Abstract

We generalize and develop the Korovkin-type approximation theory by using an appropriate abstract space. We show that our approximation is more applicable than the classical one. At the end, we display some applications.

1. Introduction

The classical Korovkin theory enables us to approximate a function by means of positive linear operators (see, e.g., [13]). In recent years, this theory has been quite improved by some efficient tools in mathematics such as the concept of statistical convergence from summability theory, the fuzzy logic theory, the complex functions theory, the theory of -calculus, and the theory of fractional analysis. The main purpose of this paper is to generalize and develop this Korovkin theory by using an appropriate abstract space. Actually, the most important motivation of this study has its roots from the paper by Yoshinaga and Tamura [4]. In the present paper, we show that our new approximation is more general and also more applicable than that of [4].

Throughout the paper the following assumptions are imposed:(i) is a Hausdorff uniform space provided with the uniform structure ;(ii) is the filter of the surroundings containing the diagonal in ;(iii) is a vector space of real-valued functions defined on including the constant-valued function ;(iv) is a compact subspace of ;(v) is a positive linear operator of into for each ;(vi) is a nonnegative regular summable matrix.

Assume further that there exists a certain real-valued function satisfying the following conditions:(i) on and for each ;(ii) for each , where is the function on defined by ;(iii)for each ,   is continuous with respect to at each point in ;(iv) for each , where the infimum is taken over ;(v)there exist , , such that and are bounded functions of , and it holds that

where the symbol denotes the classical sup-norm on the compact set . Here, we use the concept of -statistical convergence, where is a nonnegative regular summable matrix. Recall that, for a given subset of , the -density of , denoted by , is defined to be provided that the limit exists. Using this -density, we say that a sequence is -statistically convergent to if and only if for every , where (see [5]). In the case of , the Cesàro matrix, it reduces to the concept of statistical convergence introduced by Fast [6]. Of course, if we take , the identity matrix, then we get the ordinary convergence.

We should note that if, for each , is a bounded function of for which

holds, then we get the conditions in .

Then, with the above terminology, Yoshinaga and Tamura [4] proved the following approximation result (in the case of ).

Theorem A (see [4]). Let be a bounded real-valued function on and continuous at each point in . Then, if , the sequence is uniformly convergent to on .

2. Statistical Approximation Theorem

In this section, we obtain the statistical analog of Theorem A in order to get a more applicable approximation theorem.

We first need the following three lemmas.

Lemma 1 (see [4]). Let be an open subset of containing . Then, it is possible to see that for some .

Lemma 2. The sequence is -statistically bounded; that is; there exist a positive real number and a subset having -density such that

Proof. For the points given in , we can take so that . Now, choose such that and . Then, we observe, for every , that
Hence, we get, for each and for every , that
which implies that
Taking supremum over and also letting
we obtain, for every , that
Now, for a given , define the following sets:
Then, from the conditions in , we may write that
Now setting
we immediately get that
Furthermore, it follows from (8) that, for every , that is, and ,
which completes the proof.

Lemma 3. Let be a real-valued and bounded function on , and let be continuous at each diagonal point . For each , define the function on by . Assume further that and for each . Then, one has

Proof. Since is continuous at any diagonal point , for every , there exists an open neighborhood of in such that
for every . Now, if we define the set by
then we easily see that is an open subset of containing the diagonal . Also, it follows from Lemma 1 that for some . Now, setting
we get, for every , that
which in turn implies that
Thus we conclude that, for every , the inequality
holds. By Lemma 2, there exists a positive real number and a subset of having -density such that
holds for every . Now, for a given , choose an such that . Then, considering the following subsets of :
and also using (21), we have , which gives, for every , that
Taking limit as in both sides of the last inequality and also using , we obtain that
Furthermore, we may write that
Since , we get . Thus, by (24) and (25), we obtain that
which means
Therefore, the proof is completed.

Now we are ready to give our main approximation result in statistical sense.

Theorem 4. Let be a bounded real-valued function on and continuous at each point in . Then, if , one has

Proof. As in the proof of Lemma 2, we take ,   such that ,  , and , where are given in . Let
for . Then, we see that , and so for some due to the boundedness of the functions and on . Also, observe that and for each . Since and are continuous at any point and also , we easily check that the function is continuous at each point . Since is bounded on , we may write that . Then, it is not hard to see that
for every . From Lemma 3, one can get that
On the other hand, by (29), we have
which yields that
Hence we get
Taking supremum over , we immediately obtain that
Now, for a given , define the following sets:
Then, it follows from (35) that
which guarantees that, for any ,
Now letting and also using and (31), we conclude that
which is the desired result.

3. Concluding Remarks

If we take , the identity matrix, in Theorem 4, then we easily get Theorem A. Hence, one can say that Theorem 4 covers Theorem A. However, if we take , the Cesàro matrix, and also define the sequence by

then we observe that

although it is nonconvergent in the usual sense. Now, assume that is a sequence of positive linear operators satisfying all conditions of Theorem A. Then, using and , we construct new operators as follows:

In this case, we verify that our operators satisfy all conditions of Theorem 4 due to property (41). Thus, we may write that, for every ,

However, since the sequence given by (40) is nonconvergent, approximating a function by the operators is impossible. This example clearly shows that Theorem 4 is a nontrivial generalization of Theorem A.

Now we give some significant applications of Theorem 4. As usual, by we denote the space of all real-valued continuous functions on .

Corollary 5 (see Theorem  3.5 of [7]). Let be a compact Hausdorff space, and let satisfy the condition that there exist such that defining for every , it holds that and if and only if . Assume that is a sequence of positive linear operators from into itself. Assume further that, for a given nonnegative regular summable matrix ,
Then, for every , one has .

Proof. Take , and . Then, since
we observe that
where . Now, for a given , consider the following sets:
Hence, inequality (46) implies that
which gives, for every , that
By (44), we obtain that
which gives
Thus, the last equality means that condition (2) is valid for the function . As a result, all hypotheses of Theorem 4 are satisfied.

If we take , the identity matrix, in Corollary 5, then we immediately get the classical result (see, e.g., [3, page 22]).

In algebraic case, we consider the following test functions: . Then we get the next result.

Corollary 6 (see Corollary  2 of [8]). Let be a sequence of positive linear operators from into itself. If, for a given nonnegative regular summable matrix ,
then, for every , one has .

Proof. If we take , , , and , then we observe that all conditions of Theorem 4 are satisfied.

Of course, if , the Cesáro matrix, in Corollary 6, then one obtains Theorem  1 of [9]. Furthermore, taking we get the classical theorem (see [2]).

Finally, as in [4], Theorem 4 also contains the trigonometric version of Corollary 6 introduced in [10].

Acknowledgments

The authors would like to thank the referee for carefully reading the paper. The second author also thanks TUBA for their support.