Generalized Weighted Statistical Convergence of Double Sequences

Filomat 30:3 (2016), 753–762 DOI 10.2298/FIL1603753C

Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.rs/filomat

Generalized Weighted Statistical Convergence of Double Sequences and Applications Muhammed Cinara , Mikail Etb a Department b Department

of Mathematics, Mus Alparslan University, Mus, Turkey of Mathematics, Firat University and Siirt University, Turkey

Abstract. In this paper we introduce the concept generalized weighted statistical convergence of double sequences. Some relations between weighted (λ, µ)−statistical convergence and strong (Nλµ , p, q, α, β)−summablity of double sequences are examined. Furthemore, we apply our new summability method to prove a Korovkin type theorem.

1. Introduction The idea of statistical convergence was formerly defined under the name ”almost convergence” by Zygmund [37] in the first edition of his celebrated monograph published in Warsaw in 1935. The concept was formally introduced by Fast [14] and was reintroduced by Schonberg [35] and also, independently, by Buck [3]. Later the idea was associated with summability theory by Connor [6], Cakallı [7], Et et al. ([10–12, 18]), Duman and Orhan [8], Fridy [15], Is¸ık [19], Mohiuddine et al. ([1, 22–24]), Mursaleen et al. ˇ ([26–28]), Salat [32], Savas¸ ([33, 34]) and many others. Let N be the set of all natural numbers, K ⊆ N and K (n) = {k ≤ n : k ∈ K} . The natural density of K is defined by δ (K) = lim n1 |K (n)| , if the limit exists. The vertical bars indicate the number of the elements in n

enclosed set. A sequence x = (xk ) is said to be statistically convergent to L if the set K (ε) = {k ≤ n : |xk − L| ≥ ε} has natural density zero. A sequence x = (xk ) is said to be statistically Cauchy sequence if for every ε > 0 there exist a number N = N (ε) such that lim n

1 |k ≤ n : |xk − xN | ≥ ε| = 0. n

The notion of weighted statistical convergence was introduced by Karakaya and Chishti [20] as follows: Let pn be a sequence of positive real numbers such that Pn = p0 + p1 + ... + pn → ∞ as n → ∞ and pn , 0, p0 > 0. A sequence x = (xk ) is said to be weighted statistical convergent if for every ε > 0 lim

n→∞

1 k ≤ n : pk |xk − L| ≥ ε = 0. Pn

2010 Mathematics Subject Classification. Primary: 40A05; Secondary: 40C05, 46A45 Keywords. Double sequences, weighted statistical convergence, Korovkin type theorem Received: 12 August 2015; Revised: 09 December 2015; Accepted: 15 December 2015 Communicated by Ljubiˇsa D.R. Koˇcinac and Ekrem Savas¸ Email addresses: [email protected] (Muhammed Cinar), [email protected] (Mikail Et)

M. Cinar, M. Et / Filomat 30:3 (2016), 753–762

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In this case we write SN − lim x = L. We shall denote the set of all weighted statistical convergent sequences by SN . Mursaleen et al. [27] was modified the definition of weighted statistical convergence such as: A sequence x = (xk ) is said to be weighted statistical convergent if for every ε > 0 1 k ≤ Pn : pk |xk − L| ≥ ε = 0. n→∞ Pn lim

Recently Ghosal [17] was added to the definition of weighted statistical convergence the condition lim inf pn > 0. n P Let tn = P1n pk xk, n = 0, 1, 2, 3... . The sequence x = (xk ) is said to be N, pn −summable to L if lim tn = L. n→∞ k=0 A sequence x = (xk ) is said to be N, pn −statistically summable to L if st− lim tn = L [25]. In this case we n→∞

write N (st) − lim x = L. ∞ A double sequence x = x jk is said to be convergent in the Pringsheim sense if for every ε > 0 there j,k=0 exists N ∈ N such that x jk − L < ε, whenever j, k > N. In this case we write P − lim x = L [31]. ∞ A double sequence x = x jk is bounded if there exists a positive number M such that x jk < M for j,k=0

2 all i, j ∈ N. We denote the set of all double sequence by `∞ . bounded Let K ⊆ N × N and K (m, n) = j, k : j ≤ m, k ≤ n . The double natural density of K is defined by

1 |K (m, n)| , if the limit exists. mn A double sequence x = x jk is said to be statistically convergent to L if for every ε > 0 the set n o j, k , j ≤ m and k ≤ n : x jk − L ≥ ε has double natural density zero [28]. In this case we write st2 −lim x = L and we denote the set of all statistically convergent double sequence by st2 . A convergent double sequence is also st−convergent, but the converse is not true general. Also a st−convergent double sequence need not be bounded. For this consider a sequence x = x jk defined by δ2 (K) = P − lim m,n

(

if j and k are square , otherwise then st2 − lim x = 1, but x = x jk neither convergent nor bounded. n o∞ ∞ Let p = p j and q = qk k=0 be sequences of non-negative numbers that are not all zero and let x jk =

jk 1

j=0

αβ

Qn = q1 + q2 + q3 + ... + qn , q1 > 0 and Pm = p1 + p2 + ... + pm , p1 > 0. The weighted mean tmn was defined by t11 mn =

m n 1 XX p j qk x jk P m Qn j=0 k=0

t10 mn =

1 Pm

m X j=0

p j x jn , t01 mn =

n 1 X qk xmk Qn k=0

αβ where m, n ≥ 0 and α, β = (1,1) , (1, 0) or (0, 1). If tmn convergent to L as min (m, n) → ∞ then; we say that αβ a double sequence x = x jk is N, p, q, α, β −summable to L and we show that lim tmn = L. In this case we m,n→∞ write xij → L N, p, q, α, β ([4],[5]). 2. Main Results In this section we generalize the concept of weighted statistical convergence for double sequences.


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Definition 2.1. Let K be a subset of N × N. We define the double weighted density of K by 1 KPm Qn (m, n) , provided the limit exists, m,n Pm Qn n o where KPm Qn (m, n) = j, k , j ≤ Pm and k ≤ Qn : p j qk x jk − L ≥ ε , lim inf pn > 0, lim inf qm > 0. We say that a double sequence x = x jk is said to be weighted statistically convergent or SN2 − convergent to L if for every ε > 0 δN2 (K) = lim

1 m,n→∞ Pm Qn lim

n o j, k , j ≤ Pm and k ≤ Qn : p j qk x jk − L ≥ ε = 0.

In this case we write SN2 − lim x = L. Definition 2.2. Let λ = (λm ) and µ = µn be two non-decreasing sequences of positive real numbers such that each tending to ∞ and λm+1 ≤ λm + 1, λ1 = 0 µn+1 ≤ µn + 1, µ1 = 0. Let p = p j and q = qk be two sequence of non-negative numbers such that p0 > 0, q0 > 0 and Pλm =

X

pj → ∞ m → ∞

j∈Jm

Qµn =

X

qk → ∞ n → ∞

k∈In

where, Jm = [m − λm + 1, m] , In = n − µn + 1, n and we define generalized weighted mean σ11 mn =

XX 1 p j qk x jk Pλm Qµn j∈Jm k∈In

σ10 mn

1 X 1 X = qk xmk p j x jn , σ01 mn = Pλm Q µn j∈Jm

k∈In

αβ A double sequence x = x jk is said to be Nλµ , p, q, α, β −summable to L, if lim σmn = L, where m,n→∞ α, β = (1, 1) , (1, 0) or (0, 1) . h i Definition 2.3. A double sequence x = x jk is said to be strongly Nλµ , p, q, α, β −summable (or Nλµ , p, q, α, β -summable) to L, if XX 1 p j qk x jk − L . m,n→∞ Pλm Qµn lim

j∈Jm k∈In

h i In this case we write x jk → L Nλµ , p, q, α, β . If we take p j = 1 and qk = 1 for all j, k ∈ N in the above definition Nλµ , p, q, α, β −summability reduces to V, λ, µ −summability which were studied Mursaleen et al. [29]. Also if we take p j = 1, qk = 1 for all j, k ∈ N and λm = m, µn = n for all n, m ∈ N , then; Nλµ , p, q, α, β −summability reduces to (C, 1, 1) −summability.


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Definition 2.4. A double sequence x = x jk is said to be weighted λ, µ −statistically convergent (or SN(λ,µ) − convergent) to L if for every ε > 0 1 m,n→∞ Pλm Qµn

n o j, k ; j ≤ Pλ and k ≤ Qµ : p j qk x jk − L ≥ ε = 0. m n

lim

In this case we write SN double sequences by SN

(λ,µ)

(λ,µ)

− lim x = L. We denote the set of all weighted λ, µ −statistically convergent

.

2 Definition 2.5. A double sequence x = x jk is said to be Nλµ , p, q, α, β −statistically summable to L if αβ

2

st2 − lim σmn = L. In this case we write Nλµ (st) − lim x = L. m,n→∞

Theorem 2.6. Let p j qk x jk − L ≤ M for all j, k ∈ N. If a double sequence x = x jk is SN −convergent to L then; (λ,µ) 2 it is Nλµ , p, q, α, β −statistically summable, but the converse is not true. Proof. Let p j qk x jk − L ≤ M for all j, k ∈ N and x = x jk is SN KPλm Qµn (ε) =

n

(λ,µ)

−convergent to L and set

o j, k ; j ≤ Pλm and k ≤ Qµn : p j qk x jk − L ≥ ε .

Then, we can write XX 1 σαβ − L = p q x − L j k jk mn Pλm Qµn j∈Jm k∈In

≤

≤

1 Pλm Qµn M Pλm Qµn

XX

p j qk x jk − L +

j∈Jm k∈In

1 Pλm Qµn

( j,k)∈KPλm Qµn (ε) KPλm Qµn (ε) + ε → ε + 0

XX

p j qk x jk − L

j∈Jm k∈In

( j,k) 0, then; SN2 ⊂ SN

(λ,µ)

.


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Proof. Let x = x jk be SN2 −convergent to L. We may write o 1 n j, k , j ≤ Pm and k ≤ Qn : p j qk x jk − L ≥ ε Pm Qn o 1 n ≥ j, k ; j ≤ P and k ≤ Q : p q x − L ≥ ε λm µn j k jk Pm Qn n o Pλm Qµn 1 j, k ; j ≤ Pλ and k ≤ Qµ : p j qk x jk − L ≥ ε . = m n Pm Qn Pλm Qµn P Qµ − lim x = L. Since lim inf Pλmm Qnn > 0, taking limit as m, n → ∞, we get SN (λ,µ)

m,n

Theorem 2.9. If a double sequence x = x jk

h i is Nλµ , p, q, α, β −summable to L, then; it is SN

convergent to L and the inclusion is strict. h i Proof. Let x = x jk be Nλµ , p, q, α, β −summable to L. Then, for ε > 0 we have XX 1 p j qk x jk − L Pλm Qµn j∈Jm k∈In X X X X 1 1 p j qk x jk − L + p j qk x jk − L = Pλm Qµn Pλm Qµn j∈Jm k∈In j∈Jm k∈In (ε) (ε) j,k ∈K j,k 0, there is δ > 0 such that f (u, v) − f x, y < ε for all (u, v) ∈ K satisfying |u − x| < δ, v − y < δ. Hence we get o 2M n f (u, v) − f x, y < ε + 2 (u − x)2 + v − y 2 δ

(3.3)

Since Tmn is linear and positive and by (3.3) we obtain Tmn f ; x, y − f x, y = Tmn f (u, v) − f x, y ; x, y − f x, y Tmn f0 ; x, y − f0 x, y ≤ Tmn f (u, v) − f x, y ; x, y + M Tmn f0 ; x, y − f0 x, y 2 o 2M n ≤ Tmn ε + 2 (u − x)2 + v − y ; x, y + M Tmn f0 ; x, y − f0 x, y δ 2M 2 ≤ ε + M + 2 A + B2 Tmn f0 ; x, y − f0 x, y δ 4M 4M + 2 A Tmn f1 ; x, y − f1 x, y + 2 B Tmn f2 ; x, y − f2 x, y δ δ 2M + 2 Tmn f3 ; x, y − f3 x, y + ε δ where A = max |x| , B = max y . Taking supremum over x, y ∈ K we have n

o

Tmn f − f

Tmn f0 − f0

Tmn f1 − f1

Tmn f2 − f2

Tmn f3 − f3

≤ R + + + + ε, C(K) C(K) C(K) C(K) C(K) where 4M 4M 2M 2M R = max ε + M + 2 A2 + B2 , 2 A, 2 B, 2 . δ δ δ δ Hence 3 X

Tmn f ; x, y pm qn − f x, y

≤ ε + R Tmn fi ; x, y pm qn − fi x, y C(K) C(K)

(3.4)

i=0

Now replace Tmn .; x, y pm qn by Lmn .; x, y =

1 Pλm Qµn

X

Tmn .; x, y pm qn

(m,n)∈Jm ×In

in (3.4) . For a given r > 0 choose ε0 > 0 such that ε0 < r. Define the following sets n o

D = (m, n) ∈ N2 : Lmn f − f C(K) ≥ r ,

r − ε0 , i = 0, 1, 2, 3. Di = (m, n) ∈ N2 : Lmn fi − fi C(K) ≥ 4R Then, D ⊂

3 S

Di and so δ (D) ≤ δ (D0 ) + δ (D1 ) + δ (D2 ) + δ (D3 ) . Therefore, using conditions (3.4) we get

i=0

2 Nλµ (st) − lim Tmn f − f C(K) = 0 m,n

This completes the proof.


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Corollary 3.2. Let Lm,n be a sequence of positive linear operators acting from C (K) into itself. Then, for all f ∈ C (K) ,

P − lim Lmn f − f C(K) = 0 m,n

if and only if

P − lim Lmn fi − fi C(K) = 0, (i = 0, 1, 2, 3) , m,n

where f0 x, y = 1, f1 x, y = x , f2 x, y = y and f3 x, y = x2 + y2 [36]. Remark 3.3. We now construct an example of sequence of positive linear operators of two variables satisfying the conditions of Theorem 3.1, but that does not satisfy the conditions of the Korovkin Theorem. For this claim, we consider the following Brenstein operators defined as follows Bm,n

! n m P n− j P k j k k j , Cm x (1 − x)m−k Cn y j 1 − y , f f ; x, y = m n k=0 j=0

where x, y ∈ [0, 1] × [0, 1] . Let Bm,n f0 ; x, y = 1 Bm,n f1 ; x, y = x Bm,n f2 ; x, y = y x − x2 y − y2 Bm,n f3 ; x, y = x2 + y2 + + m n Then by Corllary 3.2 we can write for all f ∈ C (K)

P − lim Bm,n f − f C(K) = 0 m,n

Let the sequence Tm,n : C (K) → C (K) with Tm,n f ; x, y = (1 + umn ) Bm,n f ; x, y where umn = (−1)n for all m. Let pm = 1, qn = 1, λm = m, µn = n. The double mn ) is neither P−convergent nor statistically sequence (u 2

convertgent, but (umn ) is statistically summable Nλµ , p, q, α, β to zero. 2 y−y2 Bm,n 1; x, y = 1, Bm,n x; x, y = x, Bm,n y; x, y = y, Bm,n x2 + y2 ; x, y = x2 + y2 + x−x m + n and the double sequence Tm,n satisfies condition (3.2) for i = 0, 1, 2, 3. Hence we have

2 Nλµ (st) − lim Tmn f − f C(K) = 0. m,n

On the other hand, we get Tm,n f, 0, 0 = (1 + umn ) Bm,n f ; 0, 0 since Bm,n f ; 0, 0 = f (0, 0) and hence

Tmn f ; x, y − f x, y

Tmn f ; x, y − f x, y ≥ um,n f (0, 0) . ≥ C(K) We see that Tm,n does not satisfy classical Korovkin type theorem since lim umn and st2 − lim umn do not exists this proves the claim.


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