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Invariant mean and a Korovkin-type approximation theorem
Journal of Inequalities and Applications volume 2013, Article number: 103 (2013)
In this paper we apply this form of convergence to prove some Korovkin-type approximation theorem by using the test functions 1, , , which generalizes the results of Boyanov and Veselinov (Bull. Math. Soc. Sci. Math. Roum. 14(62):9-13, 1970).
MSC:41A65, 46A03, 47H10, 54H25.
1 Introduction and preliminaries
Let c and denote the spaces of all convergent and bounded sequences, respectively, and note that . In the theory of sequence spaces, an application of the well-known Hahn-Banach extension theorem gave rise to the concept of the Banach limit. That is, the lim functional defined on c can be extended to the whole of and this extended functional is known as the Banach limit. In 1948, Lorentz  used this notion of a generalized limit to define a new type of convergence, known as almost convergence. Later on, Raimi  gave a slight generalization of almost convergence and named it σ-convergence. Before proceeding further, we recall some notations and basic definitions used in this paper.
Let σ be a mapping of the set of positive integers into itself. A continuous linear functional φ defined on the space of all bounded sequences is called an invariant mean (or a σ-mean; cf. ) if it is non-negative, normal and .
A sequence is said to be σ-convergent to the number L if and only if all of its σ-means coincide with L, i.e., for all φ. A bounded sequence is σ-convergent (cf. ) to the number L if and only if uniformly in m, where
We denote the set of all σ-convergent sequences by and in this case we write and L is called the σ-limit of x. Note that a σ-mean extends the limit functional on c in the sense that for all if and only if σ has no finite orbits (cf. ) and .
If σ is a translation then the σ-mean is called a Banach limit and σ-convergence is reduced to the concept of almost convergence introduced by Lorentz .
If , then we get convergence, and in this case we write , where .
Remark 1.1 Note that
a convergent sequence is also σ-convergent;
a σ-convergent sequence implies convergence.
Example 1.1 Let . Define the sequence by
Then x is σ-convergent to 1/2 but not convergent.
Let be the space of all functions f continuous on . We know that is a Banach space with the norm , . Suppose that . We write for and we say that T is a positive operator if for all .
The classical Korovkin approximation theorem states the following : Let be a sequence of positive linear operators from into . Then , for all if and only if , for , where , and .
Quite recently, such type of approximation theorem has been studied in [8, 9] and  by using λ-statistical convergence, while in  lacunary statistical convergence has been used. Boyanov and Veselinov  have proved the Korovkin theorem on by using the test functions 1, , . In this paper, we generalize the result of Boyanov and Veselinov by using the notion of σ-convergence. Our results also generalize the results of Mohiuddine , in which the author has used almost convergence and the test functions 1, x, .
2 Korovkin-type approximation theorem
We prove the following σ-version of the classical Korovkin approximation theorem.
Theorem 2.1 Let be a sequence of positive linear operators from into . Then, for all ,
if and only if
Proof Since each 1, , belongs to , conditions (2.2)-(2.4) follow immediately from (2.1). Let . Then there exists a constant such that for . Therefore,
It is easy to prove that for a given there is a such that
whenever for all .
Using (2.5), (2.6), putting , we get
Now, we operate for all n to this inequality since is monotone and linear. We obtain
Note that x is fixed and so is a constant number. Therefore
Using (2.7) and (2.8), we have
Using (2.9), we obtain
Since ε is arbitrary, we can write
since for all . Now, taking
where . Now writing
Letting and using (2.2), (2.3), (2.4), we get
In the following example we construct a sequence of positive linear operators satisfying the conditions of Theorem 2.1 but not satisfying the conditions of the Korovkin theorem of Boyanov and Veselinov .
Example 2.1 Consider the sequence of classical Baskakov operators 
Let the sequence be defined by with , where is defined as above. Since
and the sequence satisfies the conditions (2.1), (2.2) and (2.3). Hence we have
On the other hand, we get since , and hence
We see that does not satisfy the classical Korovkin theorem since does not exist. Hence our Theorem 2.1 is stronger than that of Boyanov and Veselinov .
3 A consequence
Now we present a slight general result.
Theorem 3.1 Let be a sequence of positive linear operators on such that
then, for any function bounded on the real line, we have
Proof From Theorem 2.1, we have that if (3.1) holds, then
which is equivalent to
Hence, using the hypothesis, we get
that is, (3.2) holds. □
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The author declares that they have no competing interests.
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Al-Mezel, S.A. Invariant mean and a Korovkin-type approximation theorem. J Inequal Appl 2013, 103 (2013). https://doi.org/10.1186/1029-242X-2013-103
- invariant mean
- Korovkin-type approximation theorem