Lacunary Weak I-Statistical Convergence

Transcription

1 Ge. Mat. Notes, Vol. 8, No., May 05, pp ISSN 9-784; Copyigt ICSRS Publicatio, 05 vailable ee olie at ttp// Lacuay Wea I-Statistical Covegece Haize Gümüş Faculty o Eegli Educatio, Necmetti Ebaa Uivesity Eeğli Koya- 430, Tuey gumus@oya.edu.t Received / ccepted bstact I tis study, we povide a ew appoac to I statistical covegece. We itoduce a ew cocept wit I statistical covegece ad wea covegece togete ad we call it wea I statistical covegece o WS I covegece. Te we itoduce tis cocept o lacuay sequeces ad we obtai lacuay wea I- statistical covegece i.e. WS θ I covegece. WN θ I covegece is ay ote deiitio i ou study. te givig tis desciptio, we ivestigate tei elatiosip ad we ave some esults. Keywods I-statistical covegece, wea statistical covegece, lacuay sequece. Itoductio I tis aea, statistical covegece is a impotat cocept ad Zygmud [5] gave it i te ist editio o is moogap publised i Wasaw i 935. It was omally itoduced by Fast ad Steiaus [5, 4] ad late was eitoduced by Scoebeg. [3] Tis cocept as a wide applicatio aea o eample umbe teoy [4], measue teoy [0], tigoometic seies [5], summability teoy [6],

2 Lacuay Wea I-Statistical Covegece 5 etc. Fidy gave impotat popeties about statistical covegece i is study [7], Fidy ad Oa studied statistical covegece wit lacuay sequeces. [8]. Let K be a subset o te set o all atual umbes Ν ad { K} wee te vetical bas idicate te umbe o elemets i te eclosed set. Te atual desity o K is deied by δ K lim { K}. I a popety P olds o all wit δ we say tat P olds o almost all tat is a.a.. Deiitio. [4] umbe sequece is statistically coveget to povided tat o evey > 0, I tis case we wite st lim. lim { } 0 Statistical covegece was eteded to I covegece i a metic space i Kostyo, Salát ad Wilezyńsi's study. [9] Deiitio. amily o sets. K Ν I is called a ideal i ad oly i i φ I ii Fo eac iii Fo eac, B I we ave B I I ad eac B we ave B I ideal is called o-tivial i o eac Ν. { } I Deiitio.3 amily o sets Ν I ad a o-tivial ideal is called admissible i Ν F is called a ilte i Ν i ad oly i i φ F ii Fo eac iii Fo eac, B F we ave B F F ad eac B we ave B F Popositio. I is a o-tivial ideal i Ν i ad oly i is a ilte i Ν. F { M \ I} F I N Tougout te pape, I will be a admissible ideal.

3 5 Haize Gümüş Deiitio.4 eal sequece is said to be I coveget to L R i ad oly i o eac > 0 te set { Ν L } belogs to I. Te umbe L is called te I limit o te sequece. Eample. Tae o I class te I o all iite subsets o N. Te I is a admissible ideal ad I covegece coicides wit te usual covegece. I 0, Das, Savas ad Gosal [3] ave itoduced te cocept o covegece ad I lacuay statistical covegece. I statistical Deiitio.5 [3] sequece is said to be I statistically coveget to L o eac > 0 ad δ > 0, Ν { L } δ I. Eample. Let us tae te sequece y wee te ideal d, to 0 y ad 0, 0,,3,.... I wic is te ideal o desity zeo sets o Ν. Let { } Deie i a omed liea space X by, [ y ] u, o, u, o y, θ, otewise wee u X is a ied elemet wit u ad θ is te ull elemet o X. Te te sequece is I statistically coveget but it is ot statistically coveget. Now, we will give te deiitio o I lacuay statistically coveget sequeces om te pape o Das, Savas ad Gosal. But ist, we eed to emid lacuay sequece. Deiitio.6 lacuay sequece is a iceasig itege sequece θ suc tat 0 0 ad. as Te itevals detemied by θ will be deoted by, ] ad te atio will be deoted by q.

4 Lacuay Wea I-Statistical Covegece 53 Deiitio.7 [3] Let θ be a lacuay sequece. sequece is said to be I lacuay statistically coveget to L o eac > 0 ad δ > 0, Ν { L } δ I. Let s cotiue to emid impotat cocepts tat we eed o ou study. Deiitio.8 Let B be a Baac space, be a B-valued sequece ad B. Te sequece is wealy coveget to povided tat o ay i te cotiuous dual B o B, ad i tis case we wite w lim. lim 0 Deiitio.9 Let B be a Baac space, be a B-valued sequece ad B. Te sequece is wealy C₁-coveget to povided tat o ay i te cotiuous dual B o B, lim 0 I 000, Coo et al. [], ave itoduced a ew cocept o wea statistical covegece ad ave caacteized Baac spaces wit sepeable duals via statistical covegece. Peliva ad Kaaev [] ave also used te idea o wea statistical covegece i stegteig a esult o Gobeg ad Klei o compact opeatos. Badwa ad Bala ave ivestigated some elatios betwee wea coveget sequeces ad wealy statistically coveget sequeces []. Followig Coo et al. we deie wea statistical covegece as ollows Deiitio.0 [] Let B be a Baac space, be a B-valued sequece ad B. Te sequece is wealy statistically coveget to povided tat o ay i te cotiuous dual B o B te sequece is statistically coveget to i.e. lim { } 0 ad i tis case we wite W st lim. It is easy to see tat te wea statistical limit o a wealy statistically coveget sequece is uique.

5 54 Haize Gümüş I 0, Nuay [] studied wea statistical covegece by usig lacuay sequeces. Deiitio. Let B be a Baac space, be a B-valued sequece, θ be a lacuay sequece ad B. is wealy lacuay statistically coveget to o WS θ coveget to povided tat o ay i te cotiuous dual B o B, lim { } 0. Lacuay Wea I- Statistical Covegece Deiitio. Let B be a Baac space, be a B-valued sequece ad B. Te sequece is wealy I coveget to povided tat o ay i te cotiuous dual B o B, { Ν } I. Te set o all wealy I coveget sequeces is deoted by WI ad i we tae I I te ideal o all iite subsets o Ν, we ave te usual wea covegece. Eample. I d is a admissible ideal ad WI d wea statistical covegece. covegece coicides wit te Eample. Deote by I θ te class o all K Ν wit lim { K} 0. Te I θ is a admissible ideal ad WI θ covegece coicides wit te lacuay wea statistical covegece. We ow itoduce ou mai deiitios. Deiitio. Let B be a Baac space, be a B-valued sequece ad B. Te sequece is wealy I statistically coveget to povided tat o ay i te cotiuous dual B o B ad evey > 0 Ν I { } δ. ad δ > 0,

6 Lacuay Wea I-Statistical Covegece 55 Te set o all wealy I statistically coveget sequeces is deoted by WS I. Deiitio.3 Let B be a Baac space, be a B-valued sequece, B ad θ be a lacuay sequece. Te sequece is lacuay wea I statistically coveget to povided tat o ay i te cotiuous dual B o B ad evey > 0 ad δ > 0, Ν { } δ I. Te set o all lacuay wea I statistically coveget sequeces is deoted by WS θ I. Deiitio.4 Let B be a Baac space, be a B-valued sequece, B ad θ be a lacuay sequece. Te sequece is WN θ I coveget to povided tat o ay i te cotiuous dual ad evey > 0, Ν I. B o B Teoem. Let θ be a lacuay sequece. Te is WN θ I coveget to i ad oly i is WS θ I coveget to. Poo ssume tat is WN θ I coveget to ad > 0. We ca wite, Te, ad { } { } ad o ay δ > 0, Ν { } δ Ν δ. We ow tat te igt side is i ideal. So, te let side is also i ideal.

7 56 Haize Gümüş Now suppose tat is WS θ I Te tee eists a K 0 o all Ν get, Cosequetly we ave, Ν K coveget to. Sice B, is bouded. suc tat K. Give > 0, we ad. ad Ν I. K Teoem. Let θ be a lacuay sequece wit lim i q >. Te WS I covegece implies WS θ I covegece. Poo ssume tat lim i q >. Te tee eists a α > 0 suc tat α q α o all suicietly lage. Tis implies. Sice is α WS I coveget to, o evey > 0 ad suicietly lage we ave, { } { } Te o ay δ > 0 we get α { }. α δα Ν I α { } δ Ν { }. Tis poves te teoem. Teoem.3 Let θ be a lacuay sequece wit limsup q. Te WS θ I covegece implies WS I covegece. Poo I limsup q te tee is a K > 0 suc tat q K o all. Suppose tat is WS θ I coveget to ad, δ, η > 0. Deie te sets,

8 Lacuay Wea I-Statistical Covegece 57 { } { }. Ν Ν η δ R M Let I F be te ilte associated wit te ideal. I It is obvious tat FI M. I we ca sow tat FI R te we will ave te poo. Fo all M let, { }. δ Coose Ν suc tat o some. M Now, { } { } { } { } { } { } { } δ. sup K M Coosig K δ η ad i view o te act tat { } R M, te we ave FI R. Reeeces [] V.K. Badwa ad I. Bala, O wea statistical covegece, Iteatioal oual o Matematics ad Mat. Sci., ticle ID , 9 pages. []. Coo, M. Gaicev ad V. Kadets, caacteizatio o Baac spaces wit sepaable duals via wea statistical covegece,. Mat. al. ppl., 44000, 5-6. [3] P. Das, E. Savas ad S.K. Gosal, O geealizatios o cetai summability metods usig ideals, pplied Mat. Lettes, 40,

9 58 Haize Gümüş [4] P. Edös ad G. Teebaum, Su les desites de cetaies suites d'eties, Poceedigs o te Lodo Mat. Soc., , [5] H. Fast, Su la covegece statistique, Coll. Mat., 95, [6].R. Feedma ad I.. Sembe, Desities ad summability, Paciic oual o Mat., 95 98, [7].. Fidy, O statistical covegece, alysis, 5985, [8].. Fidy ad C. Oa, Lacuay statistical covegece, Pac.. Mat, 60993, [9] P. Kostyo, T. Salát ad W. Wilezyńsi, I-covegece, Real alysis Ecage, 6 000/00, [0] H.I. Mille, measue teoetical subsequece caacteizatio o statistical covegece, Tas. o te me. Mat. Soc., , [] F. Nuay, Lacuay wea statistical covegece, Mat. Boemica, 363 0, [] S. Peliva ad T. Kaaev, Some esults elated wit statistical covegece ad Beezi symbols, ou. o Mat. alysis ad ppl., , [3] I.. Scoebeg, Te itegability o cetai uctios ad elated summability metods, Te me. Mat. Motly, , [4] H. Steiaus, Su la covegece odiaie et la covegece asymptotique, Collog. Mat., 95, [5]. Zygmud, Tigoometic Seies, Cambidge Uivesity Pess, Cambidge, UK, 979.