Generalization of Statistical Korovkin Theorems

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.


Introduction
The classical Korovkin theory enables us to approximate a function by means of positive linear operators (see, e.g., [1][2][3]).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) (, U) is a Hausdorff uniform space provided with the uniform structure (, U); (ii) U is the filter of the surroundings containing the diagonal Δ = {(, ) :  ∈ } in  × ; (iii) F is a vector space of real-valued functions defined on  including the constant-valued function  0 () = 1; (iv)  is a compact subspace of ; (v)   is a positive linear operator of F into R  for each  ∈ N; (vi)  := [  ] is a nonnegative regular summable matrix.
Assume further that there exists a certain real-valued function (, ) satisfying the following conditions: (i) (, ) ≥ 0 on  ×  and (, ) = 0 for each  ∈ ; (ii)   ∈ F for each  ∈ , where   is the function on  defined by   () := (, ); (iii) for each  ∈ ,   () is continuous with respect to  at each point in ; (iv) () = inf (, ) > 0 for each  ∈ U, where the infimum is taken over ( × ) − ; (v) there exist  1 ,  2 ∈ ,  1 ̸ =  2 , such that   1 () and   2 () 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 N, the -density of , denoted by   (), is defined to be   () = lim  → ∞ ∑ ∈   provided that the limit exists.Using this -density, we say that a sequence  = (  ) is -statistically convergent to  if and only if   (()) = 0 for every  > 0, where () := { ∈ N : |  −| ≥ } (see [5]).In the case of  = holds, then we get the conditions in (V).
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  ∈ F, the sequence {  ()} is uniformly convergent to  on .

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 2. The sequence {‖  ( 0 )‖} is -statistically bounded; that is; there exist a positive real number  and a subset  ⊂ N having -density 1 such that       ( 0 )     ≤  for every  ∈ . (3) Proof.For the points  1 ,  2 given in (V), we can take  0 ∈ U so that ( 1 ,  2 ) ∉  0 .Now, choose  ∈ U such that  =  −1 and  ∘  ⊂  0 .Then, we observe, for every  ∈ , that Hence, we get, for each  ∈  and for every  ∈ N, that which implies that Taking supremum over  ∈  and also letting we obtain, for every  ∈ N, that Now, for a given  > 0, define the following sets: Then, from the conditions in (V), we may write that Now setting we immediately get that Taking limit as  → ∞ in both sides of the last inequality and also using (V), we obtain that lim Furthermore, we may write that Since   () = 1, we get   (N − ) = 0. Thus, by ( 24) and (25), we obtain that lim Now, for a given  > 0, define the following sets: Then, it follows from (35) that which guarantees that, for any  ∈ N, Now letting  → ∞ and also using (V) and (31), we conclude that which is the desired result.

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  =  1 , 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  ∈ F, However, since the sequence (  ) given by ( 40) is nonconvergent, approximating a function  ∈ F 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 .
Finally, as in [4], Theorem 4 also contains the trigonometric version of Corollary 6 introduced in [10].