ON WEAK -STATISTICAL CONVERGENCE OF ORDER

Transcription

1 UPB Sci Bu, Series A, Vo 8, Iss, 8 ISSN 3-77 ON WEAK -STATISTICAL CONVERGENCE OF ORDER Sia ERCAN, Yavuz ALTIN ad Çiğdem A BEKTAŞ 3 I the preset paper, we give the cocept of wea -statistica covergece of order ad weay [ V, ] -summabiity of order Some reatios betwee of these cocepts ad aso some reatios betwee wea -statistica covergece of order ad strog -statistica covergece of order are examied Keywords: Wea statistica covergece, Cesàro Summabiity Itroductio Zygmud [] itroduced the idea of statistica covergece i the first editio of his moograph pubished i Warsaw i 935 The otio of statistica covergece of sequeces of rea umbers was itroduced by Steihaus [] ad Fast [3] Later Schoeberg [4] itroduced the cocept of statistica covergece, idepedety I past years, differet authors studied properties of statistica covergece ad appied this cocept i various areas such as umber theory, ergodic theory, Fourier aaysis, measure theory, trigoometric series, Fuzzy set theory, iterva aaysis etc The statistica covergece depeds o desity of subsets of The atura desity of K is defied by ( K ) = im : K where : K deotes the umber of eemets of K ot exceedig c [5] Ay fiite subset of have zero atura desity ad ( K ) ( K) = A sequece x = ( x ) of umbers is said to be statisticay coverget to a umber provided that for every, im : x = I this case, we write S im x = Firat Uiversity Departmet of Mathematics, Eazig, Turey, e-mai: siaerca45@gmaicom Firat Uiversity Departmet of Mathematics, Eaziğ,Turey, e-mai: yati3@yahoocom 3 Firat Uiversity Departmet of Mathematics, Eazig, Turey, e-mai: cbetas@firatedutr

2 6 Sia Erca, Yavuz Ati ad Çiğdem A Betaş Leider [6] itroduced ( V, ) -summabiity with the hep of sequece = ( ) as foows: Let = ( ) be a o-decreasig sequece of positive umbers tedig to with + +, = The geeraized de a Vaee-Poussi mea is defied by t( x) = x I where I = [ +, ] A sequece summabe to a umber if t ( x) as ad x = x of umbers is said to be ( V, ) - [ C,] = x = ( x) : R,im x = = [ V, ] = x = ( x) : R,im x = I x = x which are strogy Cesáro summabe ad deotes the sets of sequeces strogy ( V, ) -summabe to It is beig oted that for summabiity reduces to ( C,) -summabiity =, (, ) V - Mursaee [7] itroduced the -desity of K is defied by ( K ) = im + : K ad -statistica covergece as foows: A sequece x = ( x ) of umbers is said to be -statisticay coverget to a umber provided that for every, im + : x = I this case, the umber is caed -statistica imit of the sequece x = ( x ) The statistica covergece with degree was itroduced by Gadjiev ad Orha i [8] The the statistica covergece of order ad strog p -Cesàro summabiity of order were studied by Çoa i [9] Aso Çoa ad Betaş itroduced - statistica covergece of order i [], as foows:

3 O wea -statistica covergece of order 7 Let the sequece = ( ) of rea umbers be defied as above ad The sequece x = ( x ) w is said to be -statisticay coverget of order if there is a compex umber such that im I : x =, where I = [ +, ] ( ) (,,,,) = = deote the th power ( ) of, that is, A sequece ( x ) i a ormed space X is said to be weay coverget to X provided that im ( x ) = for each X, the cotiuous dua of X I this case, we write W im x = Et et a itroduced the cocept Ŝ -statistica covergece of order i [] Coor et a itroduced wea statistica covergece ad used it to give descriptio of Baach spaces with seperabe duas i [] A sequece x i a ormed space X is said to be weay statisticay coverget to X provided that, for each, ( ( x ) ) : = for each X, the cotiuous dua of X I this case, we write WS im x = The set of a wea statisicay coverget sequeces is deoted by WS Bhardwaj et a defied wea statistica Cauchy sequeces i a ormed space X ad studied wea statistica covergece i p spaces i [3] Meeashi et a studied wea - statistica covergece, wea -statisticay Cauchy ad V, - summabiity i a ormed space X i [4] Throughout this paper wea we deote the cass of a decreasig sequece of positive rea umbers tedig to such that + +, = by Aso, uess stated otherwise, by "for a " we mea "for a N except fiite umbers of positive itegers" N where N,,, = + + for some N = {,,3,} Mai Resuts I this sectio we give the mai resuts of the paper X wi deote a = we mea ormed iear space; X its cotiuous dua ad by a sequece a o-decrasig sequece tedig to with + +, =

4 8 Sia Erca, Yavuz Ati ad Çiğdem A Betaş Defiitio A sequece ( x ) i X is said to be weay [ V, ] -summabe of order to X provided that im =, I ( x ) for each X where (,] I this case, we write W [ V, ] im x = The set of a weay [ V, ] -summabe of order sequeces wi be deoted by W [ V, ] Defiitio Let the sequece = ( ) of rea umbers be defied as above ad be give The sequece x = ( x ) w is said to be weay - statisticay coverget of order if there is X if for every im I : ( x ) =, where I, = + ad for every X I this case we write WS im x = The set of a weay - statisticay coverget of order sequeces wi be deoted by WS The weay -statisticay covergece of order is same with the weay -statisticay covergece, that is WS = WS for = Defiitio 3 A sequece ( x ) i X is said to be strogy -statisticay coverget of order to X if for every, im I : x = where (,] I this case, we write S im x = or S x The set of a strogy -statisticay coverget of order sequeces wi be deoted by S Theorem 4 Let umbers ad =, x x ( i ) For ay sequece ( ) ( ii) If WS im x ( iii ) If WS im x im Theorem 5 Let = ( ) ad ( ) N y = y be sequeces of compex x i X, if WS im x =, the must be uique = ad c beig a scaar, the WS imcx = c = ad WS y =, the WS im( x + y) = + ad et ad be such that = beog to such that for a

5 O wea -statistica covergece of order 9 ( i ) If the WS ( ii ) If the WS WS, = WS im if () im = ad im = () i Let for a N ad et () be satisfied Sice I J, for give we have Proof ad so J : ( x ) I : ( x ) J : x I : ( x ) for a N, where J [, ] = + By () as we get WS WS ( ii ) Let ( x ) WS ad () hod Sice I J, for we have J : ( x ) = + : ( x ) + I : ( x ) I : ( x + ) I : ( x + ) I : ( x + ) for a N Sice im = ad im = by () right had side of ast iequaity ted to as ( for a ) This meas that WS WS Sice () impies () we have WS = WS

6 Sia Erca, Yavuz Ati ad Çiğdem A Betaş Theorem 6 Let = ( ), (,] be ay rea umber ad im if / hods If there exists a set K = such that ( K ) = ad ( x ) im = for each X K, the ( ) x i X is WS - coverget to Proof It ca be see foowig simiar way i Theorem 3 i [4] by taig i advatage of Theorem 5 Whie the weay -statisticay covergece of order is we defied But whie it is ot we defied i geera Cosider the fuctioa defied o X by ( x) The both ad im I : ( x ) = im = im I : ( x ) = im =, such that for x = x WS -coverget both to ad This is impossibe Note that if we tae = for, the WS WS Aso if = with = the WS = WS = WS Theorem 7 Let = ( ), (,] be ay rea umber For ay sequece i X, if im if / (3) hods ad W im x = the WS im x =, however, the coverse impicatio eed ot be true Proof By Theorem 4 i [4] we have that every wea covergece sequece is i we obtai desired resut The coverse WS -covergece Usig Theorem 5 impicatio eed ot be true i geera It is ceary foowig exampe x with p be defied by Exampe 8 Let (3) hods ad x ( j) p = =, j I ad m ;, j I ad = m ;, j I ad m ;, j I ad m ;

7 O wea -statistica covergece of order where I [, ] say that = + Let K : m =, the by Theorem 6 we ca x is WS -coverget to For c K, et defie the fuctioa by j ( x) = x ( j) where x = ( x ) p Ceary, Hece W im x We have the foowig resuts from Theorem 5 Coroary 9 Let = ( ) ad ( ) x = x j = as j N ad () hods The the foowig statemets hod: ( i ) If =, the WS WS ( ii ) If =, the WS WS Coroary Let = ( ) ad ( ) = beog to such that for a for each (,] for each (,] N ad () hods The the foowig statemets hod: ( i ) If ( ii ) If =, the WS WS for each (,], =, the WS = beog to such that for a WS for each (,] Theorem S -covergece impies WS -covergece with same imit i X but the coverse impicatio eed ot be true Proof Let ( x ) i X, be a sequece such that x S I : x = The for evey, Now, for every ad each X, the expressio I : x I : x = I : x, meas S -covergece impies WS -covergece with same imit We give a exampe to show that the coverse impicatio of the above resut is ot true i geera

8 Sia Erca, Yavuz Ati ad Çiğdem A Betaş Exampe Let cosider the space p (,) defie x : (,) R by w The we have ( x) x i (,) S p L for p ad (3) hods The / p, if x [, ] =, otherwise WS L [5] By Theorem 7 we have Next we show that does ot hod For, sice S x Hece does ot hod x im I : x L p, x x = we have Theorem 3 Let The the icusio WS WS is strict for some ad such that Proof If, ceary I : ( x ) I : ( x ) for every, which gives that WS WS give the foowig exampe To show this icusio is strict we Exampe 4 Let tae = ad cosider the sequece x = ( x ) defied by S x, = m = m=,,, m Sice x, we have WS im x =, ie, x WS for But sice x S, x WS for This meas the icusio WS WS is strict for, (,], such that (, ] ad (,] Coroary 5 If a sequece is wea -statistica coverget of order to, the it is wea -statistica coverget to, that is, WS WS is strict for each (,]

9 O wea -statistica covergece of order 3 Coroary 6 ( ii) WS WS i WS = if ad oy if = = WS if ad oy if = Theorem 7 Give for = ( ), ( ) N = suppose that for a ad et The the foowig statemets hod: i If () hods, the W [ V, ] W [ V, ], ii If () hods, the W [ V, ] = W [ V, ] Proof write for a N i Suppose that for a N The I J so that we may J This gives that ( x ) ( x ) I ( x ) ( x ) J I As by () we have W [ V, ] W [ V, ] ( ii ) Let x ( x ) W [ V, ] exists some = ad suppose that () hods Sice is bouded there M such that ad so x M for a Now, sice that, ad I J for each N, we have x x x = + J J I I M x + I M x + I M x + I

10 4 Sia Erca, Yavuz Ati ad Çiğdem A Betaş for every Therefore W [ V, ] W [ V, ] Sice () impies () we have N the equaity W [ V, ] = W [ V, ] Coroary 8 Let = ( ) ad ( ) N ad () hods The the foowig statemets hod: ( i ) If ( ii ) If = beog to such that for a =, the W [ V, ] W [ V, ] for each (,], =, the W[ V, ] W [ V, ] for each (,] Coroary 9 Let = ( ) ad ( ) N ad () hods The the foowig statemets hod: ( i ) If ( ii ) If = beog to such that for a =, the W [ V, ] W [ V, ] for each (,], =, the W [ V, ] W[ V, ] for each (,] Theorem Let, (,] be rea umbers such that ( ) = such that for a N =,, ad ( i ) Let () hods, the if a sequece is W [ -summabe, ] of order, to, the it is WS -statisticay coverget of order, to, ( ii ) Let () hods, the if a sequece is WS -statisticay coverget of order, to, the it is W [ V, ] -summabe of order, to Proof ( i ) For ay sequece x ( x ) ad so that = ad, we have J J J ( x ) ( x ) ( x ) ( x ) I I ( x ) I : x I x = x + x x + x ( x )

11 O wea -statistica covergece of order 5 ( x ) I : ( x ) J Sice () hods it foows that if I : ( x ) x = x is W [ V, ] -summabe of order, to, the it is WS -statisticay coverget of order, to ( ii ) Suppose that such that WS im x = Sice is bouded there exists some M x M for a, the for every we have ( x ) = ( x ) + ( x ) J J I I M x + I M x + I = M ( x ) x + + I I M + M ( x ) ( x ) I : ( x ) + for a N Usig () we obtai that W [ V, ] im x =, wheever WS im x = Coroary Let = ( ) ad ( ) = beog to such that for a N ad () hods The the foowig statemets hod: i If =, the W [ V, ] WS for each (,], ( ii ) If =, the W[ V, ] for each (,] WS Coroary Let = ( ) ad ( ) N ad () hods The the foowig statemets hod: ( i ) If ( ii ) If =, the WS W [ V, ] for each (,], =, the WS W[ V, ] for each (,] = beog to such that for a

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