ON PRODUCT SUMMABILITY OF FOURIER SERIES USING MATRIX EULER METHOD

Transcription

1 Ieriol Jourl o Advces i Egieerig & Techology Mrch IJAET ISSN: N PRDUCT SUMMABILITY F FURIER SERIES USING MATRIX EULER METHD BPPdhy Bii Mlli 3 UMisr d 4 Mhedr Misr Depre o Mheics Rold Isiue o Techology Berhpur dish Depre o Mheics JITM Prlheudi 3 PGDepre o Mheics Berhpur Uiversiy dish 4 Pricipl NC College Auooous Jjpur dish ABSTRACT I his pper heore o produc subiliy o Fourier series usig Mrix-Euler ehod is proved EYWRDS: A - e E A - produc e Fourier series MATHEMATICS SUBJECT CLASSIFICATIN: 4B5 4B8 I INTRDUCTIN be give iiie series wih he sequece o pril sus { s } Le Le rix The he sequece o-sequece rsorio s L deies he sequece { } o he A -e o he sequece { s } s s he he series is sid o be A suble o s The codiios or regulriy o A -subiliy re esily see o be[3] I A be i sup < H where H is bsolue cos ii li I iii li 5 E E is sid o be suble he he series Le s s s E o deiie uber s 9 Vol 3 Issue pp

2 Ieriol Jourl o Advces i Egieerig & Techology Mrch IJAET ISSN: T he he series is sid o be suble o s by he A E ehod I is ow [] h E is regulr I is supposed h he ehod E s s s A is regulr hrough ou his pper Le be periodic ucio wih period iegrble i he sese o Lebesgue over - he is he Fourier series ssocied wih We use he ollowig oio hrough ou his pper x x x 7 cos b si II NWN THEREM Delig wih N p E A si si ehod o Fourier series Nigel[] proved he ollowig heore: THEREM-: p be posiive oooic o-icresig sequece o rel coss such h Le { } P p s I Φ u du s α d α s where α be posiive o-icresig ucio o he he Fourier series N p E o x he poi x suble I his pper we hve geerlied i o E A subiliy o Fourier series A is III MAIN THEREM THEREM -3: A be regulr rix d Le 9 Vol 3 Issue pp 9-96

3 Ieriol Jourl o Advces i Egieerig & Techology Mrch IJAET ISSN: Vol 3 Issue pp Φ du u α s where α is posiive o-icresig ucio o d 3 α s he he Fourier series A is suble E A he poi IV REQUIRED LEMMAS We require he ollowig Les o prove he heore LEMMA -4: PRF: For we hve si si he si si LEMMA-4: or PRF: For we hve by Jord s le si si The si si

4 Ieriol Jourl o Advces i Egieerig & Techology Mrch IJAET ISSN: Vol 3 Issue pp 9-96 V PRF F THE THEREM 3 I s is he -h pril su o he Fourier series A o he by usig Rie- Lebesgue heore ollowig Tichrch [4] we hve s - x d si si Thus he E rsor E o s is give by d x E si si I T deoe he E A rsor o s we he hve d x T si si d I order o prove he heore uder ssupio i is suicie o show h d s For < < we hve d x T d / 3 I I I sy Now d I / / d / d Usig Le - α usig 3

5 Ieriol Jourl o Advces i Egieerig & Techology Mrch IJAET ISSN: Nex s α s usig 3 I / d d usig le - Φ Φ d / / du where u/ / uα u α / d < < du α α / usig secod e-vlue heore or he iegrl i he d er s α is oooic s usig 3 s Filly 3 I d s by usig Rie Lebesgue heore d he regulriy codiio o he ehod o subiliy Thus T x s This coplees he proo o he heore VI CNCLUSIN Thus Produc Subiliy o Fourier series by Mrix-Euler ehod geerlies he N p E Produc Subiliy o Fourier series d N p q E REFERENCES - - Produc Subiliy o Fourier series [] Hrdy GH: Diverge series clredo press xord uiversiy press 949 N p E q produc subiliy o Fourier series [] NigH d usu shr : A sudy o ulr scieis vol [3] Teopli : Uber llge-eire Liere ielbilduge Prce Miyco-Fiyce [4] Tiechlch E C: The heory o ucios oxord uiversiy press Vol 3 Issue pp 9-96

6 Ieriol Jourl o Advces i Egieerig & Techology Mrch IJAET ISSN: Auhors Biogrphy B P Pdhy hs go his PhD ro Berhpur Uiversiy i He hs 4 yers o echig experiece He hs published roud 5 reserch ricles i vrious Ieriol d Niol jourls o repue He hs lso published boo Eiled Subiliy Mehods d Applicios uder Lp Lber Acdeic Publishig GbH & CoG Gery Presely he is worig s Assis Proessor i heics i Rold Isiue o Techology Berhpur dish Bii Mlli is currely worig s Ass Pro i Mheics JITM Prlheudi Gjpi dish which is uder Ceurio Uiversiy o Techology & Mgee She hs echig experiece o ore h 5yrs She hs obied her MPhil i Mheics ro Ul Uiversiy d doig reserch Ph D wor i he ield o Subiliy Theory sice 7 uder Berhpur Uiversiy U Misr is worig s culy i he PGDepre o Mheics Berhpur Uiversiy sice ls 8 yers To his credi he hs guided 3 PhDs d DSc He hs published roud 75 ppers i vrious Ieriol d Niol jourl o repue The ield o Pro Misr s reserch is Subiliy heory sequece spce Fourier series Iveory corol Mheicl Modelig He is he reviewer o Mheicl review published by Aeric Mheicl Sociey Mhedr Misr hs go his PhD i he yer 997 ro Berhpur Uiversiy He hs published roud 3 reserch ricles i vrious Ieriol d Niol jourls o repue To his credi he hs produced hree PhDs uder his guidce Presely he is worig s HD o PG Depre o Mheics i NC College Auooous Jjpur dish 96 Vol 3 Issue pp 9-96