# A Generalization of Lacunary Equistatistical Convergence of Positive Linear Operators

Authors: Yusuf Kaya; Nazmiye Gönül
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

In this paper we consider some analogs of the Korovkin approximation theorem via lacunary equistatistical convergence. In particular we study lacunary equi-statistical convergence of approximating operators onHw2 spaces, the spaces of all real valued continuous functions f dened on K=0,∞m and satisfying some special conditions.


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## Body

## 1. Introduction

Approximation theory has important applications in the theory of polynomial approximation, in various areas of functional analysis, numerical solutions of integral and differential equations [1–6]. In recent years, with the help of the concept of statistical convergence, various statistical approximation results have been proved [7]. In the usual sense, every convergent sequence is statistically convergent, but its converse is not always true. And, statistical convergent sequences do not need to be bounded.Recently, Aktuğlu and Gezer [8] generalized the idea of statistical convergence to lacunary equi-statistical convergences. In this paper, we first study some Korovkin type approximation theorems via lacunary equi- statistical convergence in Hw2 spaces. Then using the modulus of continuity, we study rates of lacunary equi-statistically convergence in Hw2.We recall here the concepts of equi-statistical convergence and lacunary equi-statistical convergence.Letf and fr  belong to C(X), which is the space of all continuous real valued functions on a compact subset X of the real numbers. {fr} is said to be equi-statistically convergent to f on X and denoted by fr↠f(equistat) if for every ε>0, the sequence of real valued functions
(1)pr,ε(x):=1r|{m≤r:fm(x)-f(x)≥ε}|
converges uniformly to the zero function on X, which means that
(2)limr→∞∥pr,ε(·)∥C(X)=0.
A lacunary sequence θ={kr} is an integer sequence such that
(3)k0=0,hr=kr-kr-1⟶∞asr⟶∞.In this paper the intervals determined byθ will  be denoted by Ir=(kr-1,kr], and the ratio kr/kr-1 will be abbreviated by qr.Letθ be a lacunary sequence then {fr}r∈ℕ is said to be lacunary equi-statistically convergent to f on X and denoted by fr↠f(θ-equistat) if for every ε>0, the sequence of real valued functions {sr,ε}n∈ℕ defined by
(4)sr,ε(x):=1hr|{m∈Ir:fm(x)-f(x)≥ε}|
uniformly converges to zero function on X, which means that
(5)limn→∞∥sr,ε(·)∥C(X)=0.A Korovkin type approximation theorem by means of lacunary equi-statistical convergence was given in [8]. We can state this theorem now. An operator L defined on a linear space of functions Y is called linear if L(αf+βg,x)=αL(f,x)+βL(g,x), for all f,g∈Y, α,β∈ℝ and is called positive, if L(f,x)≥0, for all f∈Y, f≥0. Let X be a compact subset of ℝ, and let C(X)  be the space of all continuous real valued functions on X.Lemma 1 (see [8]).
Letθ be a lacunary sequence, and let Lr:C(X)→C(X) be a sequence of positive linear operators satisfying
(6)Lr(tv,x)↠xν,(θ-equistat),v=0,1,2,
then for all f∈C(X),
(7)Lr(f,x)↠f,(θ-equistat).We turn to introducing some notation and the basic definitions used in this paper. Throughout this paperI=[0,∞). Let
(8)C(I)≔{f:fisareal-valuedcontinuousfunctiononI},
and
(9)CB(I):={f∈C(I):fisboundedfunctiononI}.
Consider the space Hw of all real-valued functions f defined on I and satisfying
(10)|f(x)-f(y)|≤w(f;|x1+x+y1+y|),
where w is the modulus of continuity defined by
(11)w(f;δ):=supx,y∈I|x-y|≤δ|f(x)-f(y)|,foranyδ>0
(see [9]). Let K:=I2=[0,∞)×[0,∞), then the norm on CB(K) is given by
(12)∥f∥:=sup(x,y)∈K|f(x,y)|,f∈CB(K),
and also denote the valued of Lf at a point (x,y)∈K is denoted by L(f;x,y) [10, 11].w
2
(
f
;
δ
1
,
δ
2
) is the type of modulus of continuity for the functions of two variables satisfying the following properties: for any real numbers δ1, δ2, δ1′, δ2′, δ1′′, and δ2′′>0,(i)
w2(f;δ1,·) and w2(f;·,δ2) are nonnegative increasing functions on [0,∞),(ii)
w2(f;δ1′+δ1′′,δ2)≤w2(f;δ1′,δ2)+w2(f;δ1′′,δ2),(iii)
w2(f;δ1,δ2′+δ2′′)≤w2(f;δ1,δ2′)+w2(f;δ1,δ2′′),(iv)
limδ1,δ2→0w2(f;δ1,δ2)=0.The spaceHw2 is of all real-valued functions f defined on K and satisfying
(13)|f(u,v)-f(x,y)|≤w2(f;|u1+u-x1+x|,|v1+v-y1+y|).
It is clear that any function in Hw2 is continuous and bounded on K.


## 2. Lacunary Equistatistical Approximation

In this section, using the concept of Lacunary equistatistical convergence, we give a Korovkin type result for a sequence of positive linear operators defined onC(Im), the space of all continuous real valued functions on the subset Im of Rm, and the real m-dimensional space. We first consider the case of m=2. Following [7] we can state the following theorem.Theorem 2.
Letθ={kr} be a lacunary sequence, and let {Lr} be a sequence of positive linear operators from Hw2 into CB(K). Lr is satisfying Lr(fv;x,y)↠fv(x,y)(θ-equistat), v=0,1,2, where fk(u,v)∈Hw2, k=0,1,2,3,
(14)f0(u,v)=1,f1(u,v)=uu+1,f2(u,v)=vv+1,f3(u,v)=(uu+1)2+(vv+1)2,
then for all f∈Hw2,
(15)Lr(f;x,y)↠f(x,y),(θ-equistat).Proof.
Let(x,y)∈K be a fixed point, f∈Hw2, and assume that (14) holds. For every ε>0, there exist δ1,δ2>0 such that |f(u,v)-f(x,y)|<ε holds for all (u,v)∈K satisfying
(16)|uu+1-xx+1|<δ1,|vv+1-yy+1|<δ2.
Let
(17)Kδ1,δ2:={(u,v)∈K:|u1+u-x1+x|<δ1,|vv+1-yy+1|<δ2}.
Hence,
(18)|f(u,v)-f(x,y)|=|f(u,v)-f(x,y)|χKδ1,δ2(u,v)+|f(u,v)-f(x,y)|χK∖Kδ1,δ2(u,v)<ε+2MχK∖Kδ1,δ2(u,v),
where χP denotes the characteristic function of the set P. Observe that
(19)χK∖Kδ1,δ2(u,v)≤1δ12(u1+u-x1+x)2+1δ22(vv+1-yy+1)2.
Using (18),  (19), and M:=∥f∥ we have
(20)|f(u,v)-f(x,y)|≤ε+2Mδ2{(u1+u-x1+x)2+(vv+1-yy+1)2},
where δ:=min{δ1,δ2}.
By the linearity and positivity of the operators{Lr} and by (18), we have
(21)Lr((f1-u1+uf0)2+(f2-v1+vf0)2;x,y)≤Lr(f3;x,y)≤-2[x1+xLr(f1;x,y)+y1+yLr(f2;x,y)]Lr(f3;x,y)≤-2[x1+xLr(f1;x,y)+y1+yLr(f2;x,y)]≤+[(x1+x)2+(y1+y)2]Lr(f0;x,y).
Hence, we get
(22)|Lr(f;x,y)-f(x,y)|≤Lr(|f(u,v)-f(x,y)|;x,y)≤+|f(x,y)||Lr(f0;x,y)-f0(x,y)|≤Lr(ε+2Mδ2{(vv+1-y1+y)2(u1+u-x1+x)2+(vv+1-y1+y)2};x,y2Mδ2)≤+M|Lr(f0;x,y)-f0(x,y)|≤Lr(ε+2Mδ2{(f2-y1+y.f0)2(f1-x1+x·f0)2+(f2-y1+y·f0)2};x,y2Mδ2)≤+M|Lr(f0;x,y)-f0(x,y)|=Lr(ε;x,y)+Lr((f2-y1+yf0)22Mδ2(f1-x1+xf0)2+(f2-y1+yf0)2;x,y2Mδ2)≤+M|Lr(f0;x,y)-f0(x,y)|≤ε+2Mδ2|Lr(f3;x,y)-f3(x,y)|≤+4Mδ2|Lr(f2;x,y)-f2(x,y)|≤+4Mδ2|Lr(f1;x,y)-f1(x,y)|≤+(ε+M+4Mδ2)|Lr(f0;x,y)-f0(x,y)|=2Mδ2|Lr(f3;x,y)-f3(x,y)|≤+4Mδ2|Lr(f2;x,y)-f2(x,y)|≤+4Mδ2|Lr(f1;x,y)-f1(x,y)|≤+ε+N|Lr(f0;x,y)-f0(x,y)|,
where N:=ε+M+4M/δ2. For a given μ>0, choose ε>0 such that ε<μ. Define the following sets:(23)Dμ(x,y):={m∈ℕ:|Lm(f;x,y)-f(x,y)|≥μ},Dμv(x,y):={m∈ℕ:|Lm(f;x,y)-f(x,y)|≥μ-ε4N},
where  v=0,1,2,3. Then from (22) we clearly have
(24)Dμ(x,y)⊆⋃v=03Dμv(x,y).
Therefore define the following real valued functions:
(25)sr,μ(x,y):=1hr|{m∈Ir:|Lm(f;x,y)-f(x,y)|≥μ}|,sr,μv(x,y):=1hr|{m∈Ir:|Lm(fv;x,y)-f(x,y)|≥μ-ε4N}|,
where  v=0,1,2,3. Then by the monotonicity and (24) we get
(26)sr,μ(x,y)≤∑v=03‍sr,μv(x,y)
for all x∈X, and this implies the inequality
(27)∥sr,μ(·)∥K≤∑v=03‍∥sr,μv(·)∥K.
Taking limit in (27) as r→∞ and using (14) we have
(28)limr→∞∥sr,μ(·)∥K=0.
Then for all f∈Hw2, we conclude that
(29)Lr(f;x,y)↠f(x,y),(θ-equistat).Now replaceI2 by Im=[0,∞)×⋯×[0,∞) and by an induction, we consider the modulus of continuity type function wm as in the function w2. Then let Hwm be the space of all real-valued functions f satisfying
(30)|f(u1,u2,…,um)-f(x1,x2,…,xm)|≤w2(f;|u1u1+1-x1x1+1|,…,|umum+1-xmxm+1|).Therefore, using a similar technique in the proof of Lemma1 one can obtain the following result immediately.Theorem 3.
Letθ={kr} be a lacunary sequence, and let {Lr} be a sequence of positive linear operators from Hwm into CB(Im). Lr is satisfying
(31)Lr(fv;x,y)↠fv(x,y),(θ-equistat),fv(x,y),5.(θ-eq)v=0,1,2,…,m+1,
where fk(u1,u2,…,um)∈Hwm, k=0,1,2,…m+1,
(32)f0(u1,u2,…,um)=1,f1(u1,u2,…,um)=u1u1+1,⋮fm(u1,u2,…,um)=umum+1,fm+1(u1,u2,…,um)=(u1u1+1)2+(u2u2+1)2+⋯+(umum+1)2.
Then for all f∈Hwm,
(33)Lr(f;u1,u2,…,um)↠f(u1,u2,…,um),(θ-equistat).
Assume that I=[0,∞), K:=I×I. One considers the following positive linear operators defined on Hw2(K):
(34)Bn(f;x,y)=1(1+x)n(1+y)n×∑k=0n∑l=0nf(kn-k+1,ln-l+1)(nk)(nl)xkyl,
where f∈Hw2, (x,y)∈K and n∈ℕ.Lemma 4.
Letθ={kr} be a lacunary sequence, and let
(35)Bn(f;x,y)=1(1+x)n(1+y)n×∑k=0n∑l=0nf(kn-k+1,ln-l+1)(nk)(nl)xkyl
be a sequence of positive linear operators from Hw2 into CB(K). If Bn is satisfying
(36)Bn(fv;x,y)↠fv(x,y),(θ-equistat),v=0,1,2,3,f0(u,v)=1,f1(u,v)=uu+1,f2(u,v)=νν+1,f3(u,v)=(uu+1)2+(νν+1)2,
then for all f∈Hw2,
(37)Bn(f;x,y)↠f(x,y),(θ-equistat).Proof.
Assume that (36) holds, and let f∈Hw2. Since
(38)(1+x)n=∑k=0n(nk)xk,(1+y)n=∑l=0n(nl)yl,
it is clear that, for all n∈ℕ,
(39)Bn(f0;x,y)=1(1+x)n(1+y)n∑k=0n∑l=0n(nk)xk(nl)yk=1(1+x)n(1+y)n(∑k=0n(nk)xk)∑l=0n(nl)yk=1.
Now, by assumption we have
(40)Bn(f0;x,y)↠f0(x,y),(θ-equistat).
Using the definition of Bn, we get
(41)Bn(f1;x,y)=1(1+x)n(1+y)n=×∑k=1nk╱(n-k+1)(k╱(n-k+1))+1(nk)xk∑l=0n(nl)yl=1(1+x)n(1+y)n∑k=1nkn+1(nk)xk∑l=0n(nl)yl=1(1+x)n∑k=0n-1k+1n+1(nk+1)xk+1=x(1+x)n∑k=0n-1k+1n+1(nk+1)xk.
Since
(42)(nk+1)=(n-1k)nk+1
we get
(43)Bn(f1;x,y)=x(1+x)n∑k=0n-1k+1n+1nk+1(n-1k)xk=x(1+x)n∑k=0n-1nn+1(n-1k)xk=x(1+x)(1+x)n-1·nn+1·∑k=0n-1(n-1k)xk=nn+1(xx+1).
So, we have
(44)|Bn(f1;x,y)-f1(x,y)|=xx+1|nn+1-1|.
The fact that limn→∞(n/(n+1))=1 and using a similar technique as in the proof of Lemma 1, we get
(45)limr→∞∥1hr|{m∈Ir:|Bn(f1;x,y)-f1(x,y)|≥ε}|∥=0.
Hence we have
(46)Bn(f2;x,y)↠f2(x,y),(θ-equistat).
Also we have
(47)Bn(f3;x,y)↠f3(x,y),(θ-equistat).
To see this, by the definition of Bn, we first write(48)Bn(f3;x,y)=1(1+x)n(1+y)n∑k=0n∑l=0n(nk)(nl)xkyl×[k2(n+1)2+l2(n+1)2]=1(1+x)n(1+y)n∑k=1nk2(n+1)2(nk)xk∑l=0n(nl)yl+1(1+x)n(1+y)n∑k=0n(nk)xk∑l=1∞l2(n+1)2(nl)yl=1(1+x)n∑k=1nk(n+1)2xk+1(1+x)n∑k=2nk(k-1)(n+1)2(nk)xk+1(1+y)n∑l=2nl(l-1)(n+1)2(nl)yl+1(1+y)n∑l=1nl(n+1)2yl.
Then,
(49)Bn(f3;x,y)=n(n-1)(n+1)2x2(x+1)2+n(n+1)2x(x+1)+n(n+1)2yy+1+y2(y+1)2n(n-1)(n+1)2
which implies that
(50)|Bn(f3;x,y)-f3(x,y)|=(x2(x+1)2+y2(y+1)2)|n(n-1)(n+1)2-1|=+n(n+1)2|xx+1+yy+1|.
Since
(51)limn→∞n(n-1)(n+1)2=1,limn→∞n(n+1)2=0,
we get
(52)limr→∞∥1hr|{m∈Ir:|Bn(f3;x,y)-f3(x,y)|≥ε}|∥=0.
Thus Bn(f3;x,y)↠f3(x,y),(θ-equistat). Therefore we obtain that for all f∈Hw2, Bn(f;x,y)↠f(x,y),(θ-equistat).


## 3. Rates of Lacunary Equistatistical Convergence

In this section we study the order of lacunary equi-statistical convergence of a sequence of positive linear operators acting onHw2(K), where K=Im. To achieve this we first consider the case of m=2.Definition 5.
A sequence{fr} is called lacunary equi-statistically convergent to a function f with rate 0<β<1 if for every ε>0,
(53)limr→∞sr,ε(x,y)r-β=0,
where sr,ε(x,y) is given in Lemma 1. In this case it is denoted by
(54)fr-f=o(r-β),(θ-equistat)onK=I×I.Lemma 6.
Let{fr} and {gr} be two sequences of functions in Hw2(K), with
(55)fr-f=o(r-β1),(θ-equistat),gr-g=o(r-β2),(θ-equistat).
Then one has
(56)(fr+gr)-(f+g)=o(r-β),(θ-equistat),
where β=min{β1,β2}.Proof.
Assume thatfr-f=o(r-β1), (θ-equistat) and gr-g=o(r-β2), (θ-equistat) on K. For all ε>0, consider the following functions:
(57)sr,ε(x,y)≔1hr|{n∈Ir:|(fn+gn)(x,y)-(f+g)(x,y)|≥ε}|,sr,ε1(x,y):=1hr|{n∈Ir:|fn(x)-f(x)|≥ε2}|,sr,ε2(x,y):=1hr|{n∈Ir:|gn(x)-g(x)|≥ε2}|.
Then we have
(58)limr→∞∥sr,ε(x,y)∥Hw2(K)r-β=limr→∞∥sr,ε1(x,y)∥r-βlimr→∞∥sr,ε(x,y)∥Hw2(K)r-β=+limr→∞∥sr,ε2(x,y)∥r-β,sr,ε(x,y)r-β≤sr,ε1(x,y)r-β+sr,ε2(x,y)r-βsr,ε(x,y)r-β≤sr,ε1(x,y)r-β1+sr,ε2(x,y)r-β2,
and hence
(59)∥sr,ε(x,y)∥Hw2(K)r-β≤∥sr,ε1(x,y)∥Hw2(K)r-β1+∥sr,ε2(x,y)∥Hw2(K)r-β2.
Taking limit as r→∞ and using the assumption complete the proof.Now we give the rate of lacunary equi-statistical convergence of a positive linear operatorsLr(f;x,y) to f(x,y) with the help of modulus of continuity.Theorem 7.
LetK=I×I, and let Lr:Hw2(K)→Hw2(K) be a sequence of positive linear operators. Assume that(i)
Lr(f0;x,y)-f0=o(r-β1),(θ-equistat) on K,(ii)
w(f;δr,x,δr,y)=o(r-β2),(θ-equistat) on K with
(60)δr,x=Lr((u1+u-x1+x)2,x),δr,y=Lr((v1+v-y1+y)2,y).Then
(61)Lr(f;x,y)-f(x,y)=o(r-β),(θ-equistat)onK,
where β=min{β1,β2}.Proof.
Letf∈Hw2(K) and x∈K. Use
(62)|Lr(f;x,y)-f(x,y)|≤Lr(|f(u,v)-f(x,y)|;x,y)≤+|f(x,y)||Lr(f0;x,y)-f(x,y)|≤Lr(w2(f;|u1+u-x1+x|,|v1+v-y1+y|);x,y)≤+|f(x,y)||Lr(f0;x,y)-f0(x,y)|≤(1+Lr(f0;x,y))w2(f;δr,x,δr,y)≤+M|Lr(f0;x,y)-f0(x,y)|=2w2(f;δr,x,δr,y)+M|Lr(f0;x,y)-f0(x,y)|≤+w2(f;δr,x,δr,y)|Lr(f0;x,y)-f0(x,y)|,
where M=∥f∥Hw2(K). Using inequality (62),  conditions (i) and (ii) we get
(63)limr→∞1r-β∥1hr|{r∈Ir:|Lr(f;x,y)-f(x,y)|≥ε}|∥=0,
so we have
(64)Lr(f;x,y)-f(x,y)=o(r-β),(θ-equistat)onK.Finally we give the rate of lacunary equi-statistical convergence for the operatorsLr(f,x) by using the Peetre's K-functional in the space Hw2(K). The Peetre K-functional of function f∈Hw2(K) is defined by
(65)K(f;δr,x,δr,y)=infg∈Hw2(K){∥f-g∥CB(K)+δ∥g∥CB(K)},
where
(66)∥f∥CB(K)=sup(x,y)∈K|f(x,y)|.Theorem 8.
Letf∈Hw2(K) and {K(f;δr,x,δr,y)} be the sequence of Peetre's K-functional. If
(67)δr,x=∥Lr((f1-x1+x);x,y)∥CB(K)+∥Lr((f1-x1+x)2;x,y)∥CB(K)∥g∥CB(K),δr,y=∥Lr((f2-y1+y);x,y)∥CB(K)+∥Lr((f2-y1+y)2;x,y)∥CB(K)∥g∥CB(K),limr→∞∥δr,x∥=0,(θ-equistat)limr→∞∥δr,y∥=0,(θ-equistat)
on x,y∈K, then
(68)∥Lr(f;x,y)-f(x,y)∥CB(K)≤K(f;δr,x,δr,y).Proof.
For eachg∈Hw2(K), we get
(69)∥Lr(g;x,y)-g(x,y)∥CB(K)≤∥g∥CB(K)(∥Lr((f1-x1+x);x,y)∥CB(K)+∥Lr((f1-x1+x)2;x,y)∥∥Lr((f1-x1+x);x,y)∥CB(K))≤+∥g∥CB(K)(∥Lr(f2-y1+y);x,y∥CB(K))≤+∥Lr((f2-y1+y)2);x,y∥=(δr,x,δr,y)∥g∥CB(K).



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