# On the Korovkin approximation theorem and Volkov-type theorems

## Abstract

In this short paper, we give a generalization of the classical Korovkin approximation theorem (Korovkin in Linear Operators and Approximation Theory, 1960), Volkov-type theorems (Volkov in Dokl. Akad. Nauk SSSR 115:17-19, 1957), and a recent result of (Taşdelen and Erençin in J. Math. Anal. Appl. 331(1):727-735, 2007).

MSC:41A36, 41A25.

## 1 Introduction

In this paper, the classical Korovkin theorem (see ) and one of the key results (Theorem 1) of  will be generalized to arbitrary compact Hausdorff spaces. For a topological space X, the space of real-valued continuous functions on X, as usual, will be denoted by $C\left(X\right)$. We note that if X is a compact Hausdorff space, then $C\left(X\right)$ is a Banach space under pointwise algebraic operations and under the norm

$\parallel f\parallel =\underset{x\in X}{sup}|f\left(x\right)|.$

Let X be a compact Hausdorff space and E be a subspace of $C\left(X\right)$. Then a linear map $A:E\to C\left(X\right)$ is called positive if $A\left(f\right)\ge \mathbf{0}$ in $C\left(X\right)$ whenever $f\ge 0$ in E. Here $f\ge \mathbf{0}$ means that $f\left(x\right)\ge 0$ in for all $x\in X$.

For more details on abstract Korovkin approximations theory, we refer to  and .

Constant-one function on a topological space X will be denoted by ${f}_{0}$, that is, ${f}_{0}\left(x\right)=1$ for all $x\in X$. If $A=\left(a,b\right)$ and $B=\left(c,d\right)$ are elements of ${\mathbb{R}}^{2}$, then the Euclidean distance between A and B, given by

$|\left(a,b\right)-\left(c,d\right)|=\sqrt{{\left(a-c\right)}^{2}+{\left(b-d\right)}^{2}},$

is denoted by $|A-B|$.

Definition 1.1 Let X and Y be compact Hausdorff spaces, Z be the product space of X and Y, and let $h\in C\left(Z×Z\right)$ and $f\in C\left(Z\right)$ be given. The module of continuity of f with respect to h is a function ${w}_{h}\left(f\right):\left[0,\mathrm{\infty }\right)\to \mathbb{R}$ defined by $w\left(f\right)\left(0\right)=0$, and

whenever $\delta >0$, with the following additional properties:

1. (i)

$w\left(f\right)$ is increasing;

2. (ii)

${lim}_{\delta \to 0}=0$.

We note that the above definition is motivated from [, p.729] and generalizes the definition which is given there.

Definition 1.2 Let X, Y, and Z be as in Definition 1.1. Let $h\in C\left(Z×Z\right)$ be given. We define ${H}_{w,h}$ as the set of all continuous functions $f\in C\left(X×Y\right)$ such that for all $\left(u,v\right),\left(x,y\right)\in X×Y$, one has

$|f\left(u,v\right)-f\left(x,y\right)|\le {w}_{h}\left(f\right)\left(|h\left(\left(u,v\right),\left(x,y\right)\right)|\right).$

When ${H}_{w,h}$ is mentioned, we always suppose that h satisfies the property for ${H}_{w,h}$ being a vector subspace of $C\left(X×X\right)$. We note that ${H}_{w,h}$ has been considered in  by taking $X=\left[0,A\right]$, $Y=\left[0,B\right]$ ($A,B>0$),

$h\left(\left(u,v\right),\left(x,y\right)\right)=\parallel \left({f}_{1}\left(u,v\right),{f}_{2}\left(u,v\right)\right)-\left({f}_{1}\left(x,y\right),{f}_{2}\left(x,y\right)\right)\parallel ,$

where

${f}_{1}\left(u,v\right)=\frac{u}{1-u}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{f}_{2}\left(u,v\right)=\frac{v}{1-v}.$

The main result of this paper will be obtained via the following lemma.

## 2 Main result

Lemma 2.1 Let X and Y be compact Hausdorff spaces and Z be a product space of X and Y. Let ${f}_{1},{f}_{2}\in C\left(Z\right)$ and $h\in C\left(Z×Z\right)$ be defined by

$h\left(\left(u,v\right),\left(x,y\right)\right)=|\left({f}_{1}\left(u,v\right),{f}_{2}\left(u,v\right)\right)-\left({f}_{1}\left(x,y\right),{f}_{2}\left(x,y\right)\right)|$

so that ${H}_{w,h}$ is a subspace $C\left(X×Y\right)$ and ${f}_{1},{f}_{2}\in {H}_{w,h}\left(Z\right)$. Let $A:{H}_{w,h}\to C\left(Z\right)$ be a positive linear map. Let $\left(u,v\right)\in Z$ be given, and define ${\phi }_{u,v},{\mathrm{\Phi }}_{u,v}\in C\left(Z\right)$ by

${\phi }_{u,v}={\left({f}_{1}\left(u,v\right){f}_{0}-{f}_{1}\right)}^{2}\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{\mathrm{\Phi }}_{u,v}={\left({f}_{2}\left(u,v\right){f}_{0}-{f}_{2}\right)}^{2}.$

Then, for all $\left(u,v\right)\in Z$, one has

$\begin{array}{rcl}0& \le & A\left({\phi }_{u,v}+{\mathrm{\Phi }}_{u,v}\right)\\ \le & {C}_{1}\left[A\left({f}_{0}\right)-{f}_{0}\right]\left(u,v\right)-{C}_{2}\left[A\left({f}_{1}+{f}_{2}\right)-\left({f}_{1}+{f}_{2}\right)\right]+\left[A\left({f}_{1}^{2}+{f}_{2}^{2}\right)-\left({f}_{1}^{2}+{f}_{2}^{2}\right)\right],\end{array}$

where

${C}_{1}=\left({f}_{1}{\left(u,v\right)}^{2}+{f}_{2}{\left(u,v\right)}^{2}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}{C}_{2}=-2\left({f}_{1}\left(u,v\right)+{f}_{2}\left(u,v\right)\right).$

Proof Note that

$0\le {\phi }_{u,v}={f}_{1}{\left(u,v\right)}^{2}{f}_{0}-2{f}_{1}\left(u,v\right){f}_{1}+{f}_{1}^{2}.$

Applying the linearity and positivity of A, we have

$0\le A\left({\phi }_{u,v}\right)={f}_{1}{\left(u,v\right)}^{2}A\left({f}_{0}\right)-2{f}_{1}\left(u,v\right)A\left({f}_{1}\right)+A\left({f}_{1}^{2}\right).$

Then one can have

$\begin{array}{rcl}0& \le & A\left({\phi }_{u,v}\right)\left(u,v\right)\\ =& {f}_{1}{\left(u,v\right)}^{2}A\left({f}_{0}\right)\left(u,v\right)-2{f}_{1}\left(u,v\right)A\left({f}_{1}\right)\left(u,v\right)+A\left({f}_{1}^{2}\right)\left(u,v\right)\\ =& {f}_{1}^{2}\left(u,v\right)\left[A\left({f}_{0}\right)\left(u,v\right)-{f}_{0}\left(u,v\right)+{f}_{0}\left(u,v\right)\right]\\ -2{f}_{1}\left(u,v\right)\left[A\left({f}_{1}\right)\left(u,v\right)-{f}_{1}\left(u,v\right)+{f}_{1}\left(u,v\right)\right]\\ +\left[A\left({f}_{1}^{2}\right)\left(u,v\right)-{f}_{1}{\left(u,v\right)}^{2}+{f}_{1}{\left(u,v\right)}^{2}\right]\\ =& {f}_{1}^{2}\left(u,v\right)\left[A\left({f}_{0}\right)-{f}_{0}\right]\left(u,v\right)-2{f}_{1}\left(u,v\right)\left[A\left({f}_{1}\right)-{f}_{1}\right]\left(u,v\right)+\left[A\left({f}_{1}^{2}\right)-{f}_{1}^{2}\right]\left(u,v\right).\end{array}$

Similarly, we have

$\begin{array}{rcl}A\left({\mathrm{\Phi }}_{u,v}\right)\left(u,v\right)& =& {f}_{2}^{2}\left(u,v\right)\left[A\left({f}_{0}\right)-{f}_{0}\right]\left(u,v\right)\\ -2{f}_{2}\left(u,v\right)\left[A\left({f}_{2}\right)-{f}_{2}\right]\left(u,v\right)+\left[A\left({f}_{2}^{2}\right)-{f}_{2}^{2}\right]\left(u,v\right).\end{array}$

Now applying A, which is linear, to ${\phi }_{u,v}+{\mathrm{\Phi }}_{u,v}$ completes the proof. □

Lemma 2.2 Let X and Y be compact Hausdorff spaces and ${f}_{1}$, ${f}_{2}$, and h be defined as in Lemma  2.1. Let $f\in {H}_{w,h}$ be given. For each $ϵ>0$, there exists $\delta >0$ such that

$|f\left(u,v\right)-f\left(x,y\right)|<ϵ+\frac{2\parallel f\parallel }{{\delta }^{2}}{h}^{2}\left(\left(u,v\right),\left(x,y\right)\right).$

Proof Let $ϵ>0$ be given. Since $w\left(f\right):\left[0,\mathrm{\infty }\right)\to \mathbb{R}$ is continuous, there exists $\delta >0$ such that $w\left(f,{\delta }^{\prime }\right)=w\left(f\right)\left({\delta }^{\prime }\right)<ϵ$ for all $0\le {\delta }^{\prime }<\delta$. This implies, since

that

${\left[\left({\phi }_{u,v}+{\mathrm{\Phi }}_{u,v}\right)\right]}^{\frac{1}{2}}\left(x,y\right)=|h\left(\left(u,v\right)-h\left(x,y\right)\right)|<\delta \phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}|f\left(u,v\right)-f\left(x,y\right)|<ϵ,$

where ${\phi }_{u,v}$ and ${\mathrm{\Phi }}_{u,v}$ are defined as in Lemma 2.1. If ${\left[\left({\phi }_{u,v}+{\mathrm{\Phi }}_{u,v}\right)\right]}^{\frac{1}{2}}\left(x,y\right)\ge \delta$, then

$|f\left(u,v\right)-f\left(x,y\right)|\le 2\parallel f\parallel \le 2\parallel f\parallel \frac{\left[\left({\phi }_{u,v}+{\mathrm{\Phi }}_{u,v}\right)\right]\left(x,y\right)}{{\delta }^{2}}.$

Hence, for all $\left(u,v\right)\in Z$, we have

$|f\left(u,v\right)-f|\le ϵ+2\parallel f\parallel \frac{\left[\left({\phi }_{u,v}+{\mathrm{\Phi }}_{u,v}\right)\right]}{{\delta }^{2}}.$

This completes the proof. □

Lemma 2.3 Suppose that the hypotheses of Lemma  2.2 are satisfied. Let $f\in {H}_{w,h}$ and $ϵ>0$ be given. Then there exists $C>0$ such that

$\parallel A\left(f\right)-f\parallel <ϵ+C\left(\parallel A\left({f}_{0}\right)-{f}_{0}\parallel +\parallel A\left({f}_{1}+{f}_{2}\right)-\left({f}_{1}+{f}_{2}\right)\parallel +\parallel A\left({f}_{1}^{2}+{f}_{2}^{2}\right)-\left({f}_{1}^{2}+{f}_{2}^{2}\right)\parallel \right).$

Proof Set $K:=\frac{2\parallel f\parallel }{{\delta }^{2}}$. From Lemma 2.2, there exists $\delta >0$ such that for each $\left(u,v\right)\in Z$ we have

$\begin{array}{rl}|f\left(u,v\right){f}_{0}-f|& \le ϵ+\frac{2\parallel f\parallel }{{\delta }^{2}}\left[{\phi }_{u,v}+{\mathrm{\Phi }}_{u,v}\right]\\ \le ϵ+\frac{2\parallel f\parallel }{{\delta }^{2}}\left[{f}_{1}^{2}\left(u,v\right){f}_{0}+{f}_{2}^{2}\left(u,v\right){f}_{0}-2{f}_{1}\left(u,v\right){f}_{1}-2{f}_{2}\left(u,v\right){f}_{2}+\left({f}_{1}^{2}+{f}_{2}^{2}\right)\right],\end{array}$

whence

$\begin{array}{rl}|\left[A\left(f\right)-f\left(u,v\right)A\left({f}_{0}\right)\right]\left(u,v\right)|& \le ϵA\left({f}_{0}\right)\left(u,v\right)+K\left(A\left({\phi }_{u,v}\right)+A\left({\mathrm{\Phi }}_{u,v}\right)\right)\\ =ϵ+ϵ\left[A\left({f}_{0}\right)-{f}_{0}\right]\left(u,v\right)+KA\left({\phi }_{u,v}+{\mathrm{\Phi }}_{u,v}\right).\end{array}$

In particular, we have

$\begin{array}{rl}|A\left(f\right)-f|\left(u,v\right)& \le |\left[A\left(f\right)-f\left(u,v\right)A\left({f}_{0}\right)\right]\left(u,v\right)|+|f\left(u,v\right)||\left(A\left({f}_{0}\right)-{f}_{0}\right)\left(u,v\right)|\\ \le ϵ+KA\left({\phi }_{u,v}+{\mathrm{\Phi }}_{u,v}\right)\left(u,v\right)+\left(\parallel f\parallel +ϵ\right)\parallel A\left({f}_{0}\right)-{f}_{0}\parallel .\end{array}$

Now, applying Lemma 2.1 and taking

$C=2K+\parallel f\parallel ,$

we have what is to be shown. □

We note that in the above theorem C depends only on $\parallel f\parallel$ and ϵ, and is independent of the positive linear operator A.

We are now in a position to state the main result of the paper.

Theorem 2.4 Let X and Y be compact Hausdorff spaces and Z be the product space of X and Y. Let ${f}_{1},{f}_{2}\in C\left(Z\right)$, and $h\in C\left(Z×Z\right)$ be defined by

$h\left(\left(u,v\right),\left(x,y\right)\right)=\parallel \left({f}_{1}\left(u,v\right),{f}_{2}\left(u,v\right)\right)-\left({f}_{1}\left(x,y\right),{f}_{2}\left(x,y\right)\right)\parallel$

so that ${H}_{w,h}$ is a subspace $C\left(X×Y\right)$ and ${f}_{1},{f}_{2}\in {H}_{w,h}\left(Z\right)$. Let ${\left({A}_{n}\right)}_{n\in \mathbb{N}}$ be a sequence of positive operators from ${H}_{w,h}$ into $C\left(X×Y\right)$ satisfying:

1. (i)

$\parallel {A}_{n}\left({f}_{0}\right)-{f}_{0}\parallel \to 0$;

2. (ii)

$\parallel {A}_{n}\left({f}_{1}\right)-{f}_{1}\parallel \to 0$;

3. (iii)

$\parallel {A}_{n}\left({f}_{2}\right)-{f}_{2}\parallel \to 0$;

4. (iv)

$\parallel {A}_{n}\left({f}_{1}^{2}+{f}_{2}^{2}\right)-\left({f}_{1}^{2}+{f}_{2}^{2}\right)\parallel \to 0$.

Then, for all $f\in {H}_{w,h}$, we have

$\parallel {A}_{n}\left(f\right)-f\parallel \to 0.$

Proof Let $f\in {H}_{w,h}$ and $ϵ>0$ be given. By Lemma 2.3, there exists $C>0$ (depending only on $\parallel f\parallel$ and $ϵ>0$) such that for each n,

$\parallel {A}_{n}\left(f\right)-f\parallel \le ϵ+C\left(\parallel {A}_{n}\left({f}_{0}\right)-{f}_{0}\parallel +\parallel {A}_{n}\left({f}_{1}+{f}_{2}\right)-\left({f}_{1}+{f}_{2}\right)\parallel +\parallel {A}_{n}\left({f}_{1}^{2}+{f}_{2}^{2}\right)-\left({f}_{1}^{2}+{f}_{2}^{2}\right)\parallel \right).$

Since $ϵ>0$ is arbitrary and the last three terms of the inequality converge to zero by the assumption, we have

${A}_{n}\left(f\right)\to f.$

This completes the proof. □

Note also that in Theorem 1 of  it is not necessary to take a double sequence of positive operators: as the above result reveals, one can take $\left({A}_{n}\right)$ instead of $\left({A}_{n,m}\right)$.

Remarks

1. (1)

If $X=\left[0,1\right]$, and $Y=\left\{y\right\}$ and ${f}_{1},{f}_{2}\in C\left(X×Y\right)$ are defined by

${f}_{u,v}=u\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{f}_{2}=0,$

then Theorem 2.4 becomes the classical Korovkin theorem.

1. (2)

If one takes $X=\left[0,A\right]$, $Y=\left[0,B\right]$ ($0), and ${f}_{1}$ and ${f}_{2}$ are defined by

${f}_{1}\left(u,v\right)=\frac{u}{1-u}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{f}_{2}\left(u,v\right)=\frac{v}{1-v},$

then the above theorem becomes Theorem 1 of .

1. (3)

For linear positive operators of two variables, Theorem 2.4 generalizes the result of Volkov in .

2. (4)

We believe that the above theorem can be generalized to n-fold copies by taking $Z={X}_{1}×{X}_{2}×\cdots ×{X}_{n}$ instead of $Z=X×Y$, where ${X}_{1},{X}_{2},\dots ,{X}_{n}$ are compact Hausdorff spaces.

3. (5)

The above theorem is also true if one replaces $C\left(X\right)$ by ${C}_{b}\left(X\right)$, the space of bounded continuous functions, in the case of an arbitrary topological space X.

## References

1. Korovkin PP: Linear Operators and Approximation Theory. Hindustan Publish Co., Delhi; 1960.

2. Taşdelen F, Erençin A: The generalization of bivariate MKZ operators by multiple generating functions. J. Math. Anal. Appl. 2007,331(1):727-735. 10.1016/j.jmaa.2006.09.024

3. Altomare F, Campiti M: Korovkin-Type Approximation Theory and Its Applications. de Gruyter, Berlin; 1994.

4. Lorentz GG: Approximation of Functions. 2nd edition. Chelsea, New York; 1986.

5. Volkov VI: On the convergence of sequences of linear positive operators in the space of continuous functions of two variables. Dokl. Akad. Nauk SSSR 1957, 115: 17-19. (in Russian)

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Correspondence to Nihan Uygun.

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Uygun, N. On the Korovkin approximation theorem and Volkov-type theorems. J Inequal Appl 2014, 89 (2014). https://doi.org/10.1186/1029-242X-2014-89

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• DOI: https://doi.org/10.1186/1029-242X-2014-89

### Keywords

• positive linear operator
• Korovkin theorem
• Volkov-type theorem
• modules of continuity
• compact Hausdorff spaces 