Statistical Approximation for Periodic Functions of Two Variables

Abstract

We prove a Korovkin type approximation theorem for a function of two variables by using the notion of statistical summability (𝐶,1,1). We also study the rate of statistical summability (𝐶,1,1) of positive linear operators. Finally we construct an example to show that our result is stronger than those previously proved for Pringsheim's convergence and statistical convergence.

1. Introduction and Preliminaries

In 1951, Fast [1] and Steinhaus [2] independently introduced an extension of the usual concept of sequential limit which is called statistical convergence.

The number sequence𝑥is said to be statistically convergent to the number 𝐿 provided that for each𝜖>0, lim_𝑛1𝑛𝑘≤𝑛;𝑥_𝑘−ℓ≥𝜖=0, (1) where|{𝑘≤𝑛∶𝑘∈𝐾}|denotes the number of elements of𝐾not exceeding𝑛. In this case we write 𝑠𝑡-lim𝑥_𝑘=ℓ.

The notion of statistical convergence of double sequences𝑥=(𝑥_𝑗𝑘)has been introduced and studied in [3, 4] independently in the same year, 2003.

Let𝐾⊆ℕ×ℕbe a two-dimensional set of positive integers and let𝐾(𝑛,𝑚)be the numbers of(𝑖,𝑗)in𝐾such that𝑖≤𝑛and𝑗≤𝑚.Then the two-dimensional analogue of natural density can be defined as follows.

The lower asymptotic density of a set𝐾⊆ℕ×ℕ is defined as 𝛿₂(𝐾)=lim_𝑛,𝑚𝐾()inf𝑛,𝑚.𝑛𝑚 (2) In this case the sequence(𝐾(𝑛,𝑚)/𝑛𝑚)has a limit in Pringsheim’s sense then we say that𝐾has a double natural density and is defined as 𝑃-lim_𝑛,𝑚𝐾(𝑛,𝑚)𝑛𝑚=𝛿₂(𝐾). (3)

A real double sequence𝑥=(𝑥_𝑗𝑘)is said to be statistically convergent to the number 𝑙 if for each𝜖>0, the set 𝑖,𝑗,𝑗≤𝑚,𝑘≤𝑛∶𝑥_𝑗𝑘−ℓ≥𝜖 (4) has double natural density zero. In this case we write𝑠𝑡₂-lim_𝑗,𝑘𝑥_𝑗𝑘=ℓ.

If𝑥is statistically convergent, then𝑥need not be convergent. Also it is not necessarily bounded. For example, let𝑥=(𝑥_𝑗𝑘)be defined as 𝑥_𝑗𝑘=𝑗𝑘,if𝑗and𝑘aresquares,1,otherwise. (5) It is easy to see that𝑠𝑡₂-lim𝑥_𝑗𝑘=1, since the cardinality of the set{(𝑗,𝑘)∶|𝑥_𝑗𝑘−1|≥𝜖}≤𝑗√𝑘for every𝜖>0. But𝑥is neither convergent nor bounded.

Móricz [5] introduced the notion of statistical summability(𝐶,1,1).A double sequence𝑥=(𝑥_𝑗𝑘)is said to be statistically summable(𝐶,1,1)to the numberℓif for every𝜖>0, 𝛿₂(𝑚,𝑛)∈ℕ×ℕ∶𝜎_𝑚𝑛−ℓ≥𝜖=0, (6) where 𝜎_𝑚𝑛=1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑗=0𝑘=0𝑥_𝑗𝑘 (7) is the(𝐶,1,1)mean of𝑥=(𝑥_𝑗𝑘).Thus, the double sequence𝑥is statistically summable(𝐶,1,1)to 𝑙 if and only if the sequence𝜎=(𝜎_𝑚𝑛)is statistically convergent to ℓ. In this case we write 𝑠𝑡₂(𝐶,1,1)-lim_𝑗,𝑘𝑥_𝑗𝑘=ℓ. Note that if a double sequence is bounded then𝑠𝑡₂-lim_𝑗,𝑘𝑥_𝑗𝑘=ℓ implies𝑠𝑡₂-lim_𝑚,𝑛𝜎_𝑚𝑛=ℓ.

Korovkin type approximation theorems (cf. [6–10]) are useful tools to check whether a given sequence(𝐿_𝑛)_𝑛≥1of positive linear operators on𝐶[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 test functions 1, 𝑥, and 𝑥² in the space𝐶[0,1]as well as for test functions 1, cos, and sin in the space of all continuous 2𝜋-periodic functions on the real line.

We know that𝐶[0,1]is a Banach space with norm 𝑓_∞∶=sup_[]𝑥∈0,1𝑓(𝑥),𝑓∈𝐶[0,1]. (8)

We denote by𝐶_2𝜋([0,1])the space of all2𝜋-periodic functions𝑓∈𝐶([0,1])which is a Banach space with 𝑓_2𝜋=sup_[]𝑡∈0,1𝑓(𝑡). (9)

After the paper of Gadjiev and Orhan [11], many papers have appeared in the literature concerning the Korovkin type approximation theorems via different statistical summability methods and for different sets of test functions. At present we are concerned about applications of such summability methods for double sequences to prove two-dimensional version of Korovkin theorem. For example, in [12, 13] the authors used the notion of statistical𝐴-summability of double sequences; in [13–16], the authors have used, respectively, statistical convergence and𝐴-statistical convergence of double sequences; and in [17, 18], the authors used almost summability. For some more related work, we refer to [19–22].

In this paper, we present the Korovkin type approximation theorem for periodic functions via statistical summability(𝐶,1,1)and also study the rate of statistical summability(𝐶,1,1)of a double sequence of positive linear operators defined from𝐶^∗(ℝ²)into𝐶^∗(ℝ²),where𝐶^∗(ℝ²)is the space of all2𝜋-periodic and real valued continuous functions onℝ²equipped with the norm 𝑓_𝐶(ℝ2)∶=sup_{𝑥,𝑦∈ℝ2}𝑓𝑥,𝑦,𝑓∈𝐶ℝ². (10)

2. Main Result

First, we state the result due to [15] for𝐴-statistical convergence of double sequences.

Theorem 1. Let(𝐿_𝑚𝑛)be a double sequence of positive linear operators acting from𝐶^∗(ℝ²)into𝐶^∗(ℝ²).Then, for all𝑓∈𝐶^∗(ℝ²)𝑠𝑡^𝐴₂-lim_{𝑚,𝑛→∞}𝐿_𝑚𝑛𝑓−𝑓_{𝐶∗(ℝ2)}=0 (11) if and only if 𝑠𝑡^𝐴₂-lim_{𝑚,𝑛→∞}𝐿_𝑚𝑛𝑓_𝑖−𝑓_{𝑖𝐶∗(ℝ2)}=0,𝑖=0,1,2,3,4, (12) where𝑓₀(𝑥,𝑦)=1,??𝑓₁(𝑥,𝑦)=sin𝑥,𝑓₂(𝑥,𝑦)=sin𝑦,?𝑓₃(𝑥,𝑦)=cos𝑥, and𝑓₄(𝑥,𝑦)=cos𝑦.

If we replace the matrix𝐴by the identity four-dimensional matrix in the above theorem, then we immediately get the following result in Pringsheim’s sense.

Corollary 2. Let(𝐿_𝑚𝑛)be a double sequence of positive linear operators acting from𝐶^∗(ℝ²)into𝐶^∗(ℝ²).Then, for all𝑓∈𝐶^∗(ℝ²)𝑃-lim_{𝑚,𝑛→∞}𝐿_𝑚𝑛𝑓−𝑓_{𝐶∗(ℝ2)}=0 (13) if and only if 𝑃-lim_{𝑚,𝑛→∞}𝐿_𝑚𝑛𝑓_𝑖−𝑓_{𝑖𝐶∗(ℝ2)}=0,𝑖=0,1,2,3,4. (14) We prove the following result.

Theorem 3. Let(𝑇_𝑗𝑘)be a double sequence of positive linear operators acting from𝐶^∗(ℝ²)into𝐶^∗(ℝ²).Then, for all𝑓∈𝐶^∗(ℝ²)𝑠𝑡₂(𝐶,1,1)-lim_{𝑗,𝑘→∞}𝑇_𝑗𝑘𝑓−𝑓_{𝐶∗(ℝ2)}=0 (15) if and only if 𝑠𝑡₂(𝐶,1,1)-lim_{𝑗,𝑘→∞}𝑇_𝑗𝑘𝑓_𝑖−𝑓_{𝑖𝐶∗(ℝ2)}().=0𝑖=0,1,2,3,4 (16)

Proof. Since each of the functions 𝑓₀, 𝑓₁, 𝑓₂, 𝑓₃, and 𝑓₄ belongs to 𝐶^∗(ℝ²), necessity follows immediately from (15). Let condition (16) hold and𝑓∈𝐶^∗(ℝ²).Let𝐼and𝐽be closed subintervals each of length 2𝜋 of ℝ. Fix(𝑥,𝑦)∈𝐼×𝐽. By the continuity of𝑓at(𝑥,𝑦), it follows that for given 𝜀>0 there is a number𝛿>0such that for all(𝑢,𝜐)∈ℝ²𝑓(𝑢,𝜐)−𝑓𝑥,𝑦<𝜀, (17) whenever|𝑢−𝑥|,|𝜐−𝑦|<𝛿.Since𝑓is bounded, it follows that 𝑓(𝑢,𝜐)−𝑓𝑥,𝑦≤𝑀_𝑓=𝑓_{𝐶∗(ℝ2)}, (18) for all(𝑢,𝜐)∈ℝ².
For all(𝑢,𝜐)∈(𝑥−𝛿,2𝜋+𝑥−𝛿]×(𝑦−𝛿,2𝜋+𝑦−𝛿], it is well known that 𝑓(𝑢,𝜐)−𝑓𝑥,𝑦<𝜀+2𝑀_𝑓sin²()𝜓(),𝛿/2𝑢,𝜐 (19) where𝜓(𝑢,𝜐)=sin²((𝑢−𝑥)/2)+sin²((𝜐−𝑦)/2).Since the function𝑓∈𝐶^∗(ℝ²)is2𝜋-periodic, the inequality (19) holds for(𝑢,𝜐)∈ℝ². Then, we obtain 𝑇_𝑗𝑘𝑓;𝑥,𝑦−𝑓𝑥,𝑦≤𝑇_𝑗𝑘𝑓()+𝑓𝑇𝑢,𝜐−𝑓𝑥,𝑦;𝑥,𝑦𝑥,𝑦_𝑗𝑘𝑓₀;𝑥,𝑦−𝑓₀≤𝑇𝑥,𝑦_𝑗𝑘𝜀+2𝑀_𝑓sin²()𝜓()𝛿/2𝑢,𝜐;𝑥,𝑦+𝑀_𝑓𝑇_𝑗𝑘𝑓₀;𝑥,𝑦−𝑓₀𝑥,𝑦≤𝜀+𝜀+𝑀_𝑓𝑇_𝑗𝑘𝑓₀;𝑥,𝑦−𝑓₀+𝑀𝑥,𝑦_𝑓sin²()×2𝑇𝛿/2_𝑗𝑘𝑓₀;𝑥,𝑦−𝑓₀+|𝑇𝑥,𝑦sin𝑥|_𝑗𝑘𝑓₁;𝑥,𝑦−𝑓₁+𝑇𝑥,𝑦sin𝑦_𝑗𝑘𝑓₂;𝑥,𝑦−𝑓₃+|𝑇𝑥,𝑦cos𝑥|_𝑗𝑘𝑓₃;𝑥,𝑦−𝑓₃+𝑇𝑥,𝑦cos𝑦_𝑗𝑘𝑓₄;𝑥,𝑦−𝑓₄𝑥,𝑦<𝜀+𝐾⁴_𝑖=0𝑇_𝑗𝑘𝑓_𝑖;𝑥,𝑦−𝑓_𝑖𝑥,𝑦, (20) where𝐾∶=𝜀+𝑀_𝑓+(2𝑀_𝑓/sin²(𝛿/2)). Now, takingsup_{(𝑥,𝑦)∈𝐼×𝐽}, we get 𝑇_𝑗𝑘𝑓−𝑓_{𝐶∗(ℝ2)}<𝜀+𝐾⁴_𝑖=0𝑇_𝑗𝑘𝑓_𝑖−𝑓_{𝑖𝐶∗(ℝ2)}. (21) Now for a given𝑟>0choose𝜀^′>0such that𝜀^′<𝑟. Define the following sets: 𝐷=(𝑚,𝑛)∶𝐿_𝑚𝑛𝑓−𝑓_{𝐶∗(ℝ2)},𝐷≥𝑟_𝑖=(𝑚,𝑛)∶𝐿_𝑚𝑛𝑓_𝑖−𝑓_{𝑖𝐶∗(ℝ2)}≥𝑟−𝜀^′(),5𝐾𝑖=0,1,2,3,4 (22) where𝐿_𝑚𝑛=(1/(𝑚+1)(𝑛+1))∑^𝑚_𝑗=0∑^𝑛_𝑘=0𝑇_𝑗𝑘. Then by (21) 𝐷⊆⁴_𝑖=0𝐷_𝑖, (23) and so 𝛿_𝐴⁽²⁾(𝐷)≤⁴_𝑖=0𝛿_𝐴⁽²⁾𝐷_𝑖. (24)
Now using (16), we get 𝑠𝑡₂(𝐶,1,1)-lim_{𝑗,𝑘→∞}𝑇_𝑗𝑘𝑓−𝑓_{𝐶∗(ℝ2)}=0. (25)

Example 4. Now we present an example of double sequences of positive linear operators, showing that Corollary 2 does not work but our approximation theorem works. We consider the double sequence of Fejer operators on𝐶^∗(ℝ²)𝜎_𝑚𝑛=1𝑓;𝑥,𝑦()⋅1𝑛𝜋()×𝑛𝜋_{𝜋−𝜋𝜋−𝜋}𝑓(𝑢,𝜐)𝐹_𝑚(𝑢)𝐹_𝑛(𝜐)𝑑𝑢𝑑𝜐, (26) where 𝐹_𝑚(𝑢)=sin²(𝑚(𝑢−𝑥)/2)sin²)),1((𝑢−𝑥/2𝜋^𝜋_−𝜋𝐹_𝑚(𝑢)𝑑𝑢=1. (27) Observe that 𝜎_𝑚𝑛𝑓₀;𝑥,𝑦=𝑓₀,𝜎𝑥,𝑦_𝑚𝑛𝑓₁;𝑥,𝑦=𝑚−1𝑚𝑓₁,𝜎𝑥,𝑦_𝑚𝑛𝑓₂;𝑥,𝑦=𝑛−1𝑛𝑓₂,𝜎𝑥,𝑦_𝑚𝑛𝑓₃;𝑥,𝑦=𝑚−1𝑚𝑓₃,𝜎𝑥,𝑦_𝑚𝑛𝑓₄;𝑥,𝑦=𝑛−1𝑛𝑓₄𝑥,𝑦. (28) Define a double sequence𝛼=(𝛼_𝑚𝑛)by𝛼_𝑚𝑛=(−1)^𝑚+𝑛, 𝑚,𝑛∈ℕ.
We observe that 𝛼=(𝛼_𝑚𝑛) is neither𝑃-convergent nor statistically convergent but 𝑠𝑡₂(𝐶,1,1)-lim𝛼=0. (29)
Let us define the operators𝐿_𝑚𝑛∶𝐶^∗(ℝ²)→𝐶^∗(ℝ²)by 𝐿_𝑚𝑛𝑓;𝑥,𝑦=1+𝛼_𝑚𝑛𝜎_𝑚𝑛𝑓;𝑥,𝑦. (30) Then, observe that the double sequence of positive linear operators(𝐿_𝑚𝑛)defined by (30) satisfies all hypotheses of Theorem 3. Hence, by (28), we have, for all𝑓∈𝐶^∗(ℝ²), 𝑠𝑡₂(𝐶,1,1)-lim_{𝑚,𝑛→∞}𝐿_𝑚𝑛𝑓−𝑓_{𝐶∗(ℝ2)}=0. (31) Since(𝛼_𝑚𝑛)is neither𝑃-convergent nor statistically convergent, the sequence(𝐿_𝑚𝑛)given by (30) is also neither𝑃-convergent nor statistically convergent to the function𝑓∈𝐶^∗(ℝ²). So, we conclude that Corollary 2 and Theorem 1 do not work for the operators(𝐿_𝑚𝑛)given by (30) while Theorem 3 still works. Hence, we conclude that𝑠𝑡₂(𝐶,1,1)-version is stronger than that of𝑃-version as well as statistical version.

3. Rate of Statistical Summability (𝐶,1,1)

Let(𝛽_𝑚𝑛)be a positive nonincreasing double sequence. We say that a double sequence𝑥=(𝑥_𝑚𝑛)is statistically summable(𝐶,1,1)to the number𝐿with the rate𝑜(𝛽_𝑚𝑛)if for every𝜀>0, 𝑃-lim_{𝑚,𝑛→∞}1𝛽_𝑚𝑛𝑗≤𝑚,𝑘≤𝑛∶𝜎_𝑗𝑘−ℓ≥𝜖=0. (32) In this case, we write𝑥_𝑚𝑛−𝐿=𝑠𝑡₂(𝐶,1,1)-𝑜(𝛽_𝑚𝑛)as𝑚,𝑛→∞.

Now, we recall the notion of modulus of continuity. The modulus of continuity of 𝑓∈𝐶^∗(ℝ²), denoted by 𝜔(𝑓,𝛿) for 𝛿>0, is defined by 𝜔𝑓,𝛿=sup𝑓(𝑢,𝜐)−𝑓𝑥,𝑦∶(𝑢,𝜐),𝑥,𝑦∈ℝ²,(𝑢−𝑥)²+𝜐−𝑦²≤𝛿. (33) It is well known that 𝑓(𝑢,𝜐)−𝑓𝑥,𝑦≤𝜔𝑓,(𝑢−𝑥)²+𝜐−𝑦²≤𝜔𝑓,𝛿(𝑢−𝑥)²+𝜐−𝑦²𝛿+1. (34) Then we have the following result.

Theorem 5. Let(𝑇_𝑗𝑘)be a double sequence of positive linear operators acting from 𝐶^∗(ℝ²) into 𝐶^∗(ℝ²). Let (𝛼_𝑗𝑘) and (𝛽_𝑗𝑘) be two positive non-increasing sequences. Suppose that(i)‖𝑇_𝑗𝑘(𝑓₀)−𝑓₀‖_{𝐶∗(ℝ2)}=𝑠𝑡₂(𝐶,1,1)-𝑜(𝛼_𝑚𝑛),(ii)𝜔(𝑓,𝜆_𝑗𝑘)=𝑠𝑡₂(𝐶,1,1)−𝑜(𝛽_𝑗𝑘), where𝜆_𝑚𝑗𝑘=‖𝑇_𝑗𝑘(𝜑)‖_𝐶^∗_(ℝ²₎and 𝜑(𝑢,𝜐)=sin²𝑢−𝑥2+sin²𝜐−𝑦2𝑓𝑜𝑟𝑒𝑎𝑐ℎ(𝑢,𝜐),𝑥,𝑦∈ℝ². (35)Then, for all𝑓∈𝐶^∗(ℝ²), 𝑇_𝑗𝑘𝑓−𝑓_{𝐶∗(ℝ2)}=𝑠𝑡₂(𝐶,1,1)-𝑜𝛾_𝑗𝑘, (36) where𝛾_𝑗𝑘=max{𝛼_𝑗𝑘,𝛽_𝑗𝑘}.

Proof. Let 𝑓∈𝐶^∗(ℝ²) and (𝑥,𝑦)∈−[𝜋,𝜋]×[−𝜋,𝜋]. Let𝛿>0; we have the following cases.
Case I. If𝛿<|𝑢−𝑥|≤𝜋,𝛿<|𝜐−𝑦|≤𝜋, then|𝑢−𝑥|≤𝜋|sin((𝑢−𝑥)/2)|and|𝜐−𝑦|≤𝜋|sin((𝜐−𝑦)/2)|. Therefore by (34), we have 𝑓()𝜋𝑢,𝜐−𝑓𝑥,𝑦≤𝜔𝑓,𝛿²sin²((𝑢−𝑥)/2)+sin²𝜐−𝑦/2𝛿²+1. (37)
Case II. If|𝑢−𝑥|>𝜋,|𝜐−𝑦|≤𝜋. Let𝑘be an integer such that|𝑢+2𝑘𝜋−𝑥|≤𝜋; then 𝑓()=𝑓()×𝜋𝑢,𝜐−𝑓𝑥,𝑦𝑢+2𝑘𝜋,𝜐−𝑓𝑥,𝑦≤𝜔𝑓,𝛿₂sin²((𝑢+2𝑘𝜋−𝑥)/2)+sin²𝜐−𝑦/2𝛿²𝜋+1=𝜔𝑓,𝛿²sin²((𝑢−𝑥)/2)+sin²𝜐−𝑦/2𝛿²+1. (38) Similarly, in the other two cases when|𝑢−𝑥|≤𝜋, |𝜐−𝑦|>𝜋 and |𝑢−𝑥|>𝜋, |𝜐−𝑦|>𝜋, we obtain (37).
Now, using the definition of modulus of continuity and the linearity and the positivity of the operators𝑇_𝑗𝑘,we get 𝑇_𝑗𝑘𝑓;𝑥,𝑦−𝑓𝑥,𝑦≤𝑇_𝑗𝑘𝑓()+𝑓𝑇𝑢,𝜐−𝑓𝑥,𝑦;𝑥,𝑦𝑥,𝑦_𝑗𝑘𝑓₀;𝑥,𝑦−𝑓₀𝑇𝑥,𝑦≤𝜔𝑓,𝛿_𝑗𝑘𝑓₀;𝑥,𝑦+𝜋²𝜔𝑓,𝛿𝛿²𝑇_𝑗𝑘+𝑓𝑇𝜑;𝑥,𝑦𝑥,𝑦_𝑗𝑘𝑓₀;𝑥,𝑦−𝑓₀𝑥,𝑦. (39) Taking supremum over(𝑥,𝑦)on both sides of the above inequality and let 𝛿∶=𝛿_𝑗𝑘=𝑇_𝑗𝑘𝜑_{𝐶∗(ℝ2)}. (40) We obtain 𝑇_𝑗𝑘𝑓−𝑓_{𝐶∗(ℝ2)}≤𝜔𝑓,𝛿_𝑗𝑘𝑇_𝑗𝑘𝑓₀−𝑓_{0𝐶∗(ℝ2)}+1+𝜋²×𝜔𝑓,𝛿_𝑗𝑘+𝑀𝑇_𝑗𝑘𝑓₀−𝑓_{0𝐶∗(ℝ2)}, (41) where𝑀∶=‖𝑓‖_{𝐶∗(ℝ2)}. Let 𝐿_𝑚𝑛=(1/(𝑚+1)(𝑛+1))∑^𝑚_𝑗=0∑^𝑛_𝑘=0𝑇_𝑗𝑘. Now for a given𝜀>0define the following sets: 𝐷=(𝑚,𝑛)∶𝐿_𝑚𝑛𝑓−𝑓_{𝐶∗(ℝ2)},𝐷≥𝜀₁=(𝑚,𝑛)∶𝐿_𝑚𝑛𝑓₀−𝑓_{0𝐶∗(ℝ2)}≥𝜀3,𝐷₂=(𝑚,𝑛)∶𝜔𝑓,𝛿_𝑚𝑛≥𝜀31+𝜋²,𝐷₃=(𝑚,𝑛)∶𝐿_𝑚𝑛𝑓−𝑓_{𝐶∗(ℝ2)}≥𝜀.3𝑀 (42) Then𝐷⊂𝐷₁∪𝐷₂∪𝐷₃. Further define 𝐷₄=(𝑚,𝑛)∶𝜔𝑓,𝛿_𝑚𝑛≥𝜀3,𝐷₅=(𝑚,𝑛)∶𝐿_𝑚𝑛𝑓−𝑓_{𝐶∗(ℝ2)}≥𝜀3. (43) We see that𝐷₁⊂𝐷₄∪𝐷₅. Therefore𝐷⊂⋃⁵_𝑖=2𝐷_𝑖. Therefore, since𝛾_𝑚𝑛=max{𝛼_𝑚𝑛,𝛽_𝑚𝑛}, we conclude that for every(𝑗,𝑘)∈ℕ×ℕ𝛿₂(𝐷)≤⁵_𝑖=2𝛿₂𝐷_𝑖. (44) Using conditions (i) and (ii), we get‖𝐿_𝑚𝑛(𝑓)−𝑓‖_{𝐶∗(ℝ2)}=𝑠𝑡₂(𝐶,1,1)-𝑜(𝛾_𝑚𝑛).

Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.