Invariant mean and a Korovkin-type approximation theorem
© Al-Mezel; licensee Springer. 2013
Received: 29 May 2012
Accepted: 24 February 2013
Published: 15 March 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 .
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 .
a convergent sequence is also σ-convergent;
a σ-convergent sequence implies convergence.
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.
whenever for all .
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 .
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.
that is, (3.2) holds. □
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