On Absolute Indexed Riesz Summability of Orthogonal Series

Transcription

1 Ieriol Jourl of Couiol d Alied Mheics. ISSN Volue 3 Nuer (8) eserch Idi Pulicios h: O Asolue Ideed iesz Suiliy of Orhogol Series L. D. Je S. K. Piry *. K. Ji 3 d. Sl 4 eserch Scholr Dere of Mheics veshw Uiversiy Cuc Odish Idi. Dere of Mheics Veer Suredr Si Uiversiy of Techology Burl Odish Idi. 3 Dere of Mheics DIEMS Tgi Cuc Odish Idi. 4 Dere of Mheics ABIT Cuc Odish Idi. Asrc I he rese er we hve sudy he N - suiliy of orhogol series h geerlizes he resuls of Y. Ouy [Tg J. Mh. 33 () 6-65]. Keywords: Covoluio eisz e seuece orhogol series AMS: 4 G5 4C5 L sce. N - suiliy orhogol.. INTODUCTION e give ifiie series wih is ril sus s. Le d Le re wo seueces he he covoluio. is give s

2 56 L.D. Je S.K. Piry.K. Ji d. Sl So we y wrie d. Agi we hve if he P Q seuece. Siilrly if he. The geerlized eisz rsfor of he seuece s is he s. The ifiie series is sid o e soluely sule N if coverges [8]. Siilrly he ifiie series N if Coverges. is sid o e soluely sule Le e orhogol seuece of fucio defied o iervl ssue h f L d (..) f where f d.... We Our i urose of he rese er is o sudy he N -suiliy of he orhogol series (..) for d o deduce s corollries ll resuls of Y. Ouy [5]. Throughou his er C deoes osiive cos. More over rece resuls o ordiry d sisicl versios of covergece s well s suiliy c

3 O Asolue Ideed iesz Suiliy of Orhogol Series 57 e eriched wih he choice of orhogol series. For such eriche oe y refers o he rece wors [-5-4]. The followig le due o Beo Levi (see for ele [6]) is ofe used i he heory of fucios i which re ivolved he series d iegrls d which re ivolved i [9] d [7] oo. I will eed us o rove i resuls. LEMMA-: If LE (..) E The he series f re o-egive fucios d f d f coverges los everywhere o E o fucio L E Moreover he series (7..) is lso coverge o f i he or of L(E). f... MAIN ESULT: We rove he followig heore. THEOEM-: If for he series coverges he he orhogol series everywhere. is sule N los POOF. series For he geerlized iesz rsfor we hve h of he ril sus of he orhogol

4 58 L.D. Je S.K. Piry.K. Ji d. Sl. We shll use he oio. Usig he Holder s ieuliy d orhogoliy o he ler euliy we hve h d d. Hece he series (..) C d coverges y he ssuio. Fro his fc d sice he fucios re o-egive he y he Le () he series coverges los everywhere. This colees he roof of he heore. Le us rove ow wo corollries of he Theore-. COOLLAY-: If for he series P P

5 O Asolue Ideed iesz Suiliy of Orhogol Series 59 coverges he he orhogol series is sule N N los everywhere. POOF. Afer soe eleery clculios oe c show h P P For ll d he roof follows iediely fro Theore. COOLLAY : If for he series Coverges he he orhogol series is sule N los everywhere. POOF. For he geerlized iesz rsfor of he ril sus of he orhogol series we hve h. We shll use he oio.

6 6 L.D. Je S.K. Piry.K. Ji d. Sl Usig he Holder s ieuliy d orhogoliy o he ler euliy we hve h d d. Hece he series (..) C d coverges y he ssuio. Fro his fc d sice he fucios coverges los everywhere. This colees he roof of he Corollry. EMAK-: We oe h: (i). If for ll vlues of he N -suiliy reduces o N -suiliy. (ii). If for ll vlues of he N -suiliy reduces o C -suiliy. (iii). If for ll vlues of he N -suiliy reduces o C - suiliy. (iv). If for ll vlues of he N -suiliy reduces o log -suiliy. EFEENCES [] A. A. Ds B. B. Je S. K. Piry d. K. Ji Sisicl deferred weighed suiliy d ssocied orovoi-ye roiio heore Nolier Sci. Le. A 9(3) (8)

7 O Asolue Ideed iesz Suiliy of Orhogol Series 6 [] B. B. Je S. K. Piry d U. K. Misr Sisicl deferred Cesro suiliy d is licios o roiio heores Filo (cceed) 7. [3] B. B. Je S. K. Piry d U. K. Misr Iclusio heores o geerl covergece d sisicl covergece of ( L) - suiliy usig geerlized Tueri codiios Tsui Of. J. If. Mh. Sci. 3 (7) -5. [4] B. B. Je L. N. Mishr S. K. Piry d U. K. Misr Aroiio of Sigls y Geerl Mri Suiliy wih Effecs of Gis Pheoeo Bol. Soc. Pr. M. 8 doi:.569s.v38i6.398 [5] B. B. Je Vd S. K. Piry d U. K. Misr O Geerlized Locl Proery of A -Suiliy of Fcored Fourier series I. J. Al. Al. 6 (8) 9--. [6] I. P. Nso Theory of fucios of rel vrile ( vols.) Frederic Ugr New Yor (96). [7] J. Kur S. S. Bhi Covergece of ew odified rigooeric sus i he eric sce L.J. Nolier Sci. Al. (8) [8] M. A. Srigol O soe solue Suiliy ehods Bull. Clcu Mh. Soc. 83 (99) [9] M. H. Froughi M. di Soe roeries of Lw J. Nolier Sci.Al. (9) [] P. Prid S. K. Piry H. Du B. B. Je d M. Dsh Tueri heores for Cesro suiliy of -h seueces Filo (cceed) (8). [] T. Prdh S. K. Piry B. B. Je d H. Du Sisicl deferred weighed B -suiliy d is licios o ssocied roiio heores J. Ieul. Al. 8 (8) - Aricle Id: 65. [] H. M. Srivsv B. B. Je S. K. Piry d U. K. Misr A ceri clss of weighed sisicl covergece d ssocied Korovi ye roiio heores for rigooeric fucios Mh. Mehods Al. Sci. 4 (8) [3] H. M. Srivsv B. B. Je S. K. Piry d U. K. Misr Geerlized euisisicl covergece of he deferred Norlud suiliy d is licios o ssocied roiio heores ev.. Acd. Ciec. Ecs Fis. N. Ser. A Mh. (ACSAM) (7) DOI:.7s [4] H. M. Srivsv B. B. Je S. K. Piry d U. K. Misr H. M. Srivsv B. B. Je S. K. Piry d U. K. Misr Deferred weighed A-sisicl covergece sed uo he ( )-Lgrge olyoils d is licios o roiio heores J. Al. Al. 4 (8) -6.

8 6 L.D. Je S.K. Piry.K. Ji d. Sl [5] Y. Ouy O he solue geerlized Norlud suiliy of orhogol series Tg J. Mh. 33 () 6-65.