Weighted 𝐴-Statistical Convergence for Sequences of Positive Linear Operators

Abstract

We introduce the notion of weighted 𝐴-statistical convergence of a sequence, where 𝐴 represents the nonnegative regular matrix. We also prove the Korovkin approximation theorem by using the notion of weighted 𝐴-statistical convergence. Further, we give a rate of weighted 𝐴-statistical convergence and apply the classical Bernstein polynomial to construct an illustrative example in support of our result.

1. Background, Notations, and Preliminaries

We begin this paper by recalling the definition of natural (or asymptotic) density as follows. Suppose that 𝐸⊆ℕ∶={1,2,…} and 𝐸_𝑛={𝑘≤𝑛∶𝑘∈𝐸}. Then 𝛿(𝐸)=lim_𝑛1𝑛𝐸_𝑛 (1) is called the natural density of 𝐸 provided that the limit exists, where |⋅| represents the number of elements in the enclosed set.

The term “statistical convergence” was first presented by Fast [1] which is a generalization of the concept of ordinary convergence. Actually, a root of the notion of statistical convergence can be detected by Zygmund [2] (also, see [3]), where he used the term “almost convergence” which turned out to be equivalent to the concept of statistical convergence. The notion of Fast was further investigated by Schoenberg [4], Šalát [5], Fridy [6], and Conner [7].

The following notion is due to Fast [1]. A sequence 𝑥=(𝑥_𝑘) is said to be statistically convergent to 𝐿 if 𝛿(𝐾_𝜖)=0 for every 𝜖>0, where 𝐾_𝜖∶=𝑘∈ℕ∶𝑥_𝑘−𝐿≥𝜖. (2) Equivalently, lim_𝑛𝑛⁻¹𝑘≤𝑛∶𝑥_𝑘−𝐿≥𝜖=0. (3) In symbol, we will write 𝑆-lim𝑥=𝐿. We remark that every convergent sequence is statistically convergent but not conversely.

Let 𝑋 and 𝑌 be two sequence spaces and let 𝐴=(𝑎_𝑛,𝑘) be an infinite matrix. If for each 𝑥=(𝑥_𝑘) in 𝑋 the series 𝐴_𝑛𝑥=_𝑘𝑎_𝑛,𝑘𝑥_𝑘=^∞_𝑘=1𝑎_𝑛,𝑘𝑥_𝑘 (4) converges for each 𝑛∈ℕ and the sequence 𝐴𝑥=(𝐴_𝑛𝑥) belongs to 𝑌, then we say that matrix 𝐴 maps 𝑋 into 𝑌. By the symbol (𝑋,𝑌) we denote the set of all matrices which map 𝑋 into 𝑌.

A matrix 𝐴 (or a matrix map 𝐴) is called regular if 𝐴∈(𝑐,𝑐), where the symbol 𝑐 denotes the spaces of all convergent sequences and lim_𝑛𝐴_𝑛𝑥=lim_𝑘𝑥_𝑘 (5) for all 𝑥∈𝑐. The well-known Silverman-Toeplitz theorem (see [8]) asserts that 𝐴=(𝑎_𝑛,𝑘) is regular if and only if(i)lim_𝑛𝑎_𝑛,𝑘=0 for each 𝑘;(ii)lim_𝑛∑_𝑘𝑎_𝑛,𝑘=1;(iii)sup_𝑛∑_𝑘|𝑎_𝑛,𝑘|<∞.

Kolk [9] extended the definition of statistical convergence with the help of nonnegative regular matrix 𝐴=(𝑎_𝑛,𝑘) calling it 𝐴-statistical convergence. The definition of 𝐴-statistical convergence is given by Kolk as follows. For any nonnegative regular matrix 𝐴, we say that a sequence is 𝐴-statistically convergent to 𝐿 provided that for every 𝜖>0 we have lim_{𝑛𝑘∶|𝑥𝑛−𝐿|≥𝜖}𝑎_𝑛,𝑘=𝐿. (6)

In 2009, the concept of weighted statistical convergence was defined and studied by Karakaya and Chishti [10] and further modified by Mursaleen et al. [11] in 2012. In 2013, Belen and Mohiuddine [12] presented a generalization of this notion through de la Vallée-Poussin mean. Quite recently, Esi [13] defined and studied the notion statistical summability through de la Vallée-Poussin mean in probabilistic normed spaces.

Let 𝑝=(𝑝_𝑘) be a sequence of nonnegative numbers such that 𝑝₀>0 and 𝑃_𝑛=^𝑛_𝑘=0𝑝_𝑘⟶∞as𝑛⟶∞. (7) Let 𝑡_𝑛=1𝑃_𝑛^𝑛_𝑘=0𝑝_𝑘𝑥_𝑘,𝑛=0,1,2…. (8) We say that the sequence 𝑥=(𝑥_𝑘) is (𝑁,𝑝_𝑛)-summable to 𝐿 if lim_𝑛→∞𝑡_𝑛=𝐿.

The lower and upper weighted densities of 𝐸⊆ℕ are defined by 𝛿_𝑁(𝐸)=liminf_𝑛1𝑃_𝑛𝑘≤𝑃_𝑛∶𝑘∈𝐸,𝛿_𝑁(𝐸)=limsup_𝑛1𝑃_𝑛𝑘≤𝑃_𝑛∶𝑘∈𝐸, (9)(10) respectively. We say that 𝐸 has weighted density, denoted by 𝛿_𝑁(𝐸), if the limits of both of the above densities exist and are equal; that is, one writes 𝛿_𝑁(𝐸)=lim_𝑛1𝑃_𝑛𝑘≤𝑃_𝑛∶𝑘∈𝐸. (11)

The sequence 𝑥=(𝑥_𝑘) is said to be weighted statistically convergent (or 𝑆_𝑁-𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡) to 𝐿 if, for every 𝜖>0, the set {𝑘∈ℕ∶𝑝_𝑘|𝑥_𝑘−𝐿|≥𝜖} has weighted density zero; that is, lim_𝑛1𝑃_𝑛𝑘≤𝑃_𝑛∶𝑝_𝑘𝑥_𝑘−𝐿≥𝜖=0. (12) In this case we write 𝐿=𝑆_𝑁-lim𝑥.

Remark 1. If 𝑝_𝑘=1 for all 𝑘, then (𝑁,𝑝_𝑛)-summable is reduced to (𝐶,1)-summable (or Cesàro summable) and weighted statistical convergence is reduced to statistical convergence.

On the other hand, let us recall that 𝒞[𝑎,𝑏] is the space of all functions 𝑓 continuous on [𝑎,𝑏]. We know that 𝒞[𝑎,𝑏] is a Banach space with norm 𝑓_∞∶=sup_{𝑥∈[𝑎,𝑏]}𝑓(𝑥),𝑓∈𝒞[𝑎,𝑏]. (13) Suppose that 𝐿 is a linear operator from 𝒞[𝑎,𝑏] into 𝒞[𝑎,𝑏]. It is clear that if 𝑓≥0 implies 𝐿𝑓≥0, then the linear operator 𝐿 is positive on 𝒞[𝑎,𝑏]. We denote the value of 𝐿𝑓 at a point 𝑥∈[𝑎,𝑏] by 𝐿(𝑓;𝑥). The classical Korovkin approximation theorem states the following [14].

Theorem 2. Let (𝑇_𝑛) be a sequence of positive linear operators from 𝒞[𝑎,𝑏] into 𝒞[𝑎,𝑏]. Then, lim_𝑛𝑇_𝑛(𝑓,𝑥)−𝑓(𝑥)_∞=0, (14) for all 𝑓∈𝒞[𝑎,𝑏] if and only if lim_𝑛𝑇_𝑛(𝑓_𝑖,𝑥)−𝑓_𝑖(𝑥)_∞=0, (15) where 𝑓_𝑖(𝑥)=𝑥^𝑖 and 𝑖=0,1,2.

Many mathematicians extended the Korovkin-type approximation theorems by using various test functions in several setups, including Banach spaces, abstract Banach lattices, function spaces, and Banach algebras. Firstly, Gadjiev and Orhan [15] established classical Korovkin theorem through statistical convergence and display an interesting example in support of our result. Recently, Korovkin-type theorems have been obtained by Mohiuddine [16] for almost convergence. Korovkin-type theorems were also obtained in [17] for 𝜆-statistical convergence. The authors of [18] established these types of approximation theorem in weighted 𝐿_𝑝 spaces, where 1≤𝑝<∞, through 𝐴-summability which is stronger than ordinary convergence. For these types of approximation theorems and related concepts, one can be referred to [19–27] and references therein.

2. Korovkin-Type Theorems by Weighted 𝐴-Statistical Convergence

Kolk [9] introduced the notion of 𝐴-statistical convergence by considering nonnegative regular matrix 𝐴 instead of Cesáro matrix in the definition of statistical convergence due to Fast. Inspired from the work of Kolk, we introduce the notion of weighted 𝐴-statistical convergence of a sequence and then we establish some Korovkin-type theorems by using this notion.

Definition 3. Let 𝐴=(𝑎_𝑛,𝑘) be a nonnegative regular matrix. A sequence 𝑥=(𝑥_𝑘) of real or complex numbers is said to be weighted 𝐴-statistically convergent, denoted by 𝑆^𝑁_𝐴-convergent, to 𝐿 if for every 𝜖>0lim_{𝑛𝑘∈𝐸(𝑝,𝜖)}𝑎_𝑛,𝑘=0, (16) where 𝐸𝑝,𝜖=𝑘∈ℕ∶𝑝_𝑘𝑥_𝑘−𝐿≥𝜖. (17) In symbol, we will write 𝑆^𝑁_𝐴-lim𝑥=𝐿.

Remark 4. One has the following.(i)If we take 𝐴=𝐼, where 𝐼 denotes the identity matrix, then weighted 𝐴-statistical convergence of a sequence is reduced to ordinary convergence.(ii)If we take 𝐴=(𝐶,1), where (𝐶,1) denotes the Cesáro matrix of order one, then weighted 𝐴-statistical convergence of a sequence reduces to weighted statistical convergence.(iii)If we take 𝐴=(𝐶,1) and 𝑝_𝑘=1 for all 𝑘, then weighted 𝐴-statistical convergence of a sequence reduces to statistical convergence.

Note that convergent sequence implies weighted 𝐴-statistically convergent to the same value but the converse is not true in general. For example, take 𝐴=(𝐶,1) and 𝑝_𝑘=1 for all 𝑘 and define a sequence 𝑥=(𝑥_𝑘) by 𝑥_𝑘=1,if𝑘=𝑛²,0,otherwise, (18) where 𝑛∈ℕ. Then this sequence is statistically convergent to 0 but not convergent; in this case, weighted 𝐴-statistical convergence of a sequence coincides with statistical convergence.

Theorem 5. Let 𝐴=(𝑎_𝑛,𝑘) be a nonnegative regular matrix. Consider a sequence of positive linear operators (𝑀_𝑘) from 𝒞[𝑎,𝑏] into itself. Then, for all 𝑓∈𝒞[𝑎,𝑏] bounded on whole real line, 𝑆^𝑁_𝐴-lim_𝑘𝑀_𝑘𝑓,𝑥−𝑓(𝑥)_∞=0 (19) if and only if 𝑆^𝑁_𝐴-lim_𝑘𝑀_𝑘(1,𝑥)−1_∞𝑆=0,_𝑁𝐴-lim_𝑘𝑀_𝑘(𝑣,𝑥)−𝑥_∞𝑆=0,_𝑁𝐴-lim_𝑘𝑀_𝑘𝑣²,𝑥−𝑥²_∞=0. (20)

Proof. Equation (20) directly follows from (19) because each of 1,𝑥,𝑥² belongs to 𝒞[𝑎,𝑏]. Consider a function 𝑓∈𝒞[𝑎,𝑏]. Then there is a constant 𝐶>0 such that |𝑓(𝑥)|≤𝐶 for all 𝑥∈(−∞,∞). Therefore, 𝑓(𝑣)−𝑓(𝑥)≤2𝐶,−∞<𝑣,𝑥<∞. (21) Let 𝜖>0 be given. By hypothesis there is a 𝛿∶=𝛿(𝜀)>0 such that 𝑓(𝑣)−𝑓(𝑥)<𝜖∀|𝑣−𝑥|<𝛿. (22) Solving (21) and (22) and then substituting Ω(𝑣)=(𝑣−𝑥)², one obtains 𝑓(𝑣)−𝑓(𝑥)<𝜖+2𝐶𝛿²Ω,∀|𝑣−𝑥|<𝛿. (23) Equation (23) can also be written as −𝜖−2𝐶𝛿²Ω<𝑓(𝑣)−𝑓(𝑥)<𝜖+2𝑀𝛿²Ω. (24) Operating 𝑀_𝑘(1,𝑥) to (24) since 𝑀_𝑘(𝑓,𝑥) is linear and monotone, one obtains 𝑀_𝑘(1,𝑥)−𝜖−2𝐶𝛿²Ω<𝑀_𝑘(1,𝑥)𝑓(𝑣)−𝑓(𝑥)<𝑀_𝑘(1,𝑥)𝜖+2𝐶𝛿²Ω. (25) Note that 𝑥 is fixed, so 𝑓(𝑥) is constant number. Thus, we obtain from (25) that −𝜖𝑀_𝑘(1,𝑥)−2𝐶𝛿²𝑀_𝑘(Ω,𝑥)<𝑀_𝑘𝑓,𝑥−𝑓(𝑥)𝑀_𝑘(1,𝑥)<𝜖𝑀_𝑘(1,𝑥)+2𝐶𝛿²𝑀_𝑘(Ω,𝑥). (26) The term “𝑀_𝑘(𝑓,𝑥)−𝑓(𝑥)𝑀_𝑘(1,𝑥)” in (26) can also be written as 𝑀_𝑘𝑓,𝑥−𝑓(𝑥)𝑀_𝑘(1,𝑥)=𝑀_𝑘(𝑥)(𝑥𝑀𝑓,𝑥−𝑓−𝑓)_𝑘(1,𝑥)−1. (27) Now substituting the value of 𝑀_𝑘(𝑓,𝑥)−𝑓(𝑥)𝑀_𝑘(1,𝑥) in (26), we get that 𝑀_𝑘𝑓,𝑥−𝑓(𝑥)<𝜖𝑀_𝑘(1,𝑥)+2𝐶𝛿²𝑀_𝑘()(𝑥)𝑀Ω,𝑥+𝑓_𝑘(1,𝑥)−1. (28) We can rewrite the term “𝑀_𝑘(Ω,𝑥)” in (28) as follows: 𝑀_𝑘(Ω,𝑥)=𝑀_𝑘(𝑣−𝑥)²,𝑥=𝑀_𝑘𝑣²,𝑥+2𝑥𝑀_𝑘(𝑣,𝑥)+𝑥²𝑀_𝑘()=𝑀1,𝑥_𝑘𝑣²,𝑥−𝑥²−2𝑥𝑀_𝑘(𝑣,𝑥)−𝑥+𝑥²𝑀_𝑘(1,𝑥)−1. (29) Equation (28) with the above value of 𝑀_𝑘(Ω,𝑥) becomes 𝑀_𝑘𝑓,𝑥−𝑓(𝑥)<𝜖𝑀_𝑘()+1,𝑥2𝐶𝛿²𝑀_𝑘𝑣²,𝑥−𝑥²+2𝑥𝑀_𝑘(𝑣,𝑥)−𝑥+𝑥²𝑀_𝑘(1,𝑥)−1+𝑓(𝑥)𝑀_𝑘()𝑀1,𝑥−1=𝜖_𝑘()+1,𝑥−1+𝜖2𝐶𝛿²𝑀_𝑘𝑣²,𝑥−𝑥²+2𝑥𝑀_𝑘(𝑣,𝑥)−𝑥+𝑥²𝑀_𝑘(1,𝑥)−1+𝑓(𝑥)𝑀_𝑘(1,𝑥)−1. (30) Therefore, 𝑀_𝑘𝑓,𝑥−𝑓(𝑥)≤𝜖+2𝐶𝑏²𝛿²+𝐶𝑀_𝑘()+1,𝑥−12𝐶𝛿²𝑀_𝑘𝑣²,𝑥−𝑥²+4𝐶𝑏𝛿²𝑀_𝑘(𝑣,𝑥)−𝑥, (31) where 𝑏=max|𝑥|. Taking supremum over 𝑥∈[𝑎,𝑏], one obtains 𝑀_𝑘𝑓,𝑥−𝑓(𝑥)_∞≤𝜖+2𝐶𝑏²𝛿²+𝐶𝑀_𝑘(1,𝑥)−1_∞+2𝐶𝛿²𝑀_𝑘𝑣²,𝑥−𝑥²_∞+4𝐶𝑏𝛿²𝑀_𝑘(𝑣,𝑥)−𝑥_∞, (32) or 𝑀_𝑘𝑓,𝑥−𝑓(𝑥)_∞≤𝑇𝑀_𝑘(1,𝑥)−1_∞+𝑀_𝑘𝑣²,𝑥−𝑥²_∞+𝑀_𝑘(𝑣,𝑥)−𝑥_∞, (33) where 𝑇∶=max𝜖+2𝐶𝑏²𝛿²+𝐶,2𝐶𝛿²,4𝐶𝑏𝛿². (34) Hence, 𝑝_𝑘𝑀_𝑘𝑓,𝑥−𝑓(𝑥)_∞≤𝑇𝑝_𝑘𝑀_𝑘(1,𝑥)−1_∞+𝑝_𝑘𝑀_𝑘𝑣²,𝑥−𝑥²_∞+𝑝_𝑘𝑀_𝑘(𝑣,𝑥)−𝑥_∞. (35) For a given 𝛼>0, choose 𝜖>0 such that 𝜖<𝛼, and we will define the following sets: 𝐸=𝑘∈ℕ∶𝑝_𝑘𝑀_𝑘𝑓,𝑥−𝑓(𝑥)_∞,𝐸≥𝛼₁=𝑘∈ℕ∶𝑝_𝑘𝑀_𝑘(1,𝑥)−1_∞≥𝛼−𝜖,𝐸3𝑇₂=𝑘∈ℕ∶𝑝_𝑘𝑀_𝑘(𝑣,𝑥)−𝑥_∞≥𝛼−𝜖,𝐸3𝑇₃=𝑘∈ℕ∶𝑝_𝑘𝑀_𝑘𝑣²,𝑥−𝑥²_∞≥𝛼−𝜖.3𝑇 (36) It is easy to see that 𝐸⊂𝐸₁∪𝐸₂∪𝐸₃. (37) Thus, for each 𝑛∈ℕ, we obtain from (35) that _𝑘∈𝐸𝑎_𝑛,𝑘≤_𝑘∈𝐸1𝑎_𝑛,𝑘+_𝑘∈𝐸2𝑎_𝑛,𝑘+_𝑘∈𝐸3𝑎_𝑛,𝑘. (38) Taking limit 𝑛→∞ in (38) and also (20) gives that lim_{𝑛𝑘∈𝐸}𝑎_𝑛,𝑘=0. (39) This yields that 𝑆^𝑁_𝐴-lim_𝑘𝑀_𝑘(𝑓,𝑥)−𝑓(𝑥)_∞=0, (40) for all 𝑓∈𝒞[𝑎,𝑏].

We also obtain the following Korovkin-type theorem for weighted statistical convergence by writing Cesáro matrix (𝐶,1) instead of nonnegative regular matrix 𝐴 in Theorem 5.

Theorem 6. Consider a sequence of positive linear operators (𝑀_𝑘) from 𝒞[𝑎,𝑏] into itself. Then, for all 𝑓∈𝒞[𝑎,𝑏]𝑆_𝑁-lim_𝑘𝑀_𝑘𝑓,𝑥−𝑓(𝑥)_∞=0 (41) if and only if 𝑆_𝑁-lim_𝑘𝑀_𝑘(1,𝑥)−1_∞=0,𝑆_𝑁-lim_𝑘𝑀_𝑘(𝑣,𝑥)−𝑥_∞=0,𝑆_𝑁-lim_𝑘𝑀_𝑘𝑣²,𝑥−𝑥²_∞=0. (42)(43)(44)

Proof. Following the proof of Theorem 5, one obtains 𝐸⊂𝐸₁∪𝐸₂∪𝐸₃ (45) and so 𝛿_𝑁(𝐸)⊂𝛿_𝑁𝐸₁+𝛿_𝑁𝐸₂+𝛿_𝑁𝐸₃. (46)
Equations (42)–(44) give that 𝑆_𝑁-lim_𝑘𝑀_𝑘𝑓,𝑥−𝑓(𝑥)_∞=0. (47)

Remark 7. If we replace nonnegative regular matrix 𝐴 by Cesáro matrix and choose 𝑝_𝑘=1 for all 𝑘, in Theorem 5, then we obtain Theorem 1 due to Gadjiev and Orhan [15].

Remark 8. By Theorem 2 of [10], we have that if a sequence 𝑥=(𝑥_𝑘) is weighted statistically convergent to 𝐿, then it is strongly (𝑁,𝑝_𝑛)-summable to 𝐿 provided that 𝑝_𝑘|𝑥_𝑘−𝐿| is bounded; that is, there exists a constant 𝐶 such that 𝑝_𝑘|𝑥_𝑘−𝐿|≤𝐶 for all 𝑘∈ℕ. We write 𝑁,𝑝_𝑛=𝑥=𝑥_𝑘∶lim_𝑛1𝑃_𝑛^𝑛_𝑘=0𝑝_𝑘𝑥_𝑘−𝐿=0forsome𝐿 (48) for the set of all sequences 𝑥=(𝑥_𝑘) which are strongly (𝑁,𝑝_𝑛)-summable to 𝐿.

Theorem 9. Let 𝑀_𝑘∶𝒞[𝑎,𝑏]→𝒞[𝑎,𝑏] be a sequence of positive linear operators which satisfies (43)-(44) of Theorem 6 and the following condition holds: lim_𝑘𝑀_𝑘(1,𝑥)−1_∞=0. (49) Then, lim_𝑛1𝑃_𝑛^𝑛_𝑘=0𝑝_𝑘𝑀_𝑘(𝑓,𝑥)−𝑓(𝑥)_∞=0, (50) for any 𝑓∈𝒞[𝑎,𝑏].

Proof. It follows from (49) that ‖𝑀_𝑘(1,𝑥)‖_∞≤𝐶^′, for some constant 𝐶^′>0 and for all 𝑘∈ℕ. Hence, for 𝑓∈𝒞[𝑎,𝑏], one obtains 𝑝_𝑘𝑀_𝑘𝑓,𝑥−𝑓(𝑥)_∞≤𝑝_𝑘𝑓_∞𝑀_𝑘(1,𝑥)_∞+𝑓_∞≤𝑝_𝑘𝐶𝐶^′+1. (51) Right hand side of (51) is constant, so 𝑝_𝑘‖𝑀_𝑘(𝑓,𝑥)−𝑓(𝑥)‖_∞ is bounded. Since (49) implies (42), by Theorem 6 we get that 𝑆_𝑁-lim𝑀_𝑘(𝑓,𝑥)−𝑓(𝑥)_∞=0. (52) By Remark 8, (51) and (52) together give the desired result.

3. Rate of Weighted 𝐴-Statistical Convergence

First we define the rate of weighted 𝐴-statistically convergent sequence as follows.

Definition 10. Let 𝐴=(𝑎_𝑛,𝑘) be a nonnegative regular matrix and let (𝑎_𝑘) be a positive nonincreasing sequence. Then, a sequence 𝑥=(𝑥_𝑘) is weighted 𝐴-statistically convergent to 𝐿 with the rate of 𝑜(𝑎_𝑘) if for each 𝜖>0lim_𝑛1𝑎_{𝑛𝑘∈𝐸(𝑝,𝜖)}𝑎_𝑛,𝑘=0, (53) where 𝐸𝑝,𝜖=𝑘∈ℕ∶𝑝_𝑘𝑥_𝑘−𝐿≥𝜖. (54) In symbol, we will write 𝑥_𝑘−𝐿=𝑆^𝑁_𝐴-𝑜𝑎_𝑘as𝑘⟶∞. (55)

We will prove the following auxiliary result by using the above definition.

Lemma 11. Let 𝐴=(𝑎_𝑛,𝑘) be a nonnegative regular matrix. Suppose that (𝑎_𝑘) and (𝑏_𝑘) are two positive nonincreasing sequences. Let 𝑥=(𝑥_𝑘) and 𝑦=(𝑦_𝑘) be two sequences such that 𝑥_𝑘−𝐿₁=𝑆^𝑁_𝐴-𝑜(𝑎_𝑘) and 𝑦_𝑘−𝐿₂=𝑆^𝑁_𝐴-𝑜(𝑏_𝑘). Then,(i)(𝑥_𝑘−𝐿₁)±(𝑦_𝑘−𝐿₂)=𝑆^𝑁_𝐴-𝑜(𝑐_𝑘),(ii)(𝑥_𝑘−𝐿₁)(𝑦_𝑘−𝐿₂)=𝑆^𝑁_𝐴-𝑜(𝑐_𝑘),(iii)𝛼(𝑥_𝑘−𝐿₁)=𝑆^𝑁_𝐴-𝑜(𝑎_𝑘), for any scalar 𝛼,where 𝑐_𝑘=max{𝑎_𝑘,𝑏_𝑘}.

Proof. (i) Suppose that 𝑥_𝑘−𝐿₁=𝑆^𝑁_𝐴-𝑜𝑎_𝑘,𝑦_𝑘−𝐿₂=𝑆^𝑁_𝐴-𝑜𝑏_𝑘. (56) Given 𝜖>0, define 𝐸^′=𝑘∈ℕ∶𝑝_𝑘𝑥_𝑘−𝐿₁±𝑦_𝑘−𝐿₂,𝐸≥𝜖_′′=𝑘∈ℕ∶𝑝_𝑘𝑥_𝑘−𝐿₁≥𝜖2,𝐸^′′′=𝑘∈ℕ∶𝑝_𝑘𝑦_𝑘−𝐿₂≥𝜖2. (57) It is easy to see that 𝐸^′⊂𝐸^′′∪𝐸^′′′. (58) This yields that 1𝑐_{𝑛𝑘∈𝐸′}𝑎_𝑛,𝑘≤1𝑐_{𝑛𝑘∈𝐸′′}𝑎_𝑛,𝑘+1𝑐_{𝑛𝑘∈𝐸′′′}𝑎_𝑛,𝑘 (59) holds for all 𝑛∈ℕ. Since 𝑐_𝑘=max{𝑎_𝑘,𝑏_𝑘}, (59) gives that 1𝑐_{𝑛𝑘∈𝐸′}𝑎_𝑛,𝑘≤1𝑎_{𝑛𝑘∈𝐸′′}𝑎_𝑛,𝑘+1𝑏_{𝑛𝑘∈𝐸′′′}𝑎_𝑛,𝑘. (60) Taking limit 𝑛→∞ in (60) together with (56), we obtain lim_𝑛1𝑐_{𝑛𝑘∈𝐸′}𝑎_𝑛,𝑘=0. (61) Thus, 𝑥_𝑘−𝐿₁±𝑦_𝑘−𝐿₂=𝑆^𝑁_𝐴-𝑜𝑐_𝑘. (62) Similarly, we can prove (ii) and (iii).

Recall that the modulus of continuity of 𝑓 in 𝒞[𝑎,𝑏] is defined by 𝜔𝑓,𝛿=sup𝑓(𝑥)−𝑓𝑦∶𝑥,𝑦∈[𝑎,𝑏],𝑥−𝑦<𝛿. (63) It is well known that 𝑓(𝑥)−𝑓𝑦≤𝜔𝑓,𝛿𝑥−𝑦𝛿+1. (64)

Theorem 12. Let 𝐴=(𝑎_𝑛,𝑘) be a nonnegative regular matrix. If the sequence of positive linear operators 𝑀_𝑘∶𝒞[𝑎,𝑏]→𝒞[𝑎,𝑏] satisfies the conditions(i)‖𝑀_𝑘(1;𝑥)−1‖_∞=𝑆^𝑁_𝐴-𝑜(𝑎_𝑘),(ii)𝜔(𝑓,𝜆_𝑘)=𝑆^𝑁_𝐴-𝑜(𝑏_𝑘) with 𝜆_𝑘=𝑀_𝑘(𝜑_𝑥;𝑥) and 𝜑_𝑥(𝑦)=(𝑦−𝑥)²,where (𝑎_𝑘) and (𝑏_𝑘) are two positive nonincreasing sequences, then 𝑀_𝑘(𝑓;𝑥)−𝑓(𝑥)_∞=𝑆^𝑁_𝐴-𝑜𝑐_𝑘, (65) for all 𝑓∈𝒞[𝑎,𝑏], where 𝑐_𝑘=max{𝑎_𝑘,𝑏_𝑘}.

Proof. Equation (27) can be reformed into the following form: 𝑀_𝑘𝑓;𝑥−𝑓(𝑥)≤𝑀_𝑘𝑓𝑦(𝑥+𝑓(𝑥𝑀−𝑓);𝑥)_𝑘(1;𝑥)−1≤𝑀_𝑘1+𝑦−𝑥𝛿𝜔+𝑓(𝑥𝑀;𝑥𝑓,𝛿)_𝑘(1;𝑥)−1≤𝑀_𝑘1+𝑦−𝑥²𝛿²𝜔+𝑓(𝑥𝑀;𝑥𝑓,𝛿)_𝑘()≤𝑀1;𝑥−1_𝑘()+11;𝑥𝛿²𝑀_𝑘𝜑_𝑥𝜔+𝑓(𝑥𝑀;𝑥𝑓,𝛿)_𝑘()≤𝑀1;𝑥−1_𝑘()𝜔+𝑓(𝑥𝑀1;𝑥−1𝑓,𝛿)_𝑘()+11;𝑥−1+𝜔𝑓,𝛿𝛿²𝑀_𝑘𝜑_𝑥;𝑥𝜔𝑓,𝛿. (66) Choosing 𝛿=𝜆_𝑘=𝑀_𝑘(𝜑_𝑥;𝑥), one obtains 𝑀_𝑘𝑓;𝑥−𝑓(𝑥)_∞≤𝑇𝑀_𝑘(1;𝑥)−1_∞+2𝜔𝑓,𝜆_𝑘+𝑀_𝑘(1;𝑥)−1_∞𝜔𝑓,𝜆_𝑘, (67) where 𝑇=‖𝑓‖_∞. For a given 𝜖>0, we will define the following sets: 𝐸^′₁=𝑘∈ℕ∶𝑝_𝑘𝑀_𝑘𝑓,𝑥−𝑓(𝑥)_∞,𝐸≥𝜖_′2=𝑘∈ℕ∶𝑝_𝑘𝑀_𝑘(1,𝑥)−1_∞≥𝜖,𝐸3𝑇_′3=𝑘∈ℕ∶𝑝_𝑘𝜔𝑓,𝜆_𝑘≥𝜖6,𝐸^′₄=𝑘∈ℕ∶𝑝_𝑘𝜔𝑓,𝜆_𝑘𝑀_𝑘(1,𝑥)−1_∞≥𝜖3. (68) It follows from (67) that 1𝑐_{𝑛𝑘∈𝐸′1}𝑎_𝑛,𝑘≤1𝑐_{𝑛𝑘∈𝐸′2}𝑎_𝑛,𝑘+1𝑐_{𝑛𝑘∈𝐸′3}𝑎_𝑛,𝑘+1𝑐_{𝑛𝑘∈𝐸′4}𝑎_𝑛,𝑘 (69) holds for 𝑛∈ℕ. Since 𝑐_𝑘=max{𝑎_𝑘,𝑏_𝑘}, we obtain from (69) that 1𝑐_{𝑛𝑘∈𝐸′1}𝑎_𝑛,𝑘≤1𝑎_{𝑛𝑘∈𝐸′2}𝑎_𝑛,𝑘+1𝑏_{𝑛𝑘∈𝐸′3}𝑎_𝑛,𝑘+1𝑐_{𝑛𝑘∈𝐸′4}𝑎_𝑛,𝑘. (70) Taking limit 𝑛→∞ in (70) together with Lemma 11 and our hypotheses (i) and (ii), one obtains lim_𝑛1𝑐_{𝑛𝑘∈𝐸′1}𝑎_𝑛,𝑘=0. (71) This yields 𝑀_𝑘(𝑓;𝑥)−𝑓(𝑥)_∞=𝑆^𝑁_𝐴-𝑜𝑐_𝑘. (72)

4. Example and the Concluding Remark

The operators 𝐵_𝑛∶𝒞[0;1]→𝒞[0;1] given by 𝐵_𝑛𝑓,𝑥∶=^𝑛_𝑘=0𝑝_𝑛,𝑘(𝑥)𝑓𝑘𝑛, (73) where 𝑝_𝑛,𝑘(𝑥) are the fundamental Bernstein polynomials defined by 𝑝_𝑛,𝑘(𝑥)=𝑛𝑘𝑥^𝑘(1−𝑥)^𝑛−𝑘 (74) for any 𝑥∈[0,1], any 𝑘∈{0,1,…,𝑛}, and any 𝑛∈ℕ, are called Bernstein operators and were first introduced in [28]. Let the sequence (𝐴_𝑛) be defined by 𝐴_𝑛∶𝒞[0,1]→𝒞[0,1] with 𝐴_𝑛(𝑓,𝑥)=(1+𝑥_𝑛)𝐵_𝑛(𝑓,𝑥), where 𝑥=(𝑥_𝑘) is a sequence defined by 𝑥=𝑥_𝑘=√𝑘,if𝑘=𝑛²,𝑛∈ℕ,0,otherwise. (75) That is, (𝑥_𝑘)=(1,0,0,2,0,0,0,0,3,0,…,0,4,0,0,…). Let 𝑝_𝑘=𝑘(𝑘=1,2,…) and consider a nonnegative regular matrix 𝐴=(𝐶,1). Then, 𝑝_𝑘𝑥_𝑘=(),𝑃1,0,0,8,0,0,0,0,27,0,…,0,64,0,0,…_𝑛=^𝑛_𝑘=1𝑝_𝑘=𝑛(𝑛+1)2. (76) Since lim_𝑛1𝑃_𝑛𝑘≤𝑃_𝑛∶𝑝_𝑘𝑥_𝑘−0≥𝜖≤lim_𝑛1𝑃_𝑛𝑃_𝑛=1𝑛(𝑛+1)/2=0, (77)(𝑥_𝑘) is weighted statistically convergent to 0 but not convergent. It is not difficult to see that 𝐵_𝑛(1,𝑥)=1,𝐵_𝑛()𝐵𝑡,𝑥=𝑥,_𝑛𝑡²,𝑥=𝑥²+𝑥−𝑥²𝑛 (78) and the sequence (𝐴_𝑛) satisfies conditions (20). This yields that 𝑆^𝑁_(𝐶,1)-lim𝐴_𝑛(𝑓,𝑥)−𝑓(𝑥)_∞=0. (79)

On the other hand, one obtains 𝐴_𝑛(𝑓,0)=(1+𝑥_𝑛)𝑓(0), since 𝐵_𝑛(𝑓,0)=𝑓(0), and hence 𝐴_𝑛(𝑓,𝑥)−𝑓(𝑥)_∞≥𝐴_𝑛𝑓,0−𝑓(0)=𝑥_𝑛𝑓(0). (80) It follows that (𝐴_𝑛) does not satisfy the Korovkin theorem, since (𝑥_𝑛) and hence (𝐴_𝑛) is not convergent. Finally, we conclude that Theorem 5 is stronger than Theorem 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (351/130/1434). The authors, therefore, acknowledge with thanks DSR technical and financial support.