- Open Access
On the Korovkin approximation theorem and Volkov-type theorems
© Uygun; licensee Springer. 2014
- Received: 3 September 2013
- Accepted: 5 February 2014
- Published: 20 February 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).
- positive linear operator
- Korovkin theorem
- Volkov-type theorem
- modules of continuity
- compact Hausdorff spaces
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 .
is denoted by .
We note that the above definition is motivated from [, p.729] and generalizes the definition which is given there.
The main result of this paper will be obtained via the following lemma.
Now applying A, which is linear, to completes the proof. □
This completes the proof. □
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.
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 .
- (1)If , and and are defined by
- (2)If one takes , (), and and are defined by
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.
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