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On the Korovkin approximation theorem and Volkovtype theorems
Journal of Inequalities and Applications volume 2014, Article number: 89 (2014)
Abstract
In this short paper, we give a generalization of the classical Korovkin approximation theorem (Korovkin in Linear Operators and Approximation Theory, 1960), Volkovtype theorems (Volkov in Dokl. Akad. Nauk SSSR 115:1719, 1957), and a recent result of (Taşdelen and Erençin in J. Math. Anal. Appl. 331(1):727735, 2007).
MSC:41A36, 41A25.
1 Introduction
In this paper, the classical Korovkin theorem (see [1]) and one of the key results (Theorem 1) of [2] will be generalized to arbitrary compact Hausdorff spaces. For a topological space X, the space of realvalued continuous functions on X, as usual, will be denoted by C(X). We note that if X is a compact Hausdorff space, then C(X) is a Banach space under pointwise algebraic operations and under the norm
Let X be a compact Hausdorff space and E be a subspace of C(X). Then a linear map A:E\to C(X) is called positive if A(f)\ge \mathbf{0} in C(X) whenever f\ge 0 in E. Here f\ge \mathbf{0} means that f(x)\ge 0 in ℝ for all x\in X.
For more details on abstract Korovkin approximations theory, we refer to [3] and [4].
Constantone function on a topological space X will be denoted by {f}_{0}, that is, {f}_{0}(x)=1 for all x\in X. If A=(a,b) and B=(c,d) are elements of {\mathbb{R}}^{2}, then the Euclidean distance between A and B, given by
is denoted by AB.
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(Z\times Z) and f\in C(Z) be given. The module of continuity of f with respect to h is a function {w}_{h}(f):[0,\mathrm{\infty})\to \mathbb{R} defined by w(f)(0)=0, and
whenever \delta >0, with the following additional properties:

(i)
w(f) is increasing;

(ii)
{lim}_{\delta \to 0}=0.
We note that the above definition is motivated from [[2], 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(Z\times Z) be given. We define {H}_{w,h} as the set of all continuous functions f\in C(X\times Y) such that for all (u,v),(x,y)\in X\times Y, one has
When {H}_{w,h} is mentioned, we always suppose that h satisfies the property for {H}_{w,h} being a vector subspace of C(X\times X). We note that {H}_{w,h} has been considered in [2] by taking X=[0,A], Y=[0,B] (A,B>0),
where
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(Z) and h\in C(Z\times Z) be defined by
so that {H}_{w,h} is a subspace C(X\times Y) and {f}_{1},{f}_{2}\in {H}_{w,h}(Z). Let A:{H}_{w,h}\to C(Z) be a positive linear map. Let (u,v)\in Z be given, and define {\phi}_{u,v},{\mathrm{\Phi}}_{u,v}\in C(Z) by
Then, for all (u,v)\in Z, one has
where
Proof Note that
Applying the linearity and positivity of A, we have
Then one can have
Similarly, we have
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 \u03f5>0, there exists \delta >0 such that
Proof Let \u03f5>0 be given. Since w(f):[0,\mathrm{\infty})\to \mathbb{R} is continuous, there exists \delta >0 such that w(f,{\delta}^{\prime})=w(f)({\delta}^{\prime})<\u03f5 for all 0\le {\delta}^{\prime}<\delta. This implies, since
that
where {\phi}_{u,v} and {\mathrm{\Phi}}_{u,v} are defined as in Lemma 2.1. If {[({\phi}_{u,v}+{\mathrm{\Phi}}_{u,v})]}^{\frac{1}{2}}(x,y)\ge \delta, then
Hence, for all (u,v)\in Z, we have
This completes the proof. □
Lemma 2.3 Suppose that the hypotheses of Lemma 2.2 are satisfied. Let f\in {H}_{w,h} and \u03f5>0 be given. Then there exists C>0 such that
Proof Set K:=\frac{2\parallel f\parallel}{{\delta}^{2}}. From Lemma 2.2, there exists \delta >0 such that for each (u,v)\in Z we have
whence
In particular, we have
Now, applying Lemma 2.1 and taking
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(Z), and h\in C(Z\times Z) be defined by
so that {H}_{w,h} is a subspace C(X\times Y) and {f}_{1},{f}_{2}\in {H}_{w,h}(Z). Let {({A}_{n})}_{n\in \mathbb{N}} be a sequence of positive operators from {H}_{w,h} into C(X\times Y) satisfying:

(i)
\parallel {A}_{n}({f}_{0}){f}_{0}\parallel \to 0;

(ii)
\parallel {A}_{n}({f}_{1}){f}_{1}\parallel \to 0;

(iii)
\parallel {A}_{n}({f}_{2}){f}_{2}\parallel \to 0;

(iv)
\parallel {A}_{n}({f}_{1}^{2}+{f}_{2}^{2})({f}_{1}^{2}+{f}_{2}^{2})\parallel \to 0.
Then, for all f\in {H}_{w,h}, we have
Proof Let f\in {H}_{w,h} and \u03f5>0 be given. By Lemma 2.3, there exists C>0 (depending only on \parallel f\parallel and \u03f5>0) such that for each n,
Since \u03f5>0 is arbitrary and the last three terms of the inequality converge to zero by the assumption, we have
This completes the proof. □
Note also that in Theorem 1 of [2] it is not necessary to take a double sequence of positive operators: as the above result reveals, one can take ({A}_{n}) instead of ({A}_{n,m}).
Remarks

(1)
If X=[0,1], and Y=\{y\} and {f}_{1},{f}_{2}\in C(X\times Y) 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.

(2)
If one takes X=[0,A], Y=[0,B] (0<A,B<1), and {f}_{1} and {f}_{2} are defined by
{f}_{1}(u,v)=\frac{u}{1u}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{f}_{2}(u,v)=\frac{v}{1v},
then the above theorem becomes Theorem 1 of [2].

(3)
For linear positive operators of two variables, Theorem 2.4 generalizes the result of Volkov in [5].

(4)
We believe that the above theorem can be generalized to nfold copies by taking Z={X}_{1}\times {X}_{2}\times \cdots \times {X}_{n} instead of Z=X\times Y, where {X}_{1},{X}_{2},\dots ,{X}_{n} are compact Hausdorff spaces.

(5)
The above theorem is also true if one replaces C(X) by {C}_{b}(X), the space of bounded continuous functions, in the case of an arbitrary topological space X.
References
Korovkin PP: Linear Operators and Approximation Theory. Hindustan Publish Co., Delhi; 1960.
Taşdelen F, Erençin A: The generalization of bivariate MKZ operators by multiple generating functions. J. Math. Anal. Appl. 2007,331(1):727735. 10.1016/j.jmaa.2006.09.024
Altomare F, Campiti M: KorovkinType Approximation Theory and Its Applications. de Gruyter, Berlin; 1994.
Lorentz GG: Approximation of Functions. 2nd edition. Chelsea, New York; 1986.
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: 1719. (in Russian)
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Uygun, N. On the Korovkin approximation theorem and Volkovtype theorems. J Inequal Appl 2014, 89 (2014). https://doi.org/10.1186/1029242X201489
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DOI: https://doi.org/10.1186/1029242X201489
Keywords
 positive linear operator
 Korovkin theorem
 Volkovtype theorem
 modules of continuity
 compact Hausdorff spaces