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).
Keywordspositive 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.
2 Main result
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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