Filomat 30:3 206, 753 762 DOI 02298/FIL603753C Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: http://wwwpmfiacrs/filomat Geeralized Weighted Statistical Covergece of Double Sequeces ad Applicatios Muhammed Ciar a, Mikail Et b a Departmet of Mathematics, Mus Alparsla Uiversity, Mus, Turkey b Departmet of Mathematics, Firat Uiversity ad Siirt Uiversity, Turkey Abstract I this paper we itroduce the cocept geeralized weighted statistical covergece of double sequeces Some relatios betwee weighted λ, µ statistical covergece ad strog N λµ, p, q, α, β summablity of double sequeces are examied Furthemore, we apply our ew summability method to prove a Korovki type theorem Itroductio The idea of statistical covergece was formerly defied uder the ame almost covergece by Zygmud [37] i the first editio of his celebrated moograph published i Warsaw i 935 The cocept was formally itroduced by Fast [4] ad was reitroduced by Schoberg [35] ad also, idepedetly, by Buck [3] Later the idea was associated with summability theory by Coor [6], Cakallı [7], Et et al [0 2, 8], Duma ad Orha [8], Fridy [5], Işık [9], Mohiuddie et al [, 22 24], Mursalee et al [26 28], Šalat [32], Savaş [33, 34] ad may others Let N be the set of all atural umbers, K N ad K = {k : k K The atural desity of K is defied by δ K = K, if the it exists The vertical bars idicate the umber of the elemets i eclosed set A sequece x = x k is said to be statistically coverget to L if the set K ε = {k : x k L ε has atural desity zero A sequece x = x k is said to be statistically Cauchy sequece if for every ε > 0 there exist a umber N = N ε such that k : x k x N ε = 0 The otio of weighted statistical covergece was itroduced by Karakaya ad Chishti [20] as follows: Let p be a sequece of positive real umbers such that P = p 0 + p + + p as ad p 0, p 0 > 0 A sequece x = x k is said to be weighted statistical coverget if for every ε > 0 P { k : pk x k L ε = 0 200 Mathematics Subject Classificatio Primary: 40A05; Secodary: 40C05, 46A45 Keywords Double sequeces, weighted statistical covergece, Korovki type theorem Received: 2 August 205; Revised: 09 December 205; Accepted: 5 December 205 Commuicated by Ljubiša DR Kočiac ad Ekrem Savaş Email addresses: muhammedciar23@gmailcom Muhammed Ciar, mikailet68@gmailcom Mikail Et
M Ciar, M Et / Filomat 30:3 206, 753 762 754 I this case we write S N x = L We shall deote the set of all weighted statistical coverget sequeces by S N Mursalee et al [27] was modified the defiitio of weighted statistical covergece such as: A sequece x = x k is said to be weighted statistical coverget if for every ε > 0 P { k P : p k x k L ε = 0 Recetly Ghosal [7] was added to the defiitio of weighted statistical covergece the coditio if p > 0 Let t = P p k x k, = 0,, 2, 3 The sequece x = x k is said to be N, p summable to L if t = L k=0 A sequece x = x k is said to be N, p statistically summable to L if st t = L [25] I this case we write N st x = L A double sequece x = x jk is said to be coverget i the Prigsheim sese if for every ε > 0 there j,k=0 exists N N such that xjk L < ε, wheever j, k > N I this case we write P x = L [3] A double sequece x = x jk j,k=0 is bouded if there exists a positive umber M such that xjk < M for all i, j N We deote the set of all bouded double sequece by l 2 Let K N N ad K m, = { j, k : j m, k The double atural desity of K is defied by δ 2 K = P K m,, if the it exists m, m A double sequece x = x jk is said to be statistically coverget to L if for every ε > 0 the set { j, k, j m ad k : xjk L ε has double atural desity zero [28] I this case we write st2 x = L ad we deote the set of all statistically coverget double sequece by st 2 A coverget double sequece is also st coverget, but the coverse is ot true geeral Also a st coverget double sequece eed ot be bouded For this cosider a sequece x = x jk defied by x jk = { jk if j ad k are square otherwise, the st 2 x =, but x = x jk either coverget or bouded Let p = { p j j=0 ad q = { q k k=0 be sequeces of o-egative umbers that are ot all zero ad let Q = q + q 2 + q 3 + + q, q > 0 ad P m = p + p 2 + + p m, p > 0 The weighted mea t αβ m was defied by t m = t 0 m = P m j=0 j=0 x jk k=0 p j x j, t 0 m = Q q k x mk k=0 where m, 0 ad α, β =,,, 0 or 0, If t αβ m coverget to L as mi m, the; we say that a double sequece x = x jk is N, p, q, α, β summable to L ad we show that m, tαβ m = L I this case we write x ij L N, p, q, α, β [4],[5] 2 Mai Results I this sectio we geeralize the cocept of weighted statistical covergece for double sequeces
M Ciar, M Et / Filomat 30:3 206, 753 762 755 Defiitio 2 Let K be a subset of N N We defie the double weighted desity of K by δ N2 K KPm = Q m, P m Q m,, provided the it exists, where K Pm Q m, = { j, k, j Pm ad k Q : xjk L ε, if p > 0, if q m > 0 We say that a double sequece x = x jk is said to be weighted statistically coverget or SN2 coverget to L if for every ε > 0 m, { j, k, j Pm ad k Q : xjk L ε = 0 I this case we write S N2 x = L Defiitio 22 Let λ = λ m ad µ = µ be two o-decreasig sequeces of positive real umbers such that each tedig to ad λ m+ λ m +, λ = 0 µ + µ +, µ = 0 Let p = p j ad q = qk be two sequece of o-egative umbers such that p0 > 0, q 0 > 0 ad P λm = p j m Q µ = q k k I where, J m = [m λ m +, m], I = [ µ +, ] ad we defie geeralized weighted mea σ m = x jk σ 0 m = p j x j, σ 0 m = q k x mk P λm Q µ A double sequece x = x jk is said to be Nλµ, p, q, α, β summable to L, if α, β =,,, 0 or 0, m, σαβ m = L, where Defiitio 23 A double sequece x = x jk is said to be strogly Nλµ, p, q, α, β summable or [ N λµ, p, q, α, β ] -summable to L, if m, x jk L k I I this case we write x jk L [ N λµ, p, q, α, β ] If we take p j = ad q k = for all j, k N i the above defiitio N λµ, p, q, α, β summability reduces to V, λ, µ summability which were studied Mursalee et al [29] Also if we take pj =, q k = for all j, k N ad λ m = m, µ = for all, m N, the; N λµ, p, q, α, β summability reduces to C,, summability
M Ciar, M Et / Filomat 30:3 206, 753 762 756 Defiitio 24 A double sequece x = x jk is said to be weighted λ, µ statistically coverget or SNλ,µ coverget to L if for every ε > 0 m, { j, k ; j Pλm ad k Q µ : xjk L ε = 0 I this case we write S Nλ,µ x = L We deote the set of all weighted λ, µ statistically coverget double sequeces by S Nλ,µ Defiitio 25 A double sequece x = x jk is said to be N 2 λµ, p, q, α, β statistically summable to L if st 2 m, σαβ m = L I this case we write N 2 λµ st x = L xjk Theorem 26 Let L M for all j, k N If a double sequece x = xjk is SNλ,µ coverget to L the; it is N 2 λµ, p, q, α, β statistically summable, but the coverse is ot true Proof Let xjk L M for all j, k N ad x = xjk is SNλ,µ coverget to L ad set K Pλm Q µ ε = { j, k ; j Pλm ad k Q µ : xjk L ε The, we ca write σ αβ m L = x jk L xjk L + j,k K Pλm Qµ ε M KPλm Q µ ε + ε ε + 0 j,k K Pλm Qµ ε xjk L as m, which implies that σ αβ m L For the coverse, cosider a sequece defied by x = x jk = j for all k N Let p j =, q k =, λ m = m, µ = for all j, k,, m N The, the sequece x = x jk is N 2 λµ, p, q, α, β statistically summable to zero, but x = x jk is ot SNλ,µ coverget Defiitio 27 A double sequece x = [ x jk is said to be N, p, q, α, β summable to L if ]r m, xjk L r = 0, 0 < r < j= k= ad we write x ij L [ N, p, q, α, β ] r Pλm Q µ Theorem 28 If if m, > 0, the; SN2 S Nλ,µ
Proof Let x = x jk be SN2 coverget to L We may write M Ciar, M Et / Filomat 30:3 206, 753 762 757 { j, xjk k, j Pm ad k Q : L ε { j, xjk k ; j Pλm ad k Q µ : L ε = P λ m Q µ { j, xjk k ; j Pλm ad k Q µ : L ε Pλm Q µ Sice if m, > 0, takig it as m,, we get SNλ,µ x = L Theorem 29 If a double sequece x = x jk is [ Nλµ, p, q, α, β ] summable to L, the; it is S Nλ,µ statistically coverget to L ad the iclusio is strict Proof Let x = [ x jk be Nλµ, p, q, α, β ] summable to L The, for ε > 0 we have x jk L = xjk L xjk + L j,k K Pλm Qµ ε j,k K Pλm Qµ ε { j, xjk k, j Pλm ad k Q µ : L ε ad this implies that S Nλ,µ x = L To show the iclusio is strict cosider the followig example: Let λ m = m, µ =, p j = j, q k = k for all j, k,, m N ad defie a sequece by x jk = { jk j ad k square 0 otherwise The, P m = p j = j= ad so we have but m m +, Q = 2 q k = k= + 2 { j, xjk k, j Pm ad k Q : 0 ε mm+ 2 j= + 2 xjk k= 0 as m,,
M Ciar, M Et / Filomat 30:3 206, 753 762 758 Theorem 20 Let xjk L M for all j, k N If a double sequece x = xjk is weighted λ, µ statistically coverget to L,the; it is [ N λµ, p, q, α, β ] summable to L, hece x = x jk is N, p, q, α, β summable to L Proof The first implicatio is obvious O the other had, we have j= = xjk L k= m λ m j= m λ m µ k= µ j= k= 2 x jk L k I xjk L + xjk L + Hece x is N, p, q, α, β -summable to L x jk L k I x jk L k I Defiitio 2 A double sequece x = x jk is said to be strogly weighted λ, µ r coverget if x jk L r = 0, 0 < r < m, k I I this case we write x jk L [ N λ,µ, p, q, α, β ] r Theorem 22 Let a double sequece x = x jk is strogly weighted λ, µ r coverget to L If the followig coditios are provided, the; x = x jk is weighted λ, µ statistically coverget to L Case : 0 < r < ad xjk L < Case2 : r < ad xjk L < Proof Sice xjk L r pj q k xjk L for Case ad Case2 the we ca write x jk L r x jk L xjk L j,k K Pλm Qµ ε ε KPλm Q µ ε Takig it as m,, we get x = x jk is weighted λ, µ statistically coverget to L, where KPλm Q µ ε = { j, k, j Jm ad k I : xjk L ε Theorem 23 Let a double sequece x = x jk is weighted λ, µ statistically coverget to L ad pj q k xjk L M If the followig cases are provided the; x = [ x jk is Nλ,µ, p, q, α, β ] summable to L r
M Ciar, M Et / Filomat 30:3 206, 753 762 759 Case : 0 < r < ad < M < Case2 : r < ad 0 M < Proof Suppose that x = x jk is weighted λ, µ statistically coverget to L Sice pj q k xjk L M we ca write x jk L r xjk = L r + j,k K Pλm Qµ ε j,k K Pλm Qµ ε M KPλm Q µ ε + xjk L + j,k K c P λm Qµ ε j,k K c P λm Qµ ε ε K c P P λm Q λm Q µ ε 0 as m,, µ xjk L r xjk L where K c ε = { j, k, j Jm ad k I : xjk L < ε Hece x = xjk is [ Nλ,µ, p, q, α, β ] r summable to it L 3 Applicatios Let C [a, b] be the space of all fuctios f cotiuous o [a, b] We kow that C [a, b] is a Baach space with orm f = sup f x, f [a, b] The classical Korovki type approximatio theorem states as x [a,b] follows [2]: Suppose that T be a sequece positive liear operators from C [a, b] ito C [a, b] The, T f ; x f x = 0, for all f C [a, b] if ad oly if T fi ; x f i x = 0 for i = 0,, 2, where f 0 x =, f x = x ad f 2 x = x 2 By C K we deote the space of all cotiuous real valued fuctios o ay compact subset of the real two dimesioal space This space is equipped with the supremum orm CK f = sup f, x,y K f C K Let L be a liear operator from C K ito C K The, as usual we say that L is positive liear operator provided that f 0 implies L f 0 Also we deote the value of L f at a poit K by L f ; x, y Recetly Korovki type approximatio theorems have bee studied i [2],[9],[3],[6],[22],[27],[30] Theorem 3 Let {T m is a double sequece of positive liear operators from C K ito C K The, for all f C K N 2 λµ if ad oly if N 2 λµ st m, st m, T m f f CK = 0 3 CK T m f i f i = 0 32 i = 0,, 2, 3, where f 0 =, f = x, f2 = y ad f3 = x 2 + y 2
M Ciar, M Et / Filomat 30:3 206, 753 762 760 Proof Sice each f i C K i = 0,, 2, 3 coditio 3 follows immediately 32 Assume that 32 holds Let f C K we ca write f M where M = f CK Sice f is cotiuous o K for every ε > 0, there is δ > 0 such that f u, v f < ε for all u, v K satisfyig u x < δ, v y < δ Hece we get f u, v f < ε + 2M δ 2 { u x 2 + v y 2 33 Sice T m is liear ad positive ad by 33 we obtai T m f ; x, y f x, y = Tm f u, v f ; x, y f Tm f0 ; x, y f 0 T m f u, Tm v f x, y ; x, y + M f0 ; x, y f 0 x, y T m ε + 2M { u x 2 + v y 2 ; x, y + M Tm f0 ; x, y f δ 2 0 x, y ε + M + 2M A 2 + B 2 Tm f0 ; x, y f δ 2 0 x, y + 4M δ 2 A Tm f ; x, y f + 4M δ 2 B Tm f2 ; x, y f 2 + 2M δ 2 Tm f3 ; x, y f 3 + ε where A = max x, B = max y Takig supremum over x, y K we have T m f f CK R { CK Tm f 0 f 0 + CK Tm f f + CK Tm f 2 f 2 + CK Tm f 3 f 3 + ε, where Hece R = max {ε + M + 2M δ 2 A 2 + B 2, 4M δ 2 A, 4M δ 2 B, 2M δ 2 T m f ; x, y pm q f CK ε + R Now replace T m ; x, y pm q by L m ; x, y = 3 T m fi ; x, y CK p m q f i x, y 34 i=0 m, J m I T m ; x, y pm q i 34 For a give r > 0 choose ε > 0 such that ε < r Defie the followig sets D = { m, N 2 : Lm f f CK r, D i = {m, N 2 : CK Lm f i f i r ε, i = 0,, 2, 3 4R The, D 3 D i ad so δ D δ D 0 + δ D + δ D 2 + δ D 3 Therefore, usig coditios 34 we get N 2 λµ i=0 st m, This completes the proof T m f f CK = 0
M Ciar, M Et / Filomat 30:3 206, 753 762 76 Corollary 32 Let { L m, be a sequece of positive liear operators actig from C K ito itself f C K, The, for all P m, Lm f f CK = 0 if ad oly if P m, Lm f i f i CK = 0, i = 0,, 2, 3, where f 0 =, f = x, f2 = y ad f3 = x 2 + y 2 [36] Remark 33 We ow costruct a example of sequece of positive liear operators of two variables satisfyig the coditios of Theorem 3, but that does ot satisfy the coditios of the Korovki Theorem For this claim, we cosider the followig Brestei operators defied as follows B m, f ; x, y = f k=0j=0 where [0, ] [0, ] Let B m, f0 ; x, y = B m, f ; x, y = x B m, f2 ; x, y = y B m, f3 ; x, y = x 2 + y 2 + x x2 m k m, j C k mx k x m k Cy j j y j, + y y2 The by Corllary 32 we ca write for all f C K P m, Bm, f f CK = 0 Let the sequece T m, : C K C K with T m, f ; x, y = + um B m, f ; x, y where um = for all m Let p m =, q =, λ m = m, µ = The double sequece u m is either P coverget or statistically covertget, but u m is statistically summable N 2 λµ, p, q, α, β to zero B m, ; x, y =, Bm, x; x, y = x, Bm, y; x, y = y, Bm, x 2 + y 2 ; x, y = x 2 + y 2 + x x2 m sequece T m, satisfies coditio 32 for i = 0,, 2, 3 Hece we have N 2 λµ st m, T m f f CK = 0 + y y2 ad the double O the other had, we get T m, f, 0, 0 = + um B m, f ; 0, 0 sice Bm, f ; 0, 0 = f 0, 0 ad hece T m f ; x, y f CK Tm f ; x, y f um, f 0, 0 We see that T m, does ot satisfy classical Korovki type theorem sice um ad st 2 u m do ot exists this proves the claim
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