# Statistical Approximation for Periodic Functions of Two Variables

Authors: Abdullah Alotaibi; M. Mursaleen; S. A. Mohiuddine
Journal: Journal of Function Spaces and Applications
 (2013)

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## 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.


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## Body

## 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,
(1)limn1n|{k≤n;|xk-ℓ|≥ϵ}|=0,
where|{k≤n:k∈K}|denotes the number of elements ofKnot exceedingn. In this case we write st-limxk=ℓ.The notion of statistical convergence of double sequencesx=(xjk)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
(2)δ2(K)=limn,minfK(n,m)nm.
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
(3)P-limn,mK(n,m)nm=δ2(K).A real double sequencex=(xjk)is said to be statistically convergent to the number l if for eachϵ>0, the set
(4){(i,j),j≤m,k≤n:|xjk-ℓ|≥ϵ}
has double natural density zero. In this case we writest2-limj,kxjk=ℓ.Ifxis statistically convergent, thenxneed not be convergent. Also it is not necessarily bounded. For example, letx=(xjk)be defined as
(5)xjk={jk,ifjandkaresquares,1,otherwise.
It is easy to see thatst2-limxjk=1, since the cardinality of the set{(j,k):|xjk-1|≥ϵ}≤jkfor everyϵ>0. Butxis neither convergent nor bounded.Móricz [5] introduced the notion of statistical summability(C,1,1).A double sequencex=(xjk)is said to be statistically summable(C,1,1)to the numberℓif for everyϵ>0,
(6)δ2{(m,n)∈ℕ×ℕ:|σmn-ℓ|≥ϵ}=0,
where
(7)σmn=1(m+1)(n+1)∑j=0m‍∑k=0n‍xjk
is the(C,1,1)mean ofx=(xjk).Thus, the double sequencexis statistically summable(C,1,1)to l if and only if the sequenceσ=(σmn)is statistically convergent to ℓ. In this case we write st2(C,1,1)-limj,kxjk=ℓ. Note that if a double sequence is bounded thenst2-limj,kxjk=ℓ impliesst2-limm,nσmn=ℓ.Korovkin type approximation theorems (cf. [6–10]) are useful tools to check whether a given sequence(Ln)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
(8)∥f∥∞:=supx∈[0,1]|f(x)|,f∈C[0,1].We denote byC2π([0,1])the space of all2π-periodic functionsf∈C([0,1])which is a Banach space with
(9)∥f∥2π=supt∈[0,1]|f(t)|.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
(10)∥f∥C(ℝ2):=sup(x,y)∈ℝ2|f(x,y)|,(f∈C(ℝ2)).


## 2. Main Result

First, we state the result due to [15] forA-statistical convergence of double sequences.Theorem 1.
Let(Lmn)be a double sequence of positive linear operators acting fromC*(ℝ2)intoC*(ℝ2).Then, for allf∈C*(ℝ2)(11)st2A-limm,n→∞∥Lmn(f)-f∥C*(ℝ2)=0
if and only if
(12)st2A-limm,n→∞∥Lmn(fi)-fi∥C*(ℝ2)=0,i=0,1,2,3,4,
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(Lmn)be a double sequence of positive linear operators acting fromC*(ℝ2)intoC*(ℝ2).Then, for allf∈C*(ℝ2)(13)P-limm,n→∞∥Lmn(f)-f∥C*(ℝ2)=0
if and only if
(14)P-limm,n→∞∥Lmn(fi)-fi∥C*(ℝ2)=0,i=0,1,2,3,4.
We prove the following result.Theorem 3.
Let(Tjk)be a double sequence of positive linear operators acting fromC*(ℝ2)intoC*(ℝ2).Then, for allf∈C*(ℝ2)(15)st2(C,1,1)-limj,k→∞∥Tjk(f)-f∥C*(ℝ2)=0
if and only if
(16)st2(C,1,1)-limj,k→∞∥Tjk(fi)-fi∥C*(ℝ2)=0st2(C,1,1)-limj,k→∞s(C1)(i=0,1,2,3,4).Proof.
Since each of the functionsf0, 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)∈ℝ2(17)|f(u,υ)-f(x,y)|<ε,
whenever|u-x|,|υ-y|<δ.Sincefis bounded, it follows that
(18)|f(u,υ)-f(x,y)|≤Mf=∥f∥C*(ℝ2),
for all(u,υ)∈ℝ2.
For all(u,υ)∈(x-δ,2π+x-δ]×(y-δ,2π+y-δ], it is well known that
(19)|f(u,υ)-f(x,y)|<ε+2Mfsin2(δ/2)ψ(u,υ),
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
(20)|Tjk(f;x,y)-f(x,y)|≤Tjk(|f(u,υ)-f(x,y)|;x,y)+|f(x,y)||Tjk(f0;x,y)-f0(x,y)|≤|Tjk(ε+2Mfsin2(δ/2)ψ(u,υ);x,y)|+Mf|Tjk(f0;x,y)-f0(x,y)|≤ε+((ε+Mf)|Tjk(f0;x,y)-f0(x,y)|)+Mfsin2(δ/2)×{2|Tjk(f0;x,y)-f0(x,y)|gg+|sinx||Tjk(f1;x,y)-f1(x,y)|gg+|siny||Tjk(f2;x,y)-f3(x,y)|gg+|cosx||Tjk(f3;x,y)-f3(x,y)|gg+|cosy||Tjk(f4;x,y)-f4(x,y)|}<ε+K∑i=04‍|Tjk(fi;x,y)-fi(x,y)|,
whereK:=ε+Mf+(2Mf/sin2(δ/2)). Now, takingsup(x,y)∈I×J, we get
(21)∥Tjk(f)-f∥C*(ℝ2)<ε+K∑i=04‍∥Tjk(fi)-fi∥C*(ℝ2).
Now for a givenr>0chooseε′>0such thatε′<r. Define the following sets:
(22)D={(m,n):∥Lmn(f)-f∥C*(ℝ2)≥r},Di={(m,n):∥Lmn(fi)-fi∥C*(ℝ2)≥r-ε′5K}ggggggggggg(i=0,1,2,3,4),
whereLmn=(1/(m+1)(n+1))∑j=0m‍∑k=0n‍Tjk. Then by (21)
(23)D⊆⋃i=04‍Di,
and so
(24)δA(2)(D)≤∑i=04‍δA(2)(Di).
Now using (16), we get
(25)st2(C,1,1)-limj,k→∞∥Tjk(f)-f∥C*(ℝ2)=0.Example 4.
Now we present an example of double sequences of positive linear operators, showing that Corollary2 does not work but our approximation theorem works. We consider the double sequence of Fejer operators onC*(ℝ2)(26)σmn(f;x,y)=1(nπ)·1(nπ)×∫-ππ‍∫-ππ‍f(u,υ)Fm(u)Fn(υ)dudυ,
where
(27)Fm(u)=sin2(m(u-x)/2)sin2((u-x)/2),1π∫-ππ‍Fm(u)du=1.
Observe that
(28)σmn(f0;x,y)=f0(x,y),σmn(f1;x,y)=m-1mf1(x,y),σmn(f2;x,y)=n-1nf2(x,y),σmn(f3;x,y)=m-1mf3(x,y),σmn(f4;x,y)=n-1nf4(x,y).
Define a double sequenceα=(αmn)byαmn=(-1)m+n, m,n∈ℕ.
We observe thatα=(αmn) is neitherP-convergent nor statistically convergent but
(29)st2(C,1,1)-limα=0.
Let us define the operatorsLmn:C*(ℝ2)→C*(ℝ2)by
(30)Lmn(f;x,y)=(1+αmn)σmn(f;x,y).
Then, observe that the double sequence of positive linear operators(Lmn)defined by (30) satisfies all hypotheses of Theorem 3. Hence, by (28), we have, for allf∈C*(ℝ2),
(31)st2(C,1,1)-limm,n→∞∥Lmn(f)-f∥C*(ℝ2)=0.
Since(αmn)is neitherP-convergent nor statistically convergent, the sequence(Lmn)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(Lmn)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(βmn)be a positive nonincreasing double sequence. We say that a double sequencex=(xmn)is statistically summable(C,1,1)to the numberLwith the rateo(βmn)if for everyε>0,
(32)P-limm,n→∞1βmn|{j≤m,k≤n:|σjk-ℓ|≥ϵ}|=0.
In this case, we writexmn-L=st2(C,1,1)-o(βmn)asm,n→∞.Now, we recall the notion of modulus of continuity. The modulus of continuity off∈C*(ℝ2), denoted by ω(f,δ) for δ>0, is defined by
(33)ω(f,δ)=sup{(u-x)2+(υ-y)2|f(u,υ)-f(x,y)|:(u,υ),(x,y)∈ℝ2,fjjjj(u-x)2+(υ-y)2≤δ}.
It is well known that
(34)|f(u,υ)-f(x,y)|≤ω(f,(u-x)2+(υ-y)2)≤ω(f,δ)((u-x)2+(υ-y)2δ+1).
Then we have the following result.Theorem 5.
Let(Tjk)be a double sequence of positive linear operators acting from C*(ℝ2) into C*(ℝ2). Let (αjk) and (βjk) be two positive non-increasing sequences. Suppose that(i)
∥Tjk(f0)-f0∥C*(ℝ2)=st2(C,1,1)-o(αmn),(ii)
ω(f,λjk)=st2(C,1,1)-o(βjk), whereλmjk=∥Tjk(φ)∥C*(ℝ2)and
(35)φ(u,υ)=sin2(u-x2)+sin2(υ-y2)foreach(u,υ),(x,y)∈ℝ2.Then, for allf∈C*(ℝ2),
(36)∥Tjk(f)-f∥C*(ℝ2)=st2(C,1,1)-o(γjk),
whereγjk=max{αjk,βjk}.Proof.
Letf∈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
(37)|f(u,υ)-f(x,y)|≤ω(f,δ)(π2sin2((u-x)/2)+sin2((υ-y)/2)δ2+1).
Case II. If|u-x|>π,|υ-y|≤π. Letkbe an integer such that|u+2kπ-x|≤π; then
(38)|f(u,υ)-f(x,y)|=|f(u+2kπ,υ)-f(x,y)|≤ω(f,δ)×(π2sin2((u+2kπ-x)/2)+sin2((υ-y)/2)δ2+1)=ω(f,δ)(π2sin2((u-x)/2)+sin2((υ-y)/2)δ2+1).
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 operatorsTjk,we get
(39)|Tjk(f;x,y)-f(x,y)|≤Tjk(|f(u,υ)-f(x,y)|;x,y)+|f(x,y)||Tjk(f0;x,y)-f0(x,y)|≤ω(f,δ)Tjk(f0;x,y)+π2ω(f,δ)δ2Tjk(φ;x,y)+|f(x,y)||Tjk(f0;x,y)-f0(x,y)|.
Taking supremum over(x,y)on both sides of the above inequality and let
(40)δ:=δjk=∥Tjk(φ)∥C*(ℝ2).
We obtain
(41)∥Tjk(f)-f∥C*(ℝ2)≤ω(f,δjk)∥Tjk(f0)-f0∥C*(ℝ2)+(1+π2)×ω(f,δjk)+M∥Tjk(f0)-f0∥C*(ℝ2),
whereM:=∥f∥C*(ℝ2). Let Lmn=(1/(m+1)(n+1))∑j=0m‍∑k=0n‍Tjk. Now for a givenε>0define the following sets:
(42)D={(m,n):∥Lmn(f)-f∥C*(ℝ2)≥ε},D1={(m,n):∥Lmn(f0)-f0∥C*(ℝ2)≥ε3},D2={(m,n):ω(f,δmn)≥ε3(1+π2)},D3={(m,n):∥Lmn(f)-f∥C*(ℝ2)≥ε3M}.
ThenD⊂D1∪D2∪D3. Further define
(43)D4={(m,n):ω(f,δmn)≥ε3},D5={(m,n):∥Lmn(f)-f∥C*(ℝ2)≥ε3}.
We see thatD1⊂D4∪D5. ThereforeD⊂⋃i=25‍Di. Therefore, sinceγmn=max{αmn,βmn}, we conclude that for every(j,k)∈ℕ×ℕ(44)δ2(D)≤∑i=25‍δ2(Di).
Using conditions (i) and (ii), we get∥Lmn(f)-f∥C*(ℝ2)=st2(C,1,1)-o(γmn).



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