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On the Korovkin approximation theorem and Volkov-type theorems
Journal of Inequalities and Applications volume 2014, Article number: 89 (2014)
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).
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 . We note that if X is a compact Hausdorff space, then 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 . Then a linear map is called positive if in whenever in E. Here means that in ℝ for all .
Constant-one function on a topological space X will be denoted by , that is, for all . If and are elements of , then the Euclidean distance between A and B, given by
is denoted by .
Definition 1.1 Let X and Y be compact Hausdorff spaces, Z be the product space of X and Y, and let and be given. The module of continuity of f with respect to h is a function defined by , and
whenever , with the following additional properties:
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 be given. We define as the set of all continuous functions such that for all , one has
When is mentioned, we always suppose that h satisfies the property for being a vector subspace of . We note that has been considered in  by taking , (),
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 and be defined by
so that is a subspace and . Let be a positive linear map. Let be given, and define by
Then, for all , one has
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 completes the proof. □
Lemma 2.2 Let X and Y be compact Hausdorff spaces and , , and h be defined as in Lemma 2.1. Let be given. For each , there exists such that
Proof Let be given. Since is continuous, there exists such that for all . This implies, since
where and are defined as in Lemma 2.1. If , then
Hence, for all , we have
This completes the proof. □
Lemma 2.3 Suppose that the hypotheses of Lemma 2.2 are satisfied. Let and be given. Then there exists such that
Proof Set . From Lemma 2.2, there exists such that for each we have
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 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 , and be defined by
so that is a subspace and . Let be a sequence of positive operators from into satisfying:
Then, for all , we have
Proof Let and be given. By Lemma 2.3, there exists (depending only on and ) such that for each n,
Since 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  it is not necessary to take a double sequence of positive operators: as the above result reveals, one can take instead of .
If , and and are defined by
then Theorem 2.4 becomes the classical Korovkin theorem.
If one takes , (), and and are defined by
then the above theorem becomes Theorem 1 of .
For linear positive operators of two variables, Theorem 2.4 generalizes the result of Volkov in .
We believe that the above theorem can be generalized to n-fold copies by taking instead of , where are compact Hausdorff spaces.
The above theorem is also true if one replaces by , the space of bounded continuous functions, in the case of an arbitrary topological space X.
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):727-735. 10.1016/j.jmaa.2006.09.024
Altomare F, Campiti M: Korovkin-Type 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: 17-19. (in Russian)
The author declares that they have no competing interests.
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Cite this article
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
- positive linear operator
- Korovkin theorem
- Volkov-type theorem
- modules of continuity
- compact Hausdorff spaces