Korovkin Second Theorem via 𝐵-Statistical 𝐴-Summability

Abstract

Korovkin type approximation theorems are useful tools to check whether a given sequence (𝐿_𝑛)_𝑛≥1 of positive linear operators on 𝐶[0,1] of all continuous functions on the real interval [0,1] is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, 𝑥, and 𝑥² in the space 𝐶[0,1] as well as for the functions 1, cos, and sin in the space of all continuous 2𝜋-periodic functions on the real line. In this paper, we use the notion of 𝐵-statistical 𝐴-summability to prove the Korovkin second approximation theorem. We also study the rate of 𝐵-statistical 𝐴-summability of a sequence of positive linear operators defined from 𝐶_2𝜋(ℝ) into 𝐶_2𝜋(ℝ).

1. Introduction and Preliminaries

Let ℕ be the set of all natural numbers, 𝐾⊆ℕ, and 𝐾_𝑛={𝑘≤𝑛∶𝑘∈𝐾}. Then the natural density of 𝐾 is defined by 𝛿(𝐾)=lim_𝑛1𝑛𝐾_𝑛=lim_𝑛𝐶₁𝜒_𝐾𝑛, (1) if the limit exists, where the vertical bars indicate the number of elements in the enclosed set, 𝐶₁=(𝐶,1) is the Cesàro matrix of order 1, and 𝜒_𝐾 denotes the characteristic sequence of 𝐾 given by 𝜒_𝐾𝑖=0,if𝑖∉𝐾,1,if𝑖∈𝐾. (2)

A sequence 𝑥=(𝑥_𝑘) is said to be statistically convergent to 𝐿 if for every 𝜀>0, the set 𝐾_𝜀∶={𝑘∈ℕ∶|𝑥_𝑘−𝐿|≥𝜀} has natural density zero (cf. Fast [1]); that is, for each 𝜀>0, lim_𝑛1𝑛𝑘≤𝑛∶𝑥_𝑘−𝐿≥𝜀=0. (3) In this case, we write 𝐿=st−lim𝑥. 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 𝐴=(𝑎_𝑛𝑘)^∞_𝑛,𝑘=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 𝐴 to be regular are(i)||𝐴||=sup_𝑛∑_𝑘|𝑎_𝑛𝑘|<∞; (ii)lim_𝑛𝑎_𝑛𝑘=0, for each 𝑘;(iii)lim_𝑛∑_𝑘𝑎_𝑛𝑘=1.

Freedmann and Sember [4] generalized the natural density by replacing 𝐶₁ with an arbitrary nonnegative regular matrix 𝐴. A subset 𝐾 of ℕ has 𝐴-density if 𝛿_𝐴(𝐾)=lim_{𝑛𝑘∈𝐾}𝑎_𝑛𝑘 (4) exists. Connor [5] and Kolk [6] extended the idea of statistical convergence to 𝐴-statistical convergence by using the notion of 𝐴-density.

A sequence 𝑥 is said to be 𝐴-statistically convergent to 𝐿 if 𝛿_𝐴(𝐾_𝜖)=0 for every 𝜖>0. In this case we write st_𝐴−lim𝑥_𝑘=𝐿. By the symbol st_𝐴 we denote the set of all 𝐴-statistically convergent sequences.

In [7], Edely and Mursaleen generalized these statistical summability methods by defining the statistical 𝐴-summability and studied its relationship with 𝐴-statistical convergence.

Let 𝐴=(𝑎_𝑖𝑗) be a nonnegative regular matrix. A sequence 𝑥 is said to be statistically 𝐴-summable to 𝐿 if for every 𝜖>0, 𝛿({𝑖≤𝑛∶|𝑦_𝑖−𝐿|≥𝜖})=0; that is, lim_𝑛1𝑛𝑖≤𝑛∶𝑦_𝑖−𝐿≥𝜖=0, (5) where 𝑦_𝑖=𝐴_𝑖(𝑥). Thus 𝑥 is statistically 𝐴-summable to 𝐿 if and only if 𝐴𝑥 is statistically convergent to 𝐿. In this case we write 𝐿=(𝐴)_st−lim𝑥=st−lim𝐴𝑥. By (𝐴)_st we denote the set of all statistically 𝐴-summable sequences. A more general case of statistically 𝐴-summability is discussed in [8].

Quite recently, Edely [9] defined the concept of 𝐵-statistical 𝐴-summability for nonnegative regular matrices 𝐴 and 𝐵 which generalizes all the variants and generalizations of statistical convergence, for example, lacunary statistical convergence [10], 𝜆-statistical convergence [11], 𝐴-statistical convergence [6], statistical 𝐴-summability [7], statistical (𝐶,1)-summability [12], statistical (𝐻,1)-summability [13], statistical (𝑁,𝑝)-summability [14], and so forth.

Let 𝐴=(𝑎_𝑖𝑗) and 𝐵=(𝑏_𝑛𝑘) be two nonnegative regular matrices. A sequence 𝑥=(𝑥_𝑘) of real numbers is said to be 𝐵-statistically𝐴-summable to 𝐿 if for every 𝜖>0, the set 𝐾(𝜖)={𝑖∶|𝑦_𝑖−𝐿|≥𝜖} has 𝐵-density zero, thus 𝛿_𝐵(𝐾(𝜖))=lim_{𝑛(𝜖)𝑘∈𝐾}𝑏_𝑛𝑘=lim_𝑛𝐵𝜒_𝐾(𝜖)=lim_𝑛𝑘𝑏_𝑛𝑘𝜒_𝐾(𝜖)(𝑘)=0, (6) where 𝑦_𝑖=𝐴_𝑖(𝑥)=∑_𝑗𝑎_𝑖𝑗𝑥_𝑗. In this case we denote by 𝐿=(𝐴)_st𝐵−lim𝑥=st_𝐵−lim𝐴𝑥. The set of all 𝐵-statistically 𝐴-summable sequences will be denoted by (𝐴)_st𝐵.

Remark 1. (1) If 𝐴=𝐼 (unit matrix), then (𝐴)_st𝐵 is reduced to the set of 𝐵-statistically convergent sequences which can be further reduced to lacunary statistical convergence and 𝜆-statistical convergence for particular choice of the matrix 𝐵.
(2) If 𝐵=(𝐶,1) matrix, then (𝐴)_st𝐵 is reduced to the set of statistically 𝐴-summable sequences.
(3) If 𝐴=𝐵=(𝐶,1) matrix, then (𝐴)_st𝐵 is reduced to the set of statistically (𝐶,1)-summable sequences.
(4) If 𝐵=(𝐶,1) matrix and 𝐴=(𝑎_𝑗𝑘) are defined by 𝑎_𝑗𝑘=𝑝_𝑘𝑃_𝑗if00≤𝑘≤𝑗,otherwise, (7) then (𝐴)_st𝐵 is reduced to the set of statistically (𝑁,𝑝)-summable sequences, where 𝑝=(𝑝_𝑘) is a sequence of nonnegative numbers, such that 𝑝₀>0 and 𝑃_𝑗=^𝑗_𝑘=0𝑝_𝑘⟶∞𝑗⟶∞. (8)
(5) If 𝐵=(𝐶,1) matrix and 𝐴=(𝑎_𝑗𝑘) are defined by 𝑎_𝑗𝑘=1𝑘𝑙_𝑗if00≤𝑘≤𝑗,otherwise, (9) where 𝑙_𝑗=∑^𝑗_𝑘=0(1/(𝑘+1)), then (𝐴)_st𝐵 is reduced to the set of statistically (𝐻,1)-summable sequences.
(6) If a sequence is convergent, then it is 𝐵-statistically 𝐴-summable, since 𝐴𝑥 converges and has 𝐵-density zero, but not conversely.
(7) The spaces st,st_𝐵,(𝐴)_st, and (𝐴)_st𝐵 are not comparable, even if 𝐴=𝐵(≠(𝐶,1)).
(8) If a sequence is 𝐴-summable, then it is 𝐵-statistically 𝐴-summable.
(9) If a sequence is bounded and 𝐴-statistically convergent, then it is 𝐴-summable and hence statistically 𝐴-summable ([7], see Theorem 2.1) and 𝐵-statistically 𝐴-summable but not conversely.

Example 2. (1) Let us define 𝐴=(𝑎_𝑖𝑗), 𝐵=(𝑏_𝑛𝑘), and 𝑥=(𝑥_𝑘) by 𝑎_𝑖𝑗=1;if𝑗=𝑖²,0;otherwise,𝑏_𝑛𝑘=110000⋅⋅⋅⋅⋅⋅⋅⋅2012100000⋅⋅⋅⋅⋅30130131000⋅⋅⋅⋅⋅401401401410000⋅⋅5015015015015,𝑥00⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅_𝑘=1;if𝑘isodd,0;if𝑘iseven. (10) Then ^∞_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=1;if𝑖isodd,0;if𝑖iseven. (11)
Here 𝑥∉st, 𝑥∉(𝐴)_st, 𝑥∉st_𝐴, and 𝑥∉(𝐴)_st𝐴, but 𝑥 is 𝐵-statistically 𝐴-summable to 1, since 𝛿_𝐵{𝑖∶|𝑦_𝑖−1|≥𝜖}=0. On the other hand we can see that 𝑥 is 𝐵-summable and hence 𝑥 is 𝐵-statistically 𝐵-summable, 𝐴-statistically 𝐵-summable, 𝐵-statistically convergent, and statistically 𝐵-summable.

Let 𝐹(ℝ) denote the linear space of all real-valued functions defined on ℝ. Let 𝐶(ℝ) be the space of all functions 𝑓 continuous on ℝ. We know that 𝐶(ℝ) is a Banach space with norm 𝑓_∞∶=sup_𝑥∈ℝ𝑓(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∈ℝ𝑓(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 𝑒₀(𝑥)=1,𝑒₁(𝑥)=𝑥, and 𝑒₂(𝑥)=𝑥².

Theorem II. Let (𝑇_𝑛) be a sequence of positive linear operators from 𝐶_2𝜋(ℝ) into 𝐹(ℝ). Then lim_𝑛‖𝑇_𝑛(𝑓,𝑥)−𝑓(𝑥)‖_∞=0,  for all 𝑓∈𝐶_2𝜋(ℝ) if and only if   lim_𝑛‖𝑇_𝑛(𝑓_𝑖,𝑥)−𝑓_𝑖(𝑥)‖_∞=0,  for 𝑖=0,1,2, where 𝑓₀(𝑥)=1,𝑓₁(𝑥)=cos𝑥, and 𝑓₂(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)st_𝐴−lim_𝑘→∞𝑇_𝑘𝑓;𝑥−𝑓(𝑥)_2𝜋=0 (14) if and only if st_𝐴−lim_𝑘→∞𝑇_𝑘(1;𝑥)−1_2𝜋=0,st_𝐴−lim_𝑘→∞𝑇_𝑘(cos𝑡;𝑥)−cos𝑥_2𝜋=0,st_𝐴−lim_𝑘→∞𝑇_𝑘(sin𝑡;𝑥)−sin𝑥_2𝜋=0. (15)

Recently, Karakuş and Demirci [18] proved Theorem II for statistical 𝐴-summability.

Theorem B. 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)st−lim_{𝑛→∞𝑘}𝑎_𝑛𝑘𝑇_𝑘𝑓;𝑥−𝑓(𝑥)_2𝜋=0 (16) if and only if st−lim_{𝑛→∞𝑘}𝑎_𝑛𝑘𝑇_𝑘(1;𝑥)−1_2𝜋=0,st−lim_{𝑛→∞𝑘}𝑎_𝑛𝑘𝑇_𝑘(cos𝑡;𝑥)−cos𝑥_2𝜋=0,st−lim_{𝑛→∞𝑘}𝑎_𝑛𝑘𝑇_𝑘(sin𝑡;𝑥)−sin𝑥_2𝜋=0. (17)

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, [20–28]. Further Korovkin type approximation theorems for functions of two variables are proved in [29–32]. In [29, 33] authors have used the concept of almost convergence. In this paper, we prove Korovkin second theorem by applying the notion of 𝐵-statistical 𝐴-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 𝐵-statistical 𝐴-summability of a sequence of positive linear operators defined from 𝐶_2𝜋(ℝ) into 𝐶_2𝜋(ℝ).

2. Main Result

Now, we prove Theorem II for 𝐵-statistically 𝐴-summability.

Theorem 3. Let 𝐴=(𝑎_𝑛𝑘) and 𝐵=(𝑏_𝑛𝑘) be nonnegative regular matrices, 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)st_𝐵−lim_{𝑛→∞𝑘}𝑎_𝑛𝑘𝑇_𝑘𝑓;𝑥−𝑓(𝑥)_2𝜋=0 (18) if and only if st_𝐵−lim_{𝑛→∞𝑘}𝑎_𝑛𝑘𝑇_𝑘(1;𝑥)−1_2𝜋=0,st_𝐵−lim_{𝑛→∞𝑘}𝑎_𝑛𝑘𝑇_𝑘(cos𝑡;𝑥)−cos𝑥_2𝜋=0,st_𝐵−lim_{𝑛→∞𝑘}𝑎_𝑛𝑘𝑇_𝑘(sin𝑡;𝑥)−sin𝑥_2𝜋=0. (19)

Proof. Since each of 1, cos𝑥, and sin𝑥 belongs to 𝐶_2𝜋(ℝ), conditions (19) follow immediately from (18). Let the conditions (19) hold and 𝑓∈𝐶_2𝜋(ℝ). Let 𝐼  be a closed subinterval of length 2𝜋 of   ℝ. Fix 𝑥∈𝐼. By the continuity of 𝑓 at 𝑥, it follows that for given 𝜀>0 there is a number 𝛿>0, such that for all 𝑡𝑓(𝑡)−𝑓(𝑥)<𝜀, (20) whenever |𝑡−𝑥|<𝛿. Since 𝑓 is bounded, it follows that 𝑓(𝑡)−𝑓(𝑥)≤2𝑓_2𝜋, (21) for all 𝑡∈ℝ. For all 𝑡∈(𝑥−𝛿,2𝜋+𝑥−𝛿], it is well known that 𝑓(𝑡)(𝑥2𝑓−𝑓)<𝜀+^2𝜋sin²()𝜓(𝑡),𝛿/2 (22) where 𝜓(𝑡)=sin²((𝑡−𝑥)/2). Since the function 𝑓∈𝐶_2𝜋(ℝ) is 2𝜋-periodic, the inequality (22) holds for 𝑡∈ℝ.
Now, operating 𝑇_𝑘(1;𝑥) to this inequality, we obtain 𝑇_𝑘(𝑥≤𝑓(𝑥𝑇𝑓;𝑥−𝑓)𝜀+)_𝑘()+𝑓1;𝑥−1+𝜀_2𝜋sin²()𝑇𝛿/2^𝑘()+||×𝑇1;𝑥−1cos𝑥_𝑘()+|𝑇cos𝑡;𝑥−cos𝑥sin𝑥|_𝑘()𝑓(𝑥+𝑓sin𝑡;𝑥−sin𝑥≤𝜀+𝜀+)^2𝜋sin²()×𝑇𝛿/2_𝑘(1;𝑥)−1+𝑇_𝑘()+𝑇cos𝑡;𝑥−cos𝑥_𝑘(sin𝑡;𝑥)−sin𝑥. (23) Now, taking sup_𝑥∈𝐼, we get 𝑇_𝑘𝑓;𝑥−𝑓(𝑥)_∞≤𝜀+𝐾𝑇_𝑘(1;𝑥)−1_2𝜋+𝑇_𝑘(cos𝑡;𝑥)−cos𝑥_2𝜋+𝑇_𝑘(sin𝑡;𝑥)−sin𝑥_2𝜋, (24) where 𝐾∶=𝜀+‖𝑓‖_2𝜋+(‖𝑓‖_2𝜋/sin²(𝛿/2)). Now replace 𝑇_𝑘(⋅,𝑥) by ∑_𝑘𝑎_𝑚𝑘𝑇_𝑘(⋅,𝑥) and then by 𝐵_𝑚(⋅,𝑥) in (24) on both sides. For a given 𝑟>0 choose 𝜀^′>0, such that 𝜀^′<𝑟. Define the following sets 𝐷=𝑚≤𝑛∶𝐵_𝑚𝑓,𝑥−𝑓(𝑥)_2𝜋,𝐷≥𝑟₁=𝑚≤𝑛∶𝐵_𝑚𝑓₁,𝑥−𝑓_02𝜋≥𝑟−𝜀^′,𝐷3𝐾₂=𝑚≤𝑛∶𝐵_𝑚𝑓₂,𝑥−𝑓_12𝜋≥𝑟−𝜀^′,𝐷3𝐾₃=𝑚≤𝑛∶𝐵_𝑚𝑓₃,𝑥−𝑓_22𝜋≥𝑟−𝜀^′.3𝐾 (25) Then 𝐷⊂𝐷₁∪𝐷₂∪𝐷₃, and so 𝛿_𝐵(𝐷)≤𝛿_𝐵(𝐷₁)+𝛿_𝐵(𝐷₂)+𝛿_𝐵(𝐷₃). Therefore, using conditions (19) we get (18).
This completes the proof of the theorem.

3. Rate of 𝐵-Statistical 𝐴-Summability

In this section, we study the rate of 𝐵-statistical 𝐴-summability of a sequence of positive linear operators defined from 𝐶_2𝜋(ℝ) into 𝐶_2𝜋(ℝ).

Definition 4. Let 𝐴=(𝑎_𝑖𝑗) and 𝐵=(𝑏_𝑛𝑘) be two nonnegative regular matrices. Let (𝛼_𝑛) be a positive nonincreasing sequence. We say that the sequence 𝑥=(𝑥_𝑘) is 𝐵-statistically 𝐴-summable to the number 𝐿 with the rate 𝑜(𝛼_𝑛) if for every 𝜀>0, lim_𝑛1𝛼_{𝑛(𝜖)𝑘∈𝐾}𝑏_𝑛𝑘=0, (26) where 𝐾(𝜖)={𝑖∶|𝑦_𝑖−𝐿|≥𝜖} and 𝑦_𝑖=𝐴_𝑖(𝑥)=∑_𝑗𝑎_𝑖𝑗𝑥_𝑗 as described above. In this case, we write 𝑥_𝑘−𝐿=(𝐴)_st𝐵−𝑜(𝛼_𝑛).

As usual we have the following auxiliary result whose proof is standard.

Lemma 5. Let (𝛼_𝑛) and (𝛽_𝑛) be two positive nonincreasing sequences. Let 𝑥=(𝑥_𝑘) and 𝑦=(𝑦_𝑘) be two sequences, such that 𝑥_𝑘−𝐿₁=(𝐴)_st𝐵−𝑜(𝛼_𝑛) and 𝑦_𝑘−𝐿₂=(𝐴)_st𝐵−𝑜(𝛽_𝑛). Then(i)𝑐(𝑥_𝑘−𝐿₁)=(𝐴)_st𝐵−𝑜(𝛼_𝑛), for any scalar 𝑐,(ii)(𝑥_𝑘−𝐿₁)±(𝑦_𝑘−𝐿₂)=(𝐴)_st𝐵−𝑜(𝛾_𝑛), (iii)(𝑥_𝑘−𝐿₁)(𝑦_𝑘−𝐿₂)=(𝐴)_st𝐵−𝑜(𝛼_𝑛𝛽_𝑛), where 𝛾_𝑛=max{𝛼_𝑛,𝑏_𝑛}.

Now, we recall the notion of modulus of continuity. The modulus of continuity of 𝑓∈𝐶_2𝜋(ℝ), denoted by 𝜔(𝑓,𝛿), is defined by 𝜔𝑓,𝛿=sup_{𝑥−𝑦<𝛿}𝑓(𝑥)−𝑓𝑦. (27)

It is well known that 𝑓(𝑥)−𝑓𝑦≤𝜔𝑓,𝛿𝑥−𝑦𝛿+1. (28)

Then prove the following result.

Theorem 6. Let (𝑇_𝑘) be a sequence of positive linear operators from 𝐶_2𝜋(ℝ) into 𝐶_2𝜋(ℝ). Suppose that(i)‖𝑇_𝑘(1;𝑥)−1‖_2𝜋=(𝐴)_st𝐵−𝑜(𝛼_𝑛), (ii)𝜔(𝑓,𝜆_𝑘)=(𝐴)_st𝐵−𝑜(𝛽_𝑛), where 𝜆_𝑘=𝑇_𝑘(𝜑_𝑥;𝑥) and  𝜑_𝑥(𝑦)=sin²((𝑦−𝑥)/2).Then for all 𝑓∈𝐶_2𝜋(ℝ), we have 𝑇_𝑘𝑓;𝑥−𝑓(𝑥)_2𝜋=(𝐴)_st𝐵−𝑜𝛾_𝑛, (29)where 𝛾_𝑛=max{𝛼_𝑛,𝛽_𝑛}.

Proof. Let 𝑓∈𝐶_2𝜋(ℝ) and 𝑥∈[−𝜋,𝜋]. Using (28), we have 𝑇_𝑘𝑓;𝑥−𝑓(𝑥)≤𝑇_𝑘𝑓𝑦(𝑥+𝑓(𝑥𝑇−𝑓);𝑥)_𝑘(1;𝑥)−1≤𝑇_𝑘𝑥−𝑦𝛿𝜔+𝑓(𝑥𝑇+1;𝑥𝑓,𝛿)_𝑘(1;𝑥)−1≤𝑇_𝑘𝜋1+²𝛿²sin²𝑦−𝑥2𝜔+𝑓(𝑥𝑇;𝑥𝑓,𝛿)_𝑘()≤𝑇1;𝑥−1_𝑘()+𝜋1;𝑥²𝛿²𝑇_𝑘𝜑_𝑥𝜔+𝑓(𝑥𝑇;𝑥𝑓,𝛿)_𝑘(1;𝑥)−1. (30) Put 𝛿=𝜆_𝑘=𝑇_𝑘(𝜑_𝑥;𝑥). Hence we get 𝑇_𝑘𝑓;𝑥−𝑓(𝑥)_2𝜋≤𝑓_2𝜋𝑇_𝑘(1;𝑥)−1_2𝜋+1+𝜋²𝜔𝑓,𝜆_𝑘+𝜔𝑓,𝜆_𝑘𝑇_𝑘(1;𝑥)−1_2𝜋≤𝐾𝑇_𝑘(1;𝑥)−1_2𝜋+𝜔𝑓,𝜆_𝑘+𝜔𝑓,𝜆_𝑘𝑇_𝑘(1;𝑥)−1_2𝜋, (31) where 𝐾=max{‖𝑓‖_2𝜋,1+𝜋²}. Hence 𝑇_𝑘𝑓;𝑥−𝑓(𝑥)_2𝜋≤𝐾𝑇_𝑘(1;𝑥)−1_2𝜋+𝜔𝑓,𝜆_𝑘+𝜔𝑓,𝜆_𝑘×𝑇_𝑘(1;𝑥)𝑝_𝑘−1_2𝜋. (32) Now, using Definition 4 and Conditions (i) and (ii), we get the desired result.
This completes the proof of the theorem.

4. 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∈ℕ, denote by 𝑆_𝑛(𝑓) the 𝑛th partial sum of the Fourier series of 𝑓; that is, 𝑆_𝑛𝑓(𝑥)=12𝑎₀𝑓+^𝑛_𝑘=1𝑎_𝑘𝑓cos𝑘𝑥+𝑏_𝑘𝑓sin𝑘𝑥. (33)

For any 𝑛∈ℕ, write 𝐹_𝑛𝑓1∶=𝑛+1^𝑛_𝑘=0𝑆_𝑘𝑓. (34)

A standard calculation gives that for every 𝑡∈ℝ𝐹_𝑛1𝑓;𝑥∶=2𝜋^𝜋_−𝜋𝑓(𝑡)1𝑛+1^𝑛_𝑘=0sin((2𝑘+1)(𝑥−𝑡)/2)))=1sin((𝑥−𝑡/2𝑑𝑡2𝜋^𝜋_−𝜋𝑓(𝑡)1𝑛+1^𝑛_𝑘=0sin²((𝑛+1)(𝑥−𝑡)/2)sin²))=1((𝑥−𝑡/2𝑑𝑡2𝜋^𝜋_−𝜋𝑓(𝑡)𝜑_𝑛(𝑥−𝑡)𝑑𝑡, (35) where 𝜑_𝑛(𝑥)∶=sin²((𝑛+1)(𝑥−𝑡)/2)(𝑛+1)sin²((𝑥−𝑡)/2)if𝑥isnotamultipleof2𝜋,𝑛+1if𝑥isamultipleof2𝜋. (36) The sequence (𝜑_𝑛)_𝑛∈ℕ is a positive kernel which is called the Fejér kernel, and the corresponding operators 𝐹_𝑛,𝑛≥1, are called the Fejér convolution operators. We have 𝐹_𝑛()𝐹1;𝑥=1,_𝑛()=𝑛cos𝑡;𝑥𝐹𝑛+1cos𝑥,_𝑛()=𝑛sin𝑡;𝑥𝑛+1sin𝑥. (37)

Note that the Theorems II, A, and B hold for the sequence (𝐹_𝑛). In fact, we have for every 𝑓∈𝐶_2𝜋(ℝ), lim_𝑛→∞𝐹_𝑛𝑓=𝑓. (38)

Let 𝐴=(𝑎_𝑖𝑗),𝐵=(𝑏_𝑛𝑘), and 𝑥=(𝑥_𝑘) be defined as in Example 2. Let 𝐿_𝑛∶𝐶_2𝜋(ℝ)→𝐶_2𝜋(ℝ) be defined by 𝐿_𝑛𝑓;𝑥=𝑥_𝑛𝐹_𝑛𝑓;𝑥. (39) Then 𝑥 is not statistically convergent, not 𝐴-statistically convergent, and not statistically 𝐴-summable, but it is 𝐵-statistically 𝐴−summable to 1. Since 𝑥 is 𝐵-statistically 𝐴-summable to 1, it is easy to see that the operator 𝐿_𝑛 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 𝑥 (and so 𝐿_𝑛) is not statistically convergent, not 𝐴-statistically convergent, and not statistically 𝐴-summable.

Hence our Theorem 3 is stronger than all the above three theorems.

Acknowledgments

This joint work was done when the first author visited University Putra Malaysia as a visiting scientist during August 27–September 25, 2012. The author is very grateful to the administration of UPM for providing him local hospitalities.