluj-Napoca,\\
		Romania}
	\email{hodissever@gmail.com}
	%\subjclass{41A36, 46A55}
	
	
	\maketitle
	{\small Dedicated to Professor Heiner Gonska on the occasion of his 70th\\ birthday.\\}
	
	
	\begin{abstract}
		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 $m\rightarrow\infty$. Qualitative results and rates of convergence are obtained. The general results are illustrated by examples involving classical operators.
	\end{abstract}
	
	{\small \textbf{Mathematics Subject Classification (2010):} 41A36, 46A55.}\newline
	{\small \hspace*{0.8cm}\textbf{Keywords:}Bauer simplex, positive linear operators, iterates, convergence.}
	
	
	\section{Introduction}
	\hspace*{0.5cm} Iterates of positive linear operators were investigated in many papers and from several points of view.\\
	- General criteria for their convergence can be found in [1], [2], [13], [14], [20], [21], [23].\\
	- Rates of convergence of the iterates were established in [6], [10], [16], [17], [18], [20], [21], [28].\\
	- The relationship with Korovkin theory is presented in [6], [7], [8], [22].\\
	- Iterates are essentially used for representing some strongly
	continuous semigroups of operators: see [7], [8], [17], [28].\\
	- Iterates for q-Bernstein operators are studied in [24]; the case of 
	complex operators is considered in [11].\\
	\hspace*{0.5cm} In the above papers analytical methods were used and also methods from probability theory. Results based on spectral theory can be found in [9]; fixed point theory is used in [3], [4], [30], [31], [32], [33].\\
	\hspace*{0.5cm} This paper is devoted to the study of iterates of positive linear operators on Bauer simplices. General definitions and results are presented in this introduction; see also [5], [7], [8], [25].\\
	\hspace*{0.5cm} Section 2 is devoted to the iterates of operators preserving affine functions. An example concerning a finite dimensional simplex is discussed in Section 3. Other examples are presented in Section 4.\\
	\hspace*{0.5cm} Throughout the paper we shall use the following notions.\\
	\hspace*{0.5cm} Let $E$ be a real locally convex Hausdorff space and $K$ a convex compact subset of $E$. Let $C(K)$ be the space of all continuous real-valued functions on $K$, endowed with the usual ordering and the supremum norm. By Herv\'{e}'s \\ theorem [5, Th.I.4.3], [7, p.57], $C(K)$ contains a strictly convex function if and only if $K$ is metrizable. Throughout the paper we shall suppose that $K$ is metrizable.\\
	\hspace*{0.5cm} The set of all probability Radon measures on $K$ will be denoted by $M^{+}_{1}(K)$. For each $x\in K,\ \varepsilon_{x}$ stands for the probability Radon measure concentrated on $\{x\}$.\\
	\hspace*{0.5cm} The \textit{Choquet-Meyer} ordering $<$ on $M_{1}^{+}(K)$ is defined as follows: for every $\mu ,\nu\in M_{1}^{+}(K),\  \mu <\nu $ if $\mu(f)\leq\nu(f)$ for every convex function $f\in C(K).$ A measure $\mu$ which is maximal with respect to $<$ will be simply called a \textit{maximal measure}.\\
	%Let $\mu,\nu\in M^{+}_{1}(K)$. We write $\mu <\nu$ if $\mu(f)\leq\nu(f)$ for each convex function $f\in C(K)$. A measure $\mu$ which is maximal with respect to the \textit{Choquet-Meyer} ordering $<$ will be simply called a \textit{maximal measure}.\\
	\hspace*{0.5cm} Let $A(K)$ be the set of all affine functions for all $h\in C(K)$. The\\ \textit{barycenter} of $\mu\in M_{1}^{+}(K)$ is the point $r\in K$ for which $\mu(h)=h(r),\ h\in A(K)$; in this case $\mu(f)\geq f(r)$ for each convex function $f\in C(K)$.\\
	\hspace*{0.5cm} There are several equivalent properties defining a Choquet simplex. We need the following one:\\
	\hspace*{0.5cm} $K$ is called a \textit{Choquet simplex} if for every $x\in K$ there exists a unique maximal measure $\mu_{x}\in M_{1}^{+}(K)$ having $x$ as barycenter.\\
	\hspace*{0.5cm} The set of the extreme points of $K$ will be denoted by $ex(K)$.\\
	\hspace*{0.5cm} A Choquet simplex $K$ such that $ex(K)$ is closed will be called a \textit{Bauer simplex}. In this case $\mu_{x}$ is supported by $ex(K)$; moreover, if $\mu_{x}=\varepsilon_{x}$, then $x\in ex(K)$.\\
	\hspace*{0.5cm} If $K$ is a Bauer simplex, then the operator $P:C(K)\longrightarrow A(K)$ defined by
	$$Pf(x)=\mu_{x}(f),\ f\in C(K),\ x\in K,$$
	is linear, positive, and $Ph=h$ for all $h\in A(K)$.\\
	\hspace*{0.5cm} $P$ is called the \textit{canonical projection} associated with the Bauer simplex $K$.\\
	\hspace*{0.5cm} Let $L:C(K)\longrightarrow C(K)$ be a positive linear operator such that $Lh=h$, for every $h\in A(K)$. For each $x\in K$ let $\nu_{x}(f):=Lf(x),\ f\in C(K)$. Then $\nu_{x}\in M_{+}^{1}(K)$ and $x$ is the barycenter of $\nu_{x}$. In particular, if $x\in ex(K)$ then $\nu_{x}=\varepsilon_{x}$, so that
	\begin{equation}
	Lf(x)=f(x),\ x\in ex(K),\ f\in C(K).
	\end{equation}
	Moreover, if $g\in C(K)$ is convex, then $\nu_{x}(g)\geq g(x),\ x\in K$, i.e.,
	\begin{equation}
	Lg\geq g.
	\end{equation}
	We shall need the following result [26], [27], [7, Th.1.5.2].
	\begin{lemma}
		Let $\mu\in M_{1}^{+}(K)$ with barycenter $r$ and let $u$ be a strictly convex function. If $\mu(u)=u(r)$, then $\mu=\varepsilon_{r}.$
	\end{lemma}
	\section{Iterates of positive linear operators preserving the affine functions}
	In the sequel, $K$ will be a metrizable Bauer simplex.
	\begin{theorem}
		Let $L:C(K)\longrightarrow C(K)$ be a positive linear operator such that $Lh=h,\ h\in A(K)$. Let $u\in C(K)$ be a strictly convex function. If
		\begin{equation}
		\lim_{m\to\infty} L^{m}f=Pf,\ f\in C(K),
		\end{equation}
		then
		\begin{equation}
		Lu(x)>u(x),\ x\in K\smallsetminus ex(K).
		\end{equation}
		\begin{proof}
			Let $x\in K$. As in the preceding section, let $\nu_{x}(f):=Lf(x),\ f\in C(K)$. By (1.2), $Lu(x)\geq u(x)$. Suppose that $Lu(x)=u(x)$. Then $\nu_{x}(u)=u(x)$, and Lemma 1.1 yields $\nu_{x}=\varepsilon_{x}$, i.e., $Lf(x)=f(x)$, $f \in C(K)$. By induction, $L^{m}f(x)=f(x),\ f\in C(K)$. Now (2.1) shows that $Pf(x)=f(x),\ f \in C(K)$. This means that $\mu_{x}=\varepsilon_{x}$, which entails $x\in ex(K)$.
		\end{proof}
	\end{theorem}
	For $K=[0,1]$ the above result was obtained in [29] and [12, Corollary 2].\\
	
	We shall prove that the converse of Th. 2.1 is also true. Having\\ applications in mind, let us consider a sequence of positive linear operators $L_{n}:C(K)\longrightarrow C(K)$ preserving the affine functions, i.e.,
	\begin{equation}
	L_{n}h=h,\ h\in A(K),\ n\geq 1.
	\end{equation}
	Fix a strictly convex function $u\in C(K)$.\\
	For $n\geq 1$ and $s\in (0,+\infty)$ define
	\begin{equation}
	a_{n}(s):=\max_{K}(Pu-u-ns(L_{n}u-u)).
	\end{equation}
	For $x\in ex (K)$ we have $Pu(x)-u(x)=L_{n}u(x)-u(x)=0$, so that $a_{n}(s)\geq 0$.
	\begin{lemma}
		If $ns\geq 1,\ m\geq 1$, then
		\begin{equation}
		0\leq Pu-L_{n}^{m}u\leq a_{n}(s)\mathbf{1}+\big(1-\frac{1}{ns}\big)^{m}(Pu-u),
		\end{equation}
		where $\mathbf{1}$ is the constant function of value $1$ defined on $K$.
		\begin{proof}
			Since $P$ preserves the affine functions, we have $u\leq Pu$ by virtue of (1.2). Moreover, $Pu\in A(K)$, and so $\displaystyle L_{n}u\leq L_{n}(Pu)=Pu$. By induction we get $\displaystyle L_{n}^{m}u\leq Pu$, and this is the first inequality in (2.5).\\
			From (2.4) we derive
			$$a_{n}(s)\mathbf{1}\geq Pu-u-ns(L_{n}u-u).$$
			This implies
			$$\frac{1}{ns}\big(Pu-a_{n}(s)\mathbf{1}\big)+\big(1-\frac{1}{ns}\big)u\leq L_{n}u.$$
			Since $\displaystyle 1-\frac{1}{ns}\geq 0$, iterating over $m\geq 1$
			$$\Big(1-\big(1-\frac{1}{ns}\big)^{m}\Big)\Big(Pu-a_{n}(s)\mathbf{1}\Big)+\big(1-\frac{1}{ns}\big)^{m}u\leq L_{n}^{m}u.$$
			This leads immediately to the second inequality in (2.5), and the lemma is proved. 
		\end{proof}
	\end{lemma}
	\begin{lemma}
		Let $n\geq 1$ be fixed, and suppose that for a given strictly convex function $u\in C(K)$ one has
		\begin{equation}
		L_{n}u(x)>u(x),\ x\in K\smallsetminus ex(K).
		\end{equation}
		Then $\displaystyle \lim_{s\to\infty}a_{n}(s)=0$.
		\begin{proof}
			Since $a_{n}\geq 0$ and $a_{n}$ is decreasing on $(0,+\infty)$, we have $\displaystyle l:=\lim_{s\to\infty}a_{n}(s)\geq 0$. Suppose that $l>0$. Let
			$$A_{s}:=\{x\in K:\ Pu(x)-u(x)-ns(L_{n}u(x)-u(x))\geq l \}.$$
			The sets $A_{s}$ are closed and the family $(A_{s})_{s>0}$ is descending. For each $s>0$, $A_{s}$ and $ex(K)$ are disjoint, so that (2.6) implies $\displaystyle\bigcap_{s>0} A_{s}=\emptyset$.
			Since $K$ is compact, there exists $t>0$ such that $A_{t}=\emptyset$. Then $a_{n}(t)<l$, a contradiction.
		\end{proof}
	\end{lemma}
	\begin{theorem}
		(i) Let $0<c<1$. Then
		\begin{equation}
		0\leq Pu-L_{n}^{m}u\leq a_{n}(m^{c})\mathbf{1}+\Big(1-\frac{1}{nm^{c}}\Big)^{m}\big(Pu-u\big),
		\end{equation}
		for all $m,n\geq 1$.\\
		(ii) If (2.6) holds for a given $n\geq 1$, then 
		\begin{equation}
		\lim_{m\to\infty}L_{n}^{m}f=Pf,\ f\in C(K).
		\end{equation}
		\begin{proof}
			(i) is a consequence of (2.5), with $s=m^{c}$. From (2.7) and Lemma 2.2 we infer that $\displaystyle \lim_{m\to\infty}L_{n}^{m}u=Pu$. This fact, combined with Corollary 3.3.4 of [7], leads to (2.8).
		\end{proof}
	\end{theorem}
	In the sequel we shall suppose that the limit
	$$T(t)f:=\lim_{n\to\infty}L_{n}^{k(n)}f$$
	exists in $C(K)$ for each $f\in C(K)$, each $t\geq 0$, and each sequence of positive integers $(k(n))_{n\geq 1}$ such that $\displaystyle \lim_{n\to\infty}\frac{k(n)}{n}=t$.\\
	\hspace*{0.5cm} Denote $a(s)=sup\{a_{n}(s):n\geq 1\},\ s>0.$
	\begin{theorem}
		(i) Let $0<c<1$. Then for all $t>0$,
		\begin{equation}
		0\leq Pu-T(t)u\leq a(t^{c})\textbf{1}+(Pu-u)exp(-t^{1-c}).
		\end{equation} 
		\quad\quad \quad \quad \quad\quad\quad(ii) If $\displaystyle \lim_{s\to\infty}a(s)=0$, then 
		\begin{equation}
		\lim_{t\to\infty}T(t)f=Pf,\ f\in C(K).
		\end{equation}
		\begin{proof}
			Let $t>0$ be fixed. If $nt^{c}\geq 1$, from (2.5) we get 
			$$0\leq Pu-L_{n}^{k(n)}u\leq a(t^{c})\mathbf{1}+\Big(1-\frac{1}{nt^{c}}\Big)^{k(n)}(Pu-u).$$
			Choosing $k(n)$ such that $\displaystyle\lim_{n\to\infty}\frac{k(n)}{n}=t$, we obtain (2.9).\\
			If $\displaystyle\lim_{s\to\infty}a(s)=0$, (2.9) yields
			$$\lim_{t\to\infty}T(t)u=Pu.$$
			Another application of [7, Cor. 3.3.4] concludes the proof.
		\end{proof}
	\end{theorem}
	\section{An example and a quantitative result}
	Let $K$ be the canonical simplex of $\mathbb{R}^{d}$, that is
	$$K=\{x\in\mathbb{R}^{d}:x_{1},\ldots x_{d}\geq 0,\ x_{1}+\ldots+x_{d}\leq 1 \}.$$
	The canonical projection associated with $K$ is defined, for every $f\in C(K)$ and $x\in K$, by 
	\begin{equation}
	Pf(x)=(1-x_{1}-\ldots-x_{d})f(0)+x_{1}f(v_{1})+\ldots+x_{d}f(v_{d}),
	\end{equation}
	where $0,\ v_{1}:=(1,0,\ldots,0),\ldots, v_{d}:=(0,\ldots,0,1)$ are the vertices of $K$.\\
	Let $f\in C(K)$; suppose that there exists a constant $Q_{f}>0$ such that for all $x\in K$,
	\begin{equation}
	|f(x)-f(0)|\leq Q_{f}\sum_{i=1}^{d}x_{i},
	\end{equation}
	\begin{equation}
	|f(x)-f(v_{j})|\leq Q_{f}\Big(1-2x_{j}+\sum_{i=1}^{d}x_{i}\Big),\ j=1,\ldots,d.
	\end{equation}
	Then, for $x\in K$,
	$$|f(x)-Pf(x)|=|f(x)-\Big(1-\sum_{i=1}^{d}x_{i}\Big)f(0)-\sum_{i=1}^{d}x_{i}f(v_{i})|=$$
	$$=|\Big(1-\sum_{i=1}^{d}x_{i}\Big)(f(x)-f(0))+\sum_{i=1}^{d}x_{i}(f(x)-f(v_{i}))|\leq$$
	$$Q_{f}\Big(\big(1-\sum_{i=1}^{d}x_{i}\big)\sum_{i=1}^{d}x_{i}+\sum_{i=1}^{d}x_{i}\big(1-2x_{i}+\sum_{j=1}^{d}x_{j}\big)\Big)=$$
	$$=2Q_{f}\Big(\sum_{i=1}^{d}x_{i}-\sum_{i=1}^{d}x_{i}^{2}\Big).$$
	Consider the strictly convex function $\displaystyle u\in C(K),\ u(x)=\sum_{i=1}^{d}x_{i}^{2},\ x\in K$. Then $\displaystyle Pu(x)=\sum_{i=1}^{d}x_{i}$, so that for the above function $f$ we have 
	\begin{equation}
	|f(x)-Pf(x)|\leq 2Q_{f}(Pu(x)-u(x)),\ x\in K.
	\end{equation}
	Let $L_{n}:C(K)\longrightarrow C(K)$ be a positive linear operator preserving affine functions. From (3.4) we get
	\begin{equation}
	|L_{n}^{m}f-Pf|\leq 2Q_{f}(Pu-L_{n}^{m}u).
	\end{equation}
	Finally, (3.5) and (2.7) yield
	$$|L_{n}^{m}f-Pf|\leq 2Q_{f}\Big(a_{n}(m^{c})\mathbf{1}+\Big(1-\frac{1}{nm^{c}}\Big)^{m}(Pu-u)\Big),$$
	i.e.,
	\begin{equation}
	|L_{n}^{m}f(x)-Pf(x)|\leq 2Q_{f}\Big[a_{n}(m^{c})+\Big(1-\frac{1}{nm^{c}}\Big)^{m}\sum_{i=1}^{d}x_{i}(1-x_{i})\Big].
	\end{equation}
	Moreover, in the context of Theorem 2.3 we derive from (3.4):
	\begin{equation}
	|T(t)f-Pf|\leq 2Q_{f}(Pu-T(t)u).
	\end{equation}
	Combined with (2.9), this gives
	\begin{equation}
	|T(t)f(x)-Pf(x)|\leq 2Q_{f}\Big[a(t^{c})+(exp(-t^{1-c}))\sum_{i=1}^{d}x_{i}(1-x_{i})\Big].
	\end{equation}
	\begin{remark}
		If $f\in C^{1}(K)$, i.e., $f$ has continuous partial derivatives on the interior of $K$ which can be continuously extended on $K$, then (3.2) and (3.3) are satisfied with
		$$Q_{f}:=\max\Big\{\parallel \frac{\partial f}{\partial x_{i}}\parallel_{\infty}:\ i=1,\ldots,d\Big\}.$$
	\end{remark}
	\section{Examples}
	In this section we present examples of sequences $(L_{n})_{n\geq 1}$ of operators preserving affine functions and satisfying the fundamental condition (2.6).
	\begin{example}
		Let $B_{n},\ n\geq 1$, be the \textit{Bernstein-Schnabl} operators associated with the canonical projection $P$ and the arithmetic mean \textit{Toeplitz} matrix (see [7, p. 381]). Let $u\in C(K)$ be a strictly convex function. Suppose that for a given $n\geq 1$ and a given $x\in K$ one has $B_{n}u(x)=u(x)$. According to \textit{Lemma 1.1}, we infer that $B_{n}f(x)=f(x)$, for every $f\in C(K)$. In particular, $B_{n}h^{2}(x)=h^{2}(x)$, for all $h\in A(K)$. Now [7, (6.1.16)] leads to $P(h^{2})(x)=h^{2}(x),\ h\in A(K)$. From [7, Cor. 3.3.4 and Remark to Prop. 3.3.2] we deduce that $x\in ex(K)$. So (2.6) is satisfied for the operators $B_{n}$.
	\end{example}
	\begin{example}
		Let $U_{n},\ n\geq 1$, be the genuine \textit{Bernstein-Durrmeyer} operators on a simples $K$ in $\mathbb{R}^{d}$ (see [34], [19], [35]). If $u\in C(K)$ is strictly convex, then $U_{n}u\geq B_{n}u\geq u$ [19, Th.8]. If $U_{n}u(x)=u(x)$, then $B_{n}u(x)=u(x)$, and from Ex. 4.1 we know that $x\in ex(K)$. So (2.6) is satisfied for the operators $U_{n}$.
	\end{example}
	\begin{example}
		It was proved in [28, Example 2.4] that (2.6) is satisfied for the classical Meyer-K\"{o}nig and Zeller operators on $C[0,1]$.
	\end{example}
	\begin{example}
		The case of the Bernstein-Schnabl operators on the unit interval, associated with a continuous selection of probability Borel measures on $[0,1]$, is considered in [28, Example 3.3]. The operators satisfy (2.6).
	\end{example}
	For all the operators presented in the above examples one can apply Lemma 2.2 and, consequently, one can obtain the corresponding quantitative results derived from Theorems 2.2 and 2.3.\\
	\hspace*{0.5cm} Other examples and quantitative results can be found in [18], [28], [29].
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