Some Results of Weighted Norlund-Euler. Statistical Convergence

Size: px

Start display at page:

Download "Some Results of Weighted Norlund-Euler. Statistical Convergence"

Chrystal Hines
6 years ago
Views:

1 Itatioal athmatical Foum Vol. 8 3 o HIAI td htt//d.doi.og/.988/im.3.39 Som ult o Wightd Nolud-Eul Statitical Covgc Em A. Aljimi Datmt o athmatic Uivity o iaa iaë Albaia mhalimii@yahoo.co.u Elida Hoha Datmt o athmatic Uivity o iaa iaë Albaia hohalida@yahoo.com 3 Valdt ou Datmt o Comut Scic ad Alid athmatic Collg Vizioi Aim Ahmt aciu N3 Fizaj 7 oova Coyight 3 Em A. Aljimi Elida Hoha ad Valdt ou. hi i a o acc aticl ditibutd ud th Cativ Commo Attibutio ic which mit utictd u ditibutio ad oductio i ay mdium ovidd th oigial wo i oly citd. Abtact I thi a w will di th w wightd tatitically ummbaility mthod ow a th wightd Nolud-Eul tatitical covgc. W will how om oti o thi mthod ad w hav ovd oovi ty thom. athmatic Subjct Claiicatio 4G5 ywod Nolud-Eul ty ma wightd Nolud-Eul tatitical covgc uc ac ummability

2 798 Em A. Aljimi Elida Hoha ad Valdt ou. INODUCION h ida o tatitical covgc which i cloly latd to th coct o atual dity o aymtotic dity o a ubt o th t o atual umb N wa it itoducd by Fat [3]. h wightd tatitical covgc i did by V. aaaja ad.a.chihti []. Ad th wll did vio o th tatitically wightd covgc uc i giv by ual t al. [5]. h ittig ol i ou ult lay th oduct o ummability mthod N E ad w call it Nolud-Eul ummability mthod. h coct o tatitical covgc lay a imotat ol i th ummability thoy ad uctioal aalyi. h latiohi btw th ummability thoy ad tatitical covgc ha b itoducd by Schobg [7]. Atwad th tatitical covgc ha b tudid a a ummability mthod by may ach uch a Fidy [8] Fdma t al. [9] ol [ ] Fidy ad ill [] Fidy ad Oha [34] ual t al. [5] Savaş [6] Baha [3-5]. Alo om toological oti o tatitical covgc uc ac hav b tudid by Salat [7]. Bid i [8 9] Coo howd th latio btw tatitical covgc ad uctioal aalyi. I gal tatitical covgc o wightd ma i tudid a a cla o gula mati taomatio. I thi wo w itoduc ad tudy th coct o wightd Nolud-Eul tatitical covgc. h latio amog N E -ummability ad S uc o wightd tatitically covgt uc latd to th wight E N. t by N ad { }. h th atual dity o i did lim i th limit it wh dot th cadiality o. A uc o al umb i aid to b tatitically covgt to ovidd that o vy > th t { } ha atual dity zo Fat [5] ad Stihau [4] o ach > { j } lim j

3 Wightd Nolud-Eul tatitical covgc 799 i thi ca w wit t lim o S - lim. h ymbol S dot th t o all tatitically covgt uc. t b a giv iiit i with uc o it I E taom i did a S. th atial um { } E S Ad w ay that thi ummability mthod i covgt i E S a. I thi ca w ay th i i E ummabl to a diit umb S. Hady [9] Ad w will wit S E t ad S a. b th two uc o o-zo al cotat uch that P... P Q... Q Fo th giv uc ad covolutio i did by h i o th uc { } mthod ad it i dotd by S S N t S td to S a. S i ummabl to S by galizd Nolud i

4 8 Em A. Aljimi Elida Hoha ad Valdt ou t u u i coidatio th ollowig mthod o ummability t E E S. I t E S a th w ay that th i o th uc { S } i ummabl to S by Nolud-Eul mthod ad it i dotd by S S N E ma I th w gt Eul ummability mthod t u dot by E uc which a E N E N th uc ac all togly covgt N ummabl to lim h mati i E A a N - ummability i giv by o om a i i > Now w a abl to giv th diitio o th wightd tatitical covgc N E ummability mthod. latd to th Diitio.. A uc i aid to b wightd Nolud-Eul tatitical covgt i o vy >. lim h t o wightd Nolud-Eul tatitical covgc uc i dotd by S a ollow

5 Wightd Nolud-Eul tatitical covgc 8 S lim o om I th uc i S -covgc th w alo u th otatio S. Diitio.. A uc i aid to b E ] ummabl [ N to th limit i lim ad w wit it a [ N E ] I th ca i calld th limit [ N E ] limit o. I th a w tablih th latio o S - covgc with tatitical ummability N E ad [ N E ].. AIN ESUS I ou it thom w tablih th latio btw covgc ad tatitical ummability E N. S -tatitical hom.. t o all N. I a uc i S -tatitically covgt to th it tatitically ummabl N E to but ot covly. Poo Sic i S -tatitically covgt to lim

6 8 Em A. Aljimi Elida Hoha ad Valdt ou wh N. Wit ad c. ha t E c c c c c a which imli that t. hat i i E N ummabl to ad hc tatitically ummabl E N to. Eaml t u coid that ad o all N. Alo w di th ollowig uc a ollow i i

7 Wightd Nolud-Eul tatitical covgc 83 h w hav ad O th oth had a. v v v v v v a. 4 h aml ov that cov i ot tu. Nt thom giv th latio btw S -tatitically covgt ad [ N E ] hom.. a. t a uc a valid th [ N E ] ummabl to. I th ollowig i Ca ad Ca ad. i S -tatitically covgt to b. t i S -tatitically covgt to ad 3. I th ollowig a valid th [ N E ] Ca ad Ca ad.

8 84 Em A. Aljimi Elida Hoha ad Valdt ou Poo a. Sic o ca. ad ca. th a lim wh Hc S -tatitically covgt to. b. Suo that i S -tatitically covgt to. h o > w hav N. Sic w hav S S wh

9 Wightd Nolud-Eul tatitical covgc 85 S S. Now i th S c Fo w hav S u a. Sic N. Hc E N ] [. 3. Alicatio to aoimatio thom t ] [ b a C b th ac o all uctio cotiuou o ] [ b a. W ow that ] [ b a C i Baach ac with om

10 86 Em A. Aljimi Elida Hoha ad Valdt ou u [ a b] C[ a b] h claical oovi aoimatio thom tat a ollow [6] t b a uc o oitiv lia oato om C[ a b] ito C [ a b]. h lim o all C[ a b] i oly i lim i i o i wh dh. It tatitical vio wa giv by Gadjiv ad Oha [5]. Such ty o aoimatio thom a ovd by uig th coct o almot covgc [][6] [-7] λ -tatitical covgc [93] ad tatitical lacuay ummability [3]. Boyaov ad Vliov [3] hav ovd th oovi thom o C[ by uig th tt uctio.i additio om latd a o thi toic ca b oud i [33 35]. I thi a w galiz th ult o Boyaov ad Vliov by uig th otio o tatitical ummability N E ad th am tt uctio.w alo giv a aml to jutiy that ou ult i tog tha that o Boyaov ad Vliov [3]. t CI b th Baach ac with th uiom om o all al-valud two dimioal cotiuou uctio o I [ ovidd that lim i iit. Suo that C I C I. W wit o ad w ay that i a oitiv oato i o all. h ollowig tatitical vio o Boyaov ad Vliov ult ca b obtaid a a cial ca o [4]. om A. t b a uc o oitiv lia oato om CI ito C I. ha o all CI. i oly i t lim t lim

11 Wightd Nolud-Eul tatitical covgc 87 t lim t lim Similaly o ca ov th S -tatitical vio. Now w ov th ollowig N E. tog vio by uig th otio o tatitical ummability om 3.. t b a uc o oitiv lia oato om CI ito C I. ha o all CI. t lim 3.. i oly i t lim 3.. t lim 3.. t lim 3..3 Poo. Sic ach blog to C I coditio ollow immdiatly om 3... t CI. h th it a cotat > uch that o I. ho 3..4 It i ay to ov that o a giv > th i a > uch that 3..5 whv o all I. Uig 3..4 ad 3..5 uttig ψ ψ w gt

12 88 Em A. Aljimi Elida Hoha ad Valdt ou ψ. hi i. ψ ψ Now oatig to thi iuality ic i mooto ad lia. W obtai. ψ ψ Not that i id ad o i cotat umb. ho ψ ψ ψ ψ 3..6 Alo ] [ 3..7 I ollow 3..6 ad 3..7 that ] [ ψ 3..8 Now ] [ ] [ ] [ ψ

13 Wightd Nolud-Eul tatitical covgc 89 Uig 3..8 w obtai } { ] [ ] [ ] [ ] [ h o Sic o all I.Now taig I u w gt wh 4 ma. Hc 3..6 Now lacig by m m ad th by m B i 3..6 o both id. Fo a giv > choo > uch that >. Di th ollowig t { } B m D m 4 B m D m

14 8 Em A. Aljimi Elida Hoha ad Valdt ou D m B m t 4 D 3 m B m t 4 h D D D D3 ad o D D D D3.ho uig coditio w gt t lim hi comlt th oo o th thom. c [] N.Baha V.ou ad E.Aljimi Wightd Nolud-Eul Statitical Covgc. Coc ocdig t wt Balca Coc o athmatical cic Elbaa may 3. [] V. aaaya.a. Chihti Wightd tatitical covgc Ia. J. Sci. chol. a. A Sci [3] N.. Baha A w cla o uc latd to th $l\b $ ac did by uc o Olicz uctio. J. Iual. Al. At. ID [4] Baha N.. O aymtotically Δ m lacuay tatitical uivalt uc. Al. ath. Comut. 9 o [5] Baha Naim. Et iâil. h uc ac E ad N -lacuay tatitical covgc. Baach J. ath. Aal. 7 3 o [6] Fat H. 95. Su la covgc tatitiu. Collo. ath [7] Schobg I. J h itgability o ctai uctio ad latd ummability mthod. Am. ath. othly [8] Fidy J. A O tatitical covgc. Aalyi [9] Fdma A.. & Smb I. J. 98. Diti ad ummability. Paciic J. ath [] ol. 99. h tatitical covgc i Baach ac. Acta t Commt. Uiv. atu

15 Wightd Nolud-Eul tatitical covgc 8 [] ol ati ummability o tatitically covgt uc. Aalyi [] Fidy J. A. & ill H. I. 99. A mati chaactizatio o tatitical covgc. Aalyi [3] Fidy J. A. & Oha C acuay tatitical covgc. Paciic J. ath [4] Fidy J. A. & Oha C acuay tatitical ummability. J. ath. Aalyi Al [5] ual ohammad aaaya Vata Etü üzyy Güoy Fai. Wightd tatitical covgc ad it alicatio to oovi ty aoimatio thom. Al. ath. Comut. 8 o [6] Savaş E. 99. O tog almot A-ummability with ct to a modulu ad tatitical covgc. Idia J. Pu ad Al. ath [7] Salat. 98. O tatitically covgt uc o al umb. ath. Slovaca [8] Coo J. S h tatitical ad tog -Cao covgc o uc. Aalyi [9] Coo J. S O tog mati ummability with ct to a modulu ad tatitical covgc. Caad. ath. Bull [] G Hady Divgt i it ditio Ood Uivity P [] G.A. Aataiou. ual S.A. ohiuddi Som aoimatio thom o uctio o two vaiabl though almot covgc o doubl uc J. Comut. Aal. Al []. Bc Global aoimatio thom o Szaz iaja ad Baaov oato i olyomial wight ac Idiaa Uiv. ath. J [3] B.D. Boyaov V.. Vliov A ot o th aoimatio o uctio i a iiit itval by lia oitiv oato Bull. ath. Soc. Sci. ath. oumai N.S

16 8 Em A. Aljimi Elida Hoha ad Valdt ou [4] O. Duma. Dmici S. aau_ Statitical aoimatio o iiit itval it. [5] A.D. Gadjiv C. Oha Som aoimatio thom via tatitical covgc ocy outai J. ath [6] P.P. oovi ia Oato ad Aoimatio hoy Hiduta Publihig Cooatio Dlhi 96. [7] S.A. ohiuddi A alicatio o almot covgc i aoimatio thom Al. ath. tt [8] F. oicz C. Oha aubia coditio ud which tatitical covgc ollow om tatitical ummability by wightd ma Studia Sci. ath. Huga [9]. ual A. Alotaibi Statitical ummability ad aoimatio by d la Vall-Poui ma Al. ath. tt [3]. ual A. Alotaibi Statitical lacuay ummability ad a oovi ty aoimatio thom A. Uiv. Faa [3] H.. Sivatava. ual Ai ha Galizd ui-tatitical covgc o oitiv lia oato ad aociatd aoimatio thom ath. Comut. od. doi.6/j.mcm... [3] H. Stihau Quality cotol by amlig Collo. ath [33] E. Eu_-Duma O. Duma Statitical aoimatio oti o high od oato cotuctd with th Cha Chya Sivatava olyomial Al. ath. Comut [34]. Ocu O. Dogu Wightd tatitical aoimatio by atoovich ty -Szaz iaja oato Al. ath. Comut [35] C. adu O tatitical aoimatio o a gal cla o oitiv lia oato tdd i -calculu Al. ath. Comut civd Octob 3

On Jackson's Theorem

On Jackson's Theorem It. J. Cotm. Math. Scics, Vol. 7, 0, o. 4, 49 54 O Jackso's Thom Ema Sami Bhaya Datmt o Mathmatics, Collg o Educatio Babylo Uivsity, Babil, Iaq mabhaya@yahoo.com Abstact W ov that o a uctio W [, ], 0