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

Octavian Agratini Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania agratini@math.ubbcluj.ro and Radu Precup Faculty of Mathematics and Computer Science and Institute of Advanced Studies in Science and Technology, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania & Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 400110 Cluj-Napoca, Romania r.precup@math.ubbcluj.ro

Abstract.

The starting point of this paper is the construction of a general family $(L_{n})_{n\geq 1}$ of positive linear operators of discrete type. Considering $(L_{n}^{k})_{k\geq 1}$ the sequence of iterates of one of such operators, $L_{n}$ , our goal is to find an expression of the upper edge of the error $\|L_{n}^{k}f-f^{\ast}\|$ , $f\in C[0,1]$ , where $f^{\ast}$ is the fixed point of $L_{n}.$ The estimate makes use of the error formula for the sequence of successive approximations in Banach’s fixed point theorem and the error of approximation of the operator $L_{n}.$ Examples of special operators are inserted. Some extensions to multidimensional approximation operators are also given.

Mathematics Subject Classification: 41A36, 47H10.

Keywords and phrases: Positive linear operator, Bernstein operator, Stancu operator, Cheney-Sharma operator, contraction principle, Banach fixed point theorem, Perov fixed point theorem

1. Introduction

A distinct branch of Approximation theory is the approximation of functions by using sequences of linear operators, say $(L_{n})_{n\geq 1}$ , the special feature being that of positivity. Such sequences become approximation processes if they satisfy certain additional conditions. Further we present the most common approach. Considering the Banach space $C[a,b]$ of all real-valued continuous functions defined on $[a,b]$ endowed with the uniform norm $\|\cdot\|$ ,

\|f\|=\sup\limits_{x\in[a,b]}|f(x)|,

a powerful criterion to decide if $L_{n}$ , $n\geq 1$ , defined on $C[a,b]$ forms a sequence converging to the identity operator was established by Popoviciu, Bohman and Korovkin in the fifties. In its last version, this criterion says: if $(L_{n}e_{k})_{n\geq 1}$ converges to $e_{k}$ uniformly on $[a,b]$ , $k\in\{0,1,2\}$ , for the test functions

e_{0}(x)=1,\ e_{1}(x)=x,\ e_{2}(x)=x^{2},

then $(L_{n}f)_{n\geq 1}$ converges to $f$ uniformly on $[a,b]$ for each $f\in C[a,b]$ .

For a linear positive approximation process $(L_{n})_{n\geq 1}$ it is of interest to investigate the sequences of iterates $(L_{n}^{k})_{k\geq 1}$ , where

L_{n}^{1}f=L_{n}f,\ \ \ L_{n}^{k}f=L_{n}(L_{n}^{k-1}f),\ \ \ k>1,

assuming that $k$ does not depend on $n$ . Since any interval $[a,b]$ is isomorphic to $[0,1]$ , in what follows we use only this compact interval.

Considering that $f^{\ast}\in C[0,1]$ is a fixed point of the operator $L_{n}$ , our first aim in this paper is to find the expression of the upper edge of the error

\|L_{n}^{k}f-f^{\ast}\|,\ \ \ f\in C[0,1],\mbox{ where }k\geq 1.

(1)

The estimation combines the error related to Banach’s fixed point theorem with the approximation error of the operator $L_{n}.$

An estimate of the error between any two iterates of the same operator $L_{n}$ is also obtained.

The results are applied to the Bernstein operators, Stancu operators and Cheney-Sharma operators.

The last part of this paper is devoted to an extension of some of the previous estimates to a class of multidimensional approximation operators of the type preveously considered in [1].

2. Preliminaries

2.1. Positive linear operators

The general family of operators used in our study has an expression that appears often in many papers. Thus, for each $n\in\mathbb{N}$ we consider a net $\Delta_{n}$ on $[0,1]$ ,

\Delta_{n}=(0=x_{n,0}<x_{n,1}<\ldots<x_{n,n}=1),

and a system of function $\ (\varphi_{n,l})_{l=\overline{0,n}},$ $\varphi_{n,l}\in C[0,1],$ $0\leq l\leq n,$ satisfying the relations

\varphi_{n,l}\geq 0\ \ (l=0,...,n),\quad\varphi_{n,0}(0)=\varphi_{n,n}(1)=1,

\sum_{l=0}^{n}\varphi_{n,l}=e_{0},\ \quad\sum_{l=0}^{n}x_{n,l}\varphi_{n,l}=e_% {1}.

(2)

The operators have the following form

L_{n}:C[0,1]\rightarrow C[0,1],\quad(L_{n}f)(x)=\sum_{l=0}^{n}\varphi_{n,l}(x)% f(x_{n,l}).

(3)

Clearly, $L_{n}$ , $n\geq 1$ , are positive linear operators and based on (2), their degree of exactness is one,

L_{n}e_{0}=e_{0}\ \mbox{ and }\ L_{n}e_{1}=e_{1},\

(4)

which means that they preserve the affine functions. Also,

(L_{n}f)(x)=f(x),\ \ \text{for }\ x\in\{0,1\},

that is the operators are interpolators at the ends of the compact interval $[0,1]$ .

Taking in view (4), by virtue of the classical result obtained by Shisha and Mond [9] regarding the error of approximation, on has

\left|(L_{n}f)(x)-f(x)\right|\leq 2\omega_{1}(f,\delta_{n}(x)),\ \ \ \ x\in[0,% 1],

(5)

where

\delta_{n}(x)=\left(\sum_{l=0}^{n}\varphi_{n,l}(x)x_{n,l}^{2}-x^{2}\right)^{1/2}

(6)

and $\ \omega_{1}(f,\cdot)$ represents the modulus of continuity of $f$ defined by

\omega_{1}(f,\delta)=\sup_{\begin{subarray}{c}x_{1},x_{2}\in[0,1]\\ |x_{1}-x_{2}|\leq\delta\end{subarray}}|f(x_{1})-f(x_{2})|,\ \ \ \delta\geq 0.

Based on the Popoviciu-Bohman-Korovkin theorem, if $\ \lim\limits_{n\rightarrow\infty}\sum\limits_{l=0}^{n}\varphi_{n,l}x_{n,l}^{2% }=e_{2}\$ uniformly on $[0,1]$ , then for every $f\in C[0,1]$ we have

\lim\limits_{n\rightarrow\infty}L_{n}f=f\ \ \ \text{uniformly on }[0,1].

2.2. Contraction property of the approximation operators

Defining

X_{\alpha,\beta}=\left\{f\in C([0,1]):\ f(0)=\alpha,\ f(1)=\beta\right\},\ \ % \ (\alpha,\beta)\in\mathbb{R}\times\mathbb{R},

(7)

the system $(X_{\alpha,\beta})_{(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}}$ makes up a partition of the space $C[0,1]$ . The interpolation property of our operators guarantees that each $\ X_{\alpha,\beta}$ is an invariant subset of every operator $L_{n}.$

Assume that

\ \lambda_{n}:=\min\limits_{x\in[0,1]}(\varphi_{n,0}(x)+\varphi_{n,n}(x))>0.

(8)

Then according to [2, Theorem 4] one has:

(a) $L_{n}\Big{|}_{X_{\alpha,\beta}}:X_{\alpha,\beta}\rightarrow X_{\alpha,\beta}$ is a contraction for every $(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}$ and $n\geq 1$ ; in addition

\|L_{n}f-L_{n}g\|\leq(1-\lambda_{n})\|f-g\|

(9)

for all $f,g\in X_{\alpha,\beta}.$

(b) The sequence $(L_{n}^{k})_{k\geq 1}$ of the iterates of $L_{n}$ satisfies

\lim\limits_{k\rightarrow\infty}(L_{n}^{k}f)(x)=f(0)+(f(1)-f(0))x\ \ \ \text{% uniformly on }\left[0,1\right],

(10)

for every $f\in C[0,1].\vskip 6.0pt plus 2.0pt minus 2.0pt$

Notice that the affine function $\ f^{\ast}:=f\left(0\right)+\left(f\left(1\right)-f\left(0\right)\right)e_{1}\$ is a fixed point in $X_{f\left(0\right),f\left(1\right)}$ of any operator $L_{n}$ and thus (10) is nothing else than the conclusion of Banach’s fixed point theorem (applied in the space $X_{f\left(0\right),f\left(1\right)}$ ) regarding the limit of the sequence of succesive approximations of the contraction $\ L_{n}\Big{|}_{X_{f\left(0\right),f\left(1\right)}}.$

3. Estimates for unidimensional operators

In accordance with (7), the function $f^{*}=f(0)+(f(1)-f(0))e_{1}$ belongs to $X_{f(0),f(1)}$ . Since $L_{n}$ preserves affine functions, $f^{*}$ is fixed point of $L_{n}$ for each $n\geq 1$ .

To provide the reader with an in-depth mathematical background so that this presentation is completely self-contained, we will refer to Banach fixed point theorem [3]: If $(X,d)$ is a complete metric space and a map $\Lambda:X\rightarrow X$ is a contraction, then $\Lambda$ admits a unique fixed point $x^{\ast}$ in $X$ . In addition, if $q\in(0,1)$ is a Lipschitz constant of $\Lambda,$ then for any starting element $x_{0}\in X,$ the sequence $\left(x_{k}\right)_{k\geq 1}$ given by $\ x_{k}=\Lambda\left(x_{k-1}\right)\ \left(k\geq 1\right)$ converges to $x^{\ast\text{ }}$ and satisfies the following inequality describing the speed of convergence:

d(x_{k},x^{\ast})\leq\frac{q^{k}}{1-q}d(x_{1},x_{0}),\ \ \ k\geq 1.

(11)

For our purposes, we consider $X=C[0,1]$ and $\ d(f,g)=\|f-g\|$ . Taking in view (9), we can choose $\ q=1-\lambda_{n}\in[0,1)$ . With this information it is easy to establish upper limits of approximation errors both between $L_{n}^{k}f$ and $f^{\ast}$ , and between two arbitrary terms of the sequence $\left(L_{n}^{k}f\right)_{k\geq 1}.$

Theorem 1.

Let $\ L_{n}$ , $n\geq 1$ , be the operators designated by (3) such that (8) holds. For each $n,$ the iterates of $L_{n}$ satisfy the following relation

\left\|L_{n}^{k}f-f^{\ast}\right\|\leq\frac{2(1-\lambda_{n})^{k}}{\lambda_{n}}% \sup_{x\in[0,1]}\omega_{1}(f,\delta_{n}(x)),\ \ \ f\in C[0,1],\ k\geq 1,

(12)

where $\delta_{n}(x)$ is given by (6).

Proof. Let $n$ and $f\in C[0,1]$ be arbitrarily chosen. We will use inequality (11) transcribed to our case. We assign the following values

x_{0}:=f,\ \ \ x_{k}:=L_{n}^{k}f\ \ \left(k\geq 1\right),\ \ x^{\ast}:=f^{\ast}.

Then (11) reads as follows

\left\|L_{n}^{k}f-f^{\ast}\right\|\leq\frac{(1-\lambda_{n})^{k}}{\lambda_{n}}% \|L_{n}f-f\|.

(13)

Next, from (5) one has

\left\|L_{n}f-f\right\|\leq 2\sup_{x\in[0,1]}\omega_{1}(f,\delta_{n}(x)).

(14)

Now (13) and (14) give (12). $\square$

Remark 2.

For $f\in C[0,1]$ , modulus $\omega_{1}(f,\cdot)$ is a bounded function. Consequently, for any $n$ arbitrarily fixed, relation (12) implies

\lim\limits_{k\rightarrow\infty}\|L_{n}^{k}f-f^{\ast}\|=0.

Thus we regained statement (10).

Theorem 3.

Let $L_{n}$ , $n\geq 1$ , be the operators designated by (3) such that (8) holds. For each $n,$ the iterates satisfy the following relation

\left\|L_{n}^{k_{1}}f-L_{n}^{k_{2}}f\right\|\leq\frac{(1-\lambda_{n})^{k_{2}}}% {\lambda_{n}}\left\|L_{n}f-f\right\|,\ \ \ \ k_{1}>k_{2}\geq 1,\ f\in C[0,1].

Proof. Let $f$ belong to $C[0,1],$ arbitrarily fixed. Based on (9), by induction on $k_{2}$ we obtain

\left\|L_{n}^{k_{2}+1}f-L_{n}^{k_{2}}f\right\|\leq(1-\lambda_{n})^{k_{2}}\left% \|L_{n}f-f\right\|.

For $\ k_{1}>k_{2},$ letting $\ L_{n}^{0}f=f,$ we can write successively

	$\displaystyle\left\\|L_{n}^{k_{1}}f-L_{n}^{k_{2}}f\right\\|$	$\displaystyle\leq\left\\|L_{n}^{k_{1}}f-L_{n}^{k_{1}-1}f\right\\|+\ \ldots+\left% \\|L_{n}^{k_{2}+1}f-L_{n}^{k_{2}}f\right\\|$
		$\displaystyle\leq(1-\lambda_{n})^{k_{1}-1}\left\\|L_{n}f-f\right\\|+\ \ldots+(1-% \lambda_{n})^{k_{2}}\left\\|L_{n}f-f\right\\|$
		$\displaystyle=(1-\lambda_{n})^{k_{2}}\left\\|L_{n}f-f\right\\|\sum_{j=0}^{k_{1}-% k_{2}-1}(1-\lambda_{n})^{j}$
		$\displaystyle\leq(1-\lambda_{n})^{k_{2}}\left\\|L_{n}f-f\right\\|\sum_{j=0}^{% \infty}(1-\lambda_{n})^{j}$
		$\displaystyle=\frac{(1-\lambda_{n})^{k_{2}}}{\lambda_{n}}\left\\|L_{n}f-f\right\\|.$

Thus the proof is completed. $\square$

Application 1 (Berstein operators). It is widely acknowledged that the most studied positive linear operators are the Bernstein operators. They fit with our construction, namely

x_{n,l}=\frac{l}{n},\ \ \varphi_{n,l}(x)=\binom{n}{l}x^{l}(1-x)^{n-l},\ \ 0% \leq l\leq n,\ \ \lambda_{n}=\left(\frac{1}{2}\right)^{n-1}.

Requirements (4) and (8) are fulfilled and (6) becomes

\delta_{n}(x)=\left(\frac{x(1-x)}{n}\right)^{1/2}\leq\frac{1}{2\sqrt{n}},\ \ % \ x\in[0,1].

Consequently, for these operators relation (12) is valid. Returning at (5), for the Bernstein operators, the smallest value of the constant in front of the modulus $\omega_{1}$ was established by Sikkema [10]. Bernstein operators $B_{n}$ , $n\geq 1$ , satisfy

\left|(B_{n}f)(x)-f(x)\right|\leq c\omega_{1}\left(f,\frac{1}{\sqrt{n}}\right)% ,\quad c=\frac{4306+837\sqrt{6}}{5832}\approx 1.089,

for any $f\in C[0,1]$ .

Along the same line, it is worth mentioning that Păltănea [6, Eq. (2)] obtained the relation

\left|(B_{n}f)(x)-f(x)\right|\leq\omega_{2}\left(f,\frac{1}{\sqrt{n}}\right),% \ \ \ f\in C[0,1],\ x\in(0,1),

where $\omega_{2}(f,\cdot)$ is the second modulus of smoothness associated with any bounded function $f:\left[0,1\right]\rightarrow\mathbb{R},$ defined as follows

\omega_{2}(f,\delta)=\sup_{\begin{subarray}{c}x,x\pm h\in[0,1]\\ \left|h\right|\leq\delta\end{subarray}}|f(x-h)-2f(x)+f(x+h)|,\ \ \ \delta\geq 0.

Thus, for the Bernstein operators, our statement (12) can be improved considering these optimal constants connected with the moduli $\omega_{1}$ and $\omega_{2}$ respectively.

We note that the result about the limit of the iterates of Bernstein operators was first obtained by Kelinsky and Rivlin [5], and reobtained by I.A. Rus [8] using the contraction principle.

Application 2 (Stancu operators). Based on the Markov-Pólya urn scheme, Stancu [11] introduced the positive linear operators $P_{n}^{\langle\alpha\rangle}$ , $n\geq 1$ , from $C[0,1]$ to itself, $\alpha$ being a nonnegative real parameter which can depend on $n$ . In this case we identify

x_{n,l}=\frac{l}{n},\ \ \varphi_{n,l}(x)=\binom{n}{l}\frac{x^{[l,-\alpha]}(1-x% )^{[n-l,-\alpha]}}{1^{[n,-\alpha]}},\ \ \ x\in[0,1].

In the above, $t^{[s,-\alpha]}$ stands for the generalized factorial power with the step $-\alpha$ , i.e.,

t^{[0,-\alpha]}=1,\ \ \ t^{[s,-\alpha]}=t(t+\alpha)\ldots(t+(s-1)\alpha)\ \ \ % \left(s\in\mathbb{N\setminus}\left\{0\right\}\right).

Again, identities (4) are fulfilled and

\lambda_{n}\geq(2^{n-1}1^{[n,-\alpha]})^{-1},\ \ n\geq 1.

Regarding (6) we have

\delta_{n}(x)=\left(\frac{1+n\alpha}{n(1+\alpha)}x(1-x)\right)^{1/2}\leq\frac{% 1}{2}\left(\frac{1+n\alpha}{n(1+\alpha)}\right)^{1/2},\ \ x\in[0,1].

Thus Theorem 1 can be applied.

Application 3 (Cheney-Sharma operators). Based on a combinatorial identity of Abel-Jensen, Cheney and Sharma [4] have defined a new class of positive linear operators of discrete type. Using the notations promoted in this paper, for $0\leq l\leq n,$ we identify

x_{n,l}=\frac{l}{n},

\varphi_{n,l}(x)=(1+n\beta)^{1-n}\binom{n}{l}x(x+l\beta)^{l-1}(1-x)(1-x+(n-l)% \beta)^{n-1-l},

where $\beta$ is a nonnegative real number which can depend on $n$ . The operators preserve the constant functions. In [12] it was shown that each operator reproduces the linear functions, thus (4) is fulfilled. Moreover, if $\beta=o(n^{-1})$ , the sequence of operators is an approximation process. Since

\lambda_{n}\geq(2(1+n\beta))^{1-n},\ \ \ n\geq 1,

also (8) takes place and the conclusion of Theorem 1 is achieved.

4. Estimates for multidimensional operators

For the convenience of the reader, we start the section by some preliminaries of vector analysis.

4.1. Perov’s fixed point theorem

For the convenience of the reader, we present here Perov’s fixed point theorem in a version sufficient for our approach. For a Banach space $\left(X,\left\|\cdot\right\|\right),$ we shall also denote by $\left\|\cdot\right\|$ the vector-valued norm on $\ X^{p}=X\times...\times X$ ( $p$ times) having the following meaning:

\left\|\mathbf{u}\right\|:=\left[\begin{array}[]{c}\left\|u_{1}\right\|\\ \cdot\cdot\cdot\\ \left\|u_{p}\right\|\end{array}\right]

(15)

for $\mathbf{u}=\left(u_{1},...,u_{p}\right)\in X^{p}.$

Theorem 4 (Perov).

Let $\left(X,\left\|\cdot\right\|\right)$ be a Banach space and $\mathbf{L}:X^{p}\rightarrow X^{p},$ $\mathbf{L}=\left(L_{1},...,L_{p}\right),$ $L_{i}:X^{p}\rightarrow X\ \left(i=1,...,p\right)$ be any mapping. Assume that there exists a matrix $M\in\mathcal{M}_{p\times p}\left(\mathbb{R}_{+}\right)$ which converges to zero (i.e., $M^{k}$ tends to the null matrix as $k$ tends to infinity), such that

\left\|\mathbf{Lu}-\mathbf{Lv}\right\|\leq M\left\|\mathbf{u-v}\right\|

for all $\mathbf{u},\ \mathbf{v}\in X^{p}.$ Then $\mathbf{L}$ has a unique fixed point $\mathbf{u}^{\ast}$ which is the limit of the sequence $\left(\mathbf{L}^{k}\mathbf{u}\right)_{k\geq 1}$ of successive approximations starting from any $\mathbf{u}\in X^{p}.$ In addition, one has

\left\|\mathbf{L}^{k}\mathbf{u}-\mathbf{u}^{\ast}\right\|\leq\left(I-M\right)^% {-1}M^{k}\left\|\mathbf{Lu}-\mathbf{u}\right\|

(16)

for all $k\geq 1$ and $\mathbf{u}\in X^{p}.$

We note that if a matrix $M$ converges la zero, then the matrix $I-M$ is nonsingular, the entries of $\left(I-M\right)^{-1}$ are nonnegative too, and

I+M+...+M^{k}+...=\left(I-M\right)^{-1}

(17)

(see [7]). Here $I$ stands for the unit matrix of size $p.$

Also note that for $p=1,$ Perov’s theorem reduces to Banach’s theorem in the space $X,$ where $M$ is a nonnegative constant with $M<1.$

4.2. Multidimensional approximation operators

Here following our previous paper [1], we define a class of multidimensional positive linear operators. First we construct convex combinations of $p$ ( $p\geq 1)$ operators of type (3). To this aim, let

\gamma_{i,j}\in[0,1],\ \ \ 1\leq i,j\leq p,\ \ \ \sum_{j=1}^{p}\gamma_{i,j}=1

and take $(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}$ . Then define the operator

	$\displaystyle\mathbf{L}$	$\displaystyle:$	$\displaystyle X_{\alpha,\beta}^{p}\rightarrow X_{\alpha,\beta}^{p},\quad% \mathbf{L}=(\mathbf{L}_{1},\ldots,\mathbf{L}_{p}),$		(18)
	$\displaystyle\mathbf{L}_{i}\mathbf{f}$	$\displaystyle=$	$\displaystyle\sum_{j=1}^{p}\gamma_{i,j}L_{n_{ij}}f_{j},\quad\ \mathbf{f=}\left% (f_{1},...,f_{p}\right),\ i=1,\ldots,p,$

where $X_{\alpha,\beta}$ is given by (7). The above convex combination uses $p$ operators of the form (3) having different orders $n_{ij}$ , $j=1,\ldots,p$ .

Notice that the operator $\mathbf{L}$ can be expressed in terms of two matrices:

\Gamma:=\left[\gamma_{ij}\right]_{1\leq i,j\leq p}\ \ \ \ \text{and\ \ \ }O:=% \left[L_{n_{ij}}\right]_{1\leq i,j\leq p}.

The first is a matrix of numbers, while the second is a matrix of linear operators. Then it is easy to see that the following representation formula holds:

\mathbf{Lf}=\text{diag\ }\Gamma O\left(\mathbf{f}\right)^{\bot},

where $O\left(\mathbf{f}\right)=\left[L_{n_{ij}}f_{j}\right]_{1\leq i,j\leq p}$ and $O\left(\mathbf{f}\right)^{\bot}$ is its transposed matrix.

We have the following convergence theorem whose proof is almost identical to that of the similar result from [1].

Theorem 5.

Assume that $\mathbf{L}$ is given by (18), where the operators $L_{n_{ij}},\ 1\leq i,j\leq p$ satisfy condition (8). Then for any vector-valued function $\mathbf{f}\in X_{\alpha,\beta}^{p},$ one has

\lim\limits_{k\rightarrow\infty}\mathbf{L}^{k}\mathbf{f}=\mathbf{f}^{\ast},

(19)

componentwise uniformly on $\left[0,1\right],$ where the components of the vector-valued function $\mathbf{f}^{\ast}$ are all equal with the affine function $\alpha+\left(\beta-\alpha\right)x.$

4.3. Estimates for the iterates of multidimensional operators

In this last section we extend some of the previous results to multidimensional operators.

Assume that $\mathbf{L}$ is a multidimensional operator of type (18), where the operators $L_{n_{ij}},\ 1\leq i,j\leq p$ satisfy condition (8). We first give the vector version of Theorem 3. We denote by $\delta_{n_{ij}}$ and $\lambda_{n_{ij}}$ $\left(1\leq i,j\leq p\right)$ the corresponding numbers of $L_{n_{ij}}$ defined as given by (6) and (8), respectively. Denoting

\lambda:=\max\left\{\lambda_{n_{ij}}:\ 1\leq i,j\leq p\right\}

one has $\lambda<1$ and, as in [1], the matrix $M:=\left[\lambda\gamma_{i,j}\right]_{1\leq i,j\leq p}$ is convergent to zero and $\mathbf{L}$ is a Perov contraction on $X_{\alpha,\beta}^{p},$ in the sense that

\left\|\mathbf{Lf}-\mathbf{Lg}\right\|\leq M\left\|\mathbf{f-g}\right\|

(20)

for all $\mathbf{f,g}\in X_{\alpha,\beta}^{p}.$ Here $\left\|\cdot\right\|$ stands for the vector-valued norm (15) on the space $C\left[0,1\right]^{p}.$

The vector version of Theorem 3 is the following:

Theorem 6.

The iterates of the operator $\mathbf{L}$ satisfy the following relation

\left\|\mathbf{L}^{k_{1}}\mathbf{f}-\mathbf{L}^{k_{2}}\mathbf{f}\right\|\leq% \left(I-M\right)^{-1}M^{k_{2}}\left\|\mathbf{Lf}-\mathbf{f}\right\|,\ \ \ k_{1% }>k_{2}\geq 1,\ \mathbf{f}\in X_{\alpha,\beta}^{p}.

Proof.

From (20), one has

\left\|\mathbf{L}^{k+1}\mathbf{f}-\mathbf{L}^{k}\mathbf{f}\right\|\leq M^{k}% \left\|\mathbf{Lf}-\mathbf{f}\right\|\ \ \ \left(k\geq 1\right).

This together with (17) gives

$\displaystyle\left\\|\mathbf{L}^{k_{1}}\mathbf{f}-\mathbf{L}^{k_{2}}\mathbf{f}\right\\|$	$\displaystyle\leq$	$\displaystyle\left\\|\mathbf{L}^{k_{1}}\mathbf{f}-\mathbf{L}^{k_{1}-1}\mathbf{f% }\right\\|+\ ...+\left\\|\mathbf{L}^{k_{2}+1}\mathbf{f}-\mathbf{L}^{k_{2}}% \mathbf{f}\right\\|$
	$\displaystyle\leq$	$\displaystyle\left(M^{k_{1}-1}+\ ...+M^{k_{2}}\right)\left\\|\mathbf{Lf}-% \mathbf{f}\right\\|$
	$\displaystyle=$	$\displaystyle\left(M^{k_{1}-k_{2}-1}+\ ...+I\right)M^{k_{2}}\left\\|\mathbf{Lf}% -\mathbf{f}\right\\|$
	$\displaystyle\leq$	$\displaystyle\left(I-M\right)^{-1}M^{k_{2}}\left\\|\mathbf{Lf}-\mathbf{f}\right\\|.$

∎

Finally, the estimate (16) given by Perov’s theorem immediately yields the following error result.

Theorem 7.

For every $\mathbf{f}\in X_{\alpha,\beta}^{p}$ and every $k\geq 1,$ one has

\left\|\mathbf{L}^{k}\mathbf{f}-\mathbf{f}^{\ast}\right\|\leq\left(I-M\right)^% {-1}M^{k}\left\|\mathbf{Lf}-\mathbf{f}\right\|.

References

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	$\displaystyle\left\\|L_{n}^{k_{1}}f-L_{n}^{k_{2}}f\right\\|$	$\displaystyle\leq\left\\|L_{n}^{k_{1}}f-L_{n}^{k_{1}-1}f\right\\|+\ \ldots+\left% \\|L_{n}^{k_{2}+1}f-L_{n}^{k_{2}}f\right\\|$
		$\displaystyle\leq(1-\lambda_{n})^{k_{1}-1}\left\\|L_{n}f-f\right\\|+\ \ldots+(1-% \lambda_{n})^{k_{2}}\left\\|L_{n}f-f\right\\|$
		$\displaystyle=(1-\lambda_{n})^{k_{2}}\left\\|L_{n}f-f\right\\|\sum_{j=0}^{k_{1}-% k_{2}-1}(1-\lambda_{n})^{j}$
		$\displaystyle\leq(1-\lambda_{n})^{k_{2}}\left\\|L_{n}f-f\right\\|\sum_{j=0}^{% \infty}(1-\lambda_{n})^{j}$
		$\displaystyle=\frac{(1-\lambda_{n})^{k_{2}}}{\lambda_{n}}\left\\|L_{n}f-f\right\\|.$

$\displaystyle\left\\|\mathbf{L}^{k_{1}}\mathbf{f}-\mathbf{L}^{k_{2}}\mathbf{f}\right\\|$	$\displaystyle\leq$	$\displaystyle\left\\|\mathbf{L}^{k_{1}}\mathbf{f}-\mathbf{L}^{k_{1}-1}\mathbf{f% }\right\\|+\ ...+\left\\|\mathbf{L}^{k_{2}+1}\mathbf{f}-\mathbf{L}^{k_{2}}% \mathbf{f}\right\\|$
	$\displaystyle\leq$	$\displaystyle\left(M^{k_{1}-1}+\ ...+M^{k_{2}}\right)\left\\|\mathbf{Lf}-% \mathbf{f}\right\\|$
	$\displaystyle=$	$\displaystyle\left(M^{k_{1}-k_{2}-1}+\ ...+I\right)M^{k_{2}}\left\\|\mathbf{Lf}% -\mathbf{f}\right\\|$
	$\displaystyle\leq$	$\displaystyle\left(I-M\right)^{-1}M^{k_{2}}\left\\|\mathbf{Lf}-\mathbf{f}\right\\|.$

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

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

Abstract.

1. Introduction

2. Preliminaries

2.1. Positive linear operators

2.2. Contraction property of the approximation operators

3. Estimates for unidimensional operators

Theorem 1.

Remark 2.

Theorem 3.

4. Estimates for multidimensional operators

4.1. Perov’s fixed point theorem

Theorem 4 (Perov).

4.2. Multidimensional approximation operators

Theorem 5.

4.3. Estimates for the iterates of multidimensional operators

Theorem 6.

Proof.

Theorem 7.

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