Korovkin Second Theorem via B-Statistical A-Summability

and Applied Analysis 3 f continuous on R. We know that C(R) is a Banach space with norm 󵄩󵄩󵄩f 󵄩󵄩󵄩∞ := sup x∈R 󵄨󵄨󵄨f (x) 󵄨󵄨󵄨 , f ∈ C (R) . (12) We denote by C 2π (R) the space of all 2π-periodic functions f ∈ C(R) which is a Banach space with 󵄩󵄩󵄩f 󵄩󵄩󵄩2π = sup t∈R 󵄨󵄨󵄨f (t) 󵄨󵄨󵄨 . (13) The classical Korovkin first and second theorems statewhatfollows [15, 16]: Theorem I. Let (T n ) be a sequence of positive linear operators from C[0, 1] into F[0, 1]. Then lim n ‖T n (f, x) − f(x)‖ ∞ = 0, for all f ∈ C[0, 1] if and only if lim n ‖T n (f i , x) − e i (x)‖ ∞ = 0, for i = 0, 1, 2, where e 0 (x) = 1, e 1 (x) = x, and e 2 (x) = x. Theorem II. Let (T n ) be a sequence of positive linear operators fromC 2π (R) into F(R). Then lim n ‖T n (f, x)−f(x)‖ ∞ = 0, for allf ∈ C 2π (R) if and only if lim n ‖T n (f i , x)−f i (x)‖ ∞ = 0, for i = 0, 1, 2, where f 0 (x) = 1, f 1 (x) = cosx, and f 2 (x) = sinx. We write L n (f; x) for L n (f(s); x), and we say that L is a positive operator if L(f; x) ≥ 0 for all f(x) ≥ 0. The following result was studied by Duman [17] which is A-statistical analogue of Theorem II. Theorem A. Let A = (a nk ) be a nonnegative regular matrix, and let (T k ) be a sequence of positive linear operators from C 2π (R) into C 2π (R). Then for all f ∈ C 2π (R)


Introduction and Preliminaries
Let N be the set of all natural numbers,  ⊆ N, and   = { ≤  :  ∈ }.Then the natural density of  is defined by if the limit exists, where the vertical bars indicate the number of elements in the enclosed set,  1 = (, 1) is the Cesàro matrix of order 1, and   denotes the characteristic sequence of  given by A sequence  = (  ) is said to be statistically convergent to  if for every  > 0, the set   := { ∈ N : |  − | ≥ } has natural density zero (cf.Fast [1]); that is, for each  > 0, In this case, we write  = st − lim .By the symbol st we denote the set of all statistically convergent sequences.
Statistical convergence of double sequences is studied in [2,3].
A matrix  = (  ) ∞ ,=0 is called regular if it transforms a convergent sequence into a convergent sequence leaving the limit invariant.The well-known necessary and sufficient conditions (Silverman-Toeplitz) for  to be regular are Freedmann and Sember [4] generalized the natural density by replacing  1 with an arbitrary nonnegative regular matrix .A subset  of N has -density if exists.Connor [5] and Kolk [6] extended the idea of statistical convergence to -statistical convergence by using the notion of -density.
A sequence  is said to be -statistically convergent to  if   (  ) = 0 for every  > 0. In this case we write st  −lim   = .By the symbol st  we denote the set of all -statistically convergent sequences.
In [7], Edely and Mursaleen generalized these statistical summability methods by defining the statistical summability and studied its relationship with -statistical convergence.
Let =(  ) be a nonnegative regular matrix.A sequence  is said to be statistically -summable to  if for every  > 0, ({ ≤  : where   =   ().Thus  is statistically -summable to  if and only if  is statistically convergent to .In this case we write  = () st − lim  = st − lim .By () st we denote the set of all statistically -summable sequences.A more general case of statistically -summability is discussed in [8].
Remark 1. (1) If  =  (unit matrix), then () st  is reduced to the set of -statistically convergent sequences which can be further reduced to lacunary statistical convergence and statistical convergence for particular choice of the matrix .
Let (R) denote the linear space of all real-valued functions defined on R. Let (R) be the space of all functions Abstract and Applied Analysis 3  continuous on R. We know that (R) is a Banach space with norm We denote by  2 (R) the space of all 2-periodic functions  ∈ (R) which is a Banach space with The classical Korovkin first and second theorems statewhatfollows [15,16]: Theorem II.Let (  ) be a sequence of positive linear operators from We write   (; ) for   ((); ), and we say that  is a positive operator if (; ) ≥ 0 for all () ≥ 0.
The following result was studied by Duman [17] which is -statistical analogue of Theorem II.
Theorem A. Let  = (  ) be a nonnegative regular matrix, and let (  ) be a sequence of positive linear operators from  2 (R) into  2 (R).Then for all  ∈  2 (R) if and only if Recently, Karakus ¸and Demirci [18] proved Theorem II for statistical -summability.

Theorem
Several mathematicians have worked on extending or generalizing the Korovkin's theorems in many ways and to several settings, including function spaces, abstract Banach lattices, Banach algebras, and Banach spaces.This theory is very useful in real analysis, functional analysis, harmonic analysis, measure theory, probability theory, summability theory, and partial differential equations.But the foremost applications are concerned with constructive approximation theory which uses it as a valuable tool.Even today, the development of Korovkin-type approximation theory is far frombeingcomplete.Note that the first and the second theorems of Korovkin are actually equivalent to the algebraic and the trigonometric version, respectively, of the classical Weierstrass approximation theorem [19].Recently, such type of approximation theorems has been proved by many authors by using the concept of statistical convergence and its variants, for example, [20][21][22][23][24][25][26][27][28].Further Korovkin type approximation theorems for functions of two variables are proved in [29][30][31][32].In [29,33] authors have used the concept of almost convergence.In this paper, we prove Korovkin second theorem by applying the notion of -statistical summability.We give here an example to justify that our result is stronger than Theorems II, A, and B. We also study the rate of -statistical -summability of a sequence of positive linear operators defined from  2 (R) into  2 (R).

Rate of 𝐵-Statistical 𝐴-Summability
In this section, we study the rate of -statistical summability of a sequence of positive linear operators defined from  2 (R) into  2 (R).
As usual we have the following auxiliary result whose proof is standard.
Then prove the following result.
where  = max{‖‖ 2 , 1 + Now, using Definition 4 and Conditions (i) and (ii), we get the desired result.This completes the proof of the theorem.

Example and Concluding Remark
In the following we construct an example of a sequence of positive linear operators satisfying the conditions of Theorem 3 but does not satisfy the conditions of Theorems II, A, and B. For any  ∈ N, denote by   () the th partial sum of the Fourier series of ; that is, For any  ∈ N, write ) sin . (37) Note that the Theorems II, A, and B hold for the sequence (  ).In fact, we have for every  ∈  2 (R), Then  is not statistically convergent, not -statistically convergent, and not statistically -summable, but it is statistically −summable to 1. Since  is -statistically summable to 1, it is easy to see that the operator   satisfies the conditions (19), and hence Theorem 3 holds.But on the other hand, Theorems II, A, and B do not hold for our operator defined by (39), since  (and so   ) is not statistically convergent, not -statistically convergent, and not statistically -summable.
Hence our Theorem 3 is stronger than all the above three theorems.