Generalized Weighted Norlund-Euler. Statistical Convergence
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1 It. Joural of Math. Aalysis Vol o HIAI td htt//d.doi.org/.2988/ijma Geeralized Weighted orlud-uler tatistical Covergece rem A. Aljimi Deartmet of Mathematics Uiversity of Tiraa Tiraë Albaia. Valdete ou Deartmet of Comuter cieces ad Alied Mathematics College Vizioi er Arsim Ahmet aciu r=3 Ferizaj 7 osova Coyright 24 rem A. Aljimi ad Valdete ou. This is a oe access article distributed uder the Creative Commos Attributio icese which ermits urestricted use distributio ad reroductio i ay medium rovided the origial wor is roerly cited. Abstract I this aer the otio of summability is itroduced to geeralize the cocet of orlud uler statistical covergece. We call this ew method geeralized weighted orlud-uler statistically covergece. We will show some roerties of this summability method. Mathematics ubject Classificatio 4G5 eywords orlud-uler tye meas geeralized weighted orlud-uler statistical covergece seuece saces uler summability
2 346 rem A. Aljimi ad Valdete ou. Bacgroud ad Prelimiaries The idea of statistical covergece which is closely related to the cocet of atural desity or asymtotic desity of a subset of the set of atural umbers was first itroduced by Fast [6]. The idea of orlud uler statistical covergece is defieted by. Aljimi. Hoha ad V. ou []. The weighted statistical covergece is defied by V. araaja ad T.A.Chishti [2]. Ad the well defied versio of the statistically weighted covergece seueces is give by Mursalee et al. [5]. The iterestig role i our results lay the roduct of summability methods ad we call it geeralized orlud-uler summability method. The cocet of statistical covergece lays a imortat role i the summability theory ad fuctioal aalysis. The relatioshi betwee the summability theory ad statistical covergece has bee itroduced by choeberg [7]. Afterwards the statistical covergece has bee studied as a summability method by may researchers such as Fridy [8] Freedma et al. [9] ol [ ] Fridy ad Miller [2] Fridy ad Orha [34] Mursalee et al. [5] avaş [6] Braha [3-5]. Also some toological roerties of statistical covergece seuece saces have bee studied by alat [7]. Besides i [8 9] Coor showed the relatios betwee statistical covergece ad fuctioal aalysis. I geeral statistical covergece of weighted meas is studied as a class of regular matri trasformatios. I this wor we itroduce ad study the cocet of weighted - orlud-uler statistical covergece. The relatios amog summability ad related to the weight et seuece of weighted statistically coverget seueces. ad. The the atural desity of is defied by ( lim if the limit eists where deotes the cardiality of. A seuece ( of real umbers is said to be statistically coverget to rovided that for every ( has atural desity zero; i this case we write the set
3 Geeralized weighted orlud-uler statistical covergece 347 lim. The symbol deotes the set of all statistically coverget seueces. a ifiite series with seuece of artial sums ( ad > ay real et be umber. The uler meas ( ( The series ( of the seueces ( are defied to be is said to be summable ( to the umber if as et be a give ifiite series with seuece of its If ( trasform is defied as ( ( Ad we say that this summability method is coverget if as ( th artial sum. I this case we say the series is ( summable to a defiite umber. as. Ad we will write et ad be the two seueces of o-zero real costats such that P... P Q... Q For the give seueces ad covolutio is defied by
4 348 rem A. Aljimi ad Valdete ou The series or the seuece is summable to by geeralized orlud method ad it is deoted by if t teds to as. et us use i cosideratio the followig method of summability t ( emar For = we have orlud-uler methodes t 2. emar 2 For ad we have Cesaro-uler methods ( ( If t as the we say that the series or the seuece is summable to by geeralized orlud-uler method ad it is deoted by et us deote by the seuece sace all strogly coverget seueces ( which are summable to some for ( lim The matri a A i - summability is give by
5 Geeralized weighted orlud-uler statistical covergece 349 if if a ( ow we are able to give the defiitio of the weighted statistical covergece related to the summability method. Defiitio. A seuece ( is said to be geeralized weighted orlud-uler statistical coverget if for every. ( lim The set of geeralized weighted orlud-uler statistical covergece seuece is deoted by as follows some for ( lim ( If the seuece ( is -covergece the we also use the otatio (. Defiitio 2. A seuece ( is said to be ( ] [ r summable r to the limit if ( lim r ad we write it as ] ([ r. I these case is called the r ] [ limit of.
6 35 rem A. Aljimi ad Valdete ou 2. Mai esults I cotiuatio of this article we fid the relatioshis of with ad (. Firstly let us begi the followig theorem. Theorem. If the seuece ( is -summable to the the seuece ( is coverget ad the iclusio is roer. Proof et the seuece ( be -summable to ad ( The for a give we have ( = ( + ( ( Hece we obtai that the seuece ( is -coverget to. I what follows we will show that iclusio relatio is strict. amle et us cosider that ad for all. Also we defie the followig seuece ( as follows 2 if if The we have ad ( as.
7 Geeralized weighted orlud-uler statistical covergece 35 O the other had 2 ( ( 2( ( as. Hece it ca be see that the iclusio is roer. Theorem 2. et P ad M ( for all. If ( the. Proof et ( ad ( ice P ad M ( for all the for a give we have ( = ( + ( ( M ice is arbitrary we have. I what follows we will cosider that ad with ( we will deote the statistical covergece of the seuece (. The followig Theorem shows that uder which coditios statistical covergece imlies geeralized weighted orlud uler statistical covergece.
8 352 rem A. Aljimi ad Valdete ou Theorem 3. et limif for all. If ( the ( ad the iclusio is roer. Proof For give we have ( ( ( This comletes the roof. To rove that this iclusio is roer we will show the followig eamle. amle 2 et us cosider that ad for all. Ad let us cosider that (. The it follows that ( lim but. ( ( Theorem 4. If the seuece is a bouded seuece such that limsu the from follows covergece. Proof For a give we have ( ( (
9 Geeralized weighted orlud-uler statistical covergece 353 for some costat. Hece we get that ( (. Theorem 5. If seuece limif < limsu is a bouded seuece such that < the from covergece. Proof Prof of the Theorem follows directly from Theorems 3 ad 4. is euivalet with efereces.. Aljimi. Hoha ad V.ou ome esults of Weighted orlud-uler tatistical Covergece Iteratioal Mathematical Forum Vol o V. AAAYA AD T. A. CHIHTI. WIGHTD TATITICA COVGC. Iraia Joural of ciece & Techology Trasactio A Vol. 33 o. A Braha A ew class of seueces related to the l \sb saces defied by seueces of Orlicz fuctios. J. Ieual. Al. 2 Art. ID Braha.. O asymtotically \Delta \s m lacuary statistical euivalet seueces. Al. Math. Comut. 29 (22 o Braha aim.; t Miâil. The seuece sace \sb \s (Ms ad \sb - lacuary statistical covergece. Baach J. Math. Aal. 7 (23 o Fast H. (95. ur la covergece statistiue. Collo. Math choeberg I. J. (959. The itegrability of certai fuctios ad related summability methods. Amer. Math. Mothly Fridy J. A. (985. O statistical covergece. Aalysis Freedma A.. & ember I. J. (98. Desities ad summability. Pacific J. Math
10 354 rem A. Aljimi ad Valdete ou. ol. (99. The statistical covergece i Baach saces. Acta et Commet. Uiv. Tartu ol. (993. Matri summability of statistically coverget seueces. Aalysis Fridy J. A. & Miller H. I. (99. A matri characterizatio of statistical covergece. Aalysis Fridy J. A. & Orha C. (993. acuary statistical covergece. Pacific J. Math Fridy J. A. & Orha C. (993. acuary statistical summability. J. Math. Aalysis Al. 73( Mursalee Mohammad; araaya Vata; rtür Müzeyye; Gürsoy Fai. Weighted statistical covergece ad its alicatio to orovi tye aroimatio theorem. Al. Math. Comut. 28 (22 o avaş. (992. O strog almost A- summability with resect to a modulus ad statistical covergece. Idia J. Pure ad Al. Math. 23( alat T. (98. O statistically coverget seuece of real umbers. Math. lovaca Coor J.. (988. The statistical ad strog -Cesaro covergece of seuece. Aalysis Coor J.. (989. O strog matri summability with resect to a modulus ad statistical covergece. Caad. Math. Bull eceived Jauary 5 24
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