A Study On (H, 1)(E, q) Product Summability Of Fourier Series And Its Conjugate Series

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1 Mahemaical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol.7, No.5, 7 A Sudy O (H, )(E, q) Produc Summabiliy Of Fourier Series Ad Is Cojugae Series Sheela Verma, Kalpaa Saxea * Research Scholar Deparme of Mahemaics, Gov. M.V.M. Bhopal * Deparme of Mahemaics, Gov. M.V.M. Bhopal alpaasaxea@gmail.com, sheelaverma9@gmail.com Absrac: I his paper, iroduce he cocep of (H,) (E,q) produc operaors ad esablishes wo ew heorems o (H,)(E,q) produc Summabiliy of Fourier series ad is cojugae series. The resuls obaied i he paper furher exed several ow resul o liear operaors. Keyword: (E,q) Summabiliy, (H,) Summabiliy, (H,) (E,q) Summabiliy.. INTRODUCTION I his field of Summabiliy of Fourier series & is allied series, he produc Summabiliy (E,q)(X),(X)(E,q) or E,q have be sudied by a umber of researchers lie, Mohay,R. ad Mohapara, S.(968), Kwee, B.(97), chadra, P.(977), chadra, P. ad Dishi, G.D.(98), sacha,m.p.(983), Bhagwa, Purima(987), Nigam, H.K. ad Sharama, Ajay(6), lal, S. Sigh,H.P. Tiwari, 8 Sadeep umar, ad Bariwal, Chadrashehar (), 3 Dhaal, Biod Prasad (), Rahore, H.L. ad Shrivassava, U.K. (), Nigam, H.K. ad Sharma, K.(,3), Siha, Saosh Kumar ad Shrivasava, U.K.(4), Mishara,V.N. Soavae, Vaishali(5) ad may more, uder various ype of crieria ad codiios. Afer his, so may resuls esablished o double facorable Summabiliy of double Fourier series, Bu ohig seems o have bee doe so far o sudy (H,) (E,q) produc Summabiliy of Fourier series ad is cojugaes series. Therefore, i his paper, wo heorems o (H,) (E,q) Summabiliy of Fourier series ad is cojugae series have bee proved uder a geeral codiio.. DEFINATION AND NOTATION Le f(x) be a - periodic fucio ad Lebesgue iegrable over (-,). The Fourier series of f(x) is give by f(x)~ a + (a cos x + b si x) A (x) (.) The cojugae series of Fourier series is give by We shall use he followig oaios: (b cos x a si x) B (x) (.) Φ()f(x+)+f(x-)-S ψ()f(x+)-f(x-) K (). log [ (K + )( + q) { ( v ) q si (v + v ) si }]

2 Mahemaical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol.7, No.5, 7 K () log [ (K + )( + q) { ( v ) q cos (v + v ) si }] Ad τ [ ], where τ deoes he greaes ieger o greaer ha Le u be a give ifiie series wih sequece of is h parial some sum of {S }. The (H,) rasform is defied as he h parial sum of (H,) Summabiliy ad is give by H () S log as (.3) K+ he he ifiie series u is summable o he defiie umber s by (E,q) mehod. If, (E, q) E q (+q) ( v )q v s v s as (.4) he he ifiie series u is summable o he defiie o. s by (H,)(E,q) Summabiliy mehod. If, H E q q S, as (.5) E log K+ 3. MAIN THEOREMS We prove he followig heorems, 3. Theorem. Le {p } be a posiive, moooic, o-icreasig sequece of real cosas such ha if, p p v as () (u) du o [ α( ),p ], as + (3.) Where, α() is posiive, moooic ad o-icreasig fucio of ad log O[{α()}. p ], as (3.) The he Fourier series (.) is summable (H,)(E,q) o f(x). 3. Theorem. Le {p } be a posiive, moooic, o-icreasig sequece of real cosas such ha If, p p v as ψ() ψ(u) du o [ α( ),p ], as + (3.3) where α() is a posiive, moooic ad o-icreasig fucio of, he he cojugae Fourier series (.) is summable o (H,)(E,q) o

3 Mahemaical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol.7, No.5, 7 a ay poi where his poi exiss. f (x) ψ() co () d 4. LEMMAS Lemma. () O(), for ; si si ; cos Proof: () [.log si(v+ (+)(+q) ( ) )q v ] v si. log [ ( + )( + q) ( (v + )si ) q v v si ]. log [ ( + )( + q) ( + ) ( v ) q v ]. log ( + ) + ( + ). log + +.log O() Lemma () o ( ), for ; si( ) ad si (). log [ ( + )( + q) ( si (v + ) ) q v v si ]. log [ ( + )( + q) ( v ) q v. ]. log [ ( ( + )( + q) ) ( v ) ]. log [ ( + ) ]. log O ( ) 3

4 Mahemaical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol.7, No.5, 7 Lemma 3. () O ( ), for ; si( ) ; cos 4 (). log [ ( + )( + q) ( cos (v + ) ) q v v si ]. log [ ( + )( + q) ( v ) q v cos (v + ) si ]. log [ ( + )( + q) ( v ) q v ]. log [ ( + ) ]. log O ( ) Lemmas 4. () O ( ), for, si( ) Now, Proof:- () [.log (+)(+q) ( v )q v. cos(v+ ) ] si. log [ ( + )( + q) Re { ( v ) q v e i(v+ ) }]. log [ ( + )( + q) Re { ( v ) q v e iv }] e i. log [ ( + )( + q) Re { ( v ) q v e iv }] τ. log [ ( + )( + q) Re { ( v ) q v e iv }] +. log [ ( + )( + q) Re { ( v ) q v e iv }] + τ

5 Mahemaical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol.7, No.5, 7 τ. log [ ( + )( + q) Re { ( v ) q v e iv }] τ. log [ ( + )( + q) ( v ) q v ] e iv τ. log ( + )( + q ) ( ) ( v ) τ. log ( ( + ) ).log q v O ( ) Ad. log [ ( + )( + q) Re { ( v ) q v e iv }] τ. log ( + )( + q) ( ) max m ( v ) q v e iv τ. log ( + q)τ ( + )( + q) τ. log ( + ) τ O ( ) 5. PROOF OF MAIN THEOREMS 5. Proof of heorem Followig Tichmash [8] ad usig Riema-Lebesgue heorem, S (f; x) of he series (.) is give by 5

6 Mahemaical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol.7, No.5, 7 S (f; x) f(x) () si ( + ) si d Therefore usig (.), he (E,q), rasform E q of S (f; x) is give by q E f(x) ( + q) () { ( v ) si ( + ) q v si } d Now deoig (H,)(E,q) rasform of S (f; x) by H E q we wrie H q E f(x). log [ ( + )( + q) () { ( v ) q v si ( + ) si } d] () ()d (5.) we have o show ha, uder he hypohesis of heorem We cosider, Now, () ()d () ()d o(), as For < <, We have [ () + () + () ] ()d I + I + I 3 (say) (5.) I () () d O() [ () d] by lemma O() [o { }] by (3.) α().p o { } α(). p o { } usig (3.) log o(), as (5.3) I () () d 6

7 Mahemaical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol.7, No.5, 7 O [ () ( ) d ] by lemma O ( ) [{ ()} O ( ) [o { α( ).p } + () d] + o ( ) d] by (3.) α( ).p Puig u i secod erm, O ( ) [o { } + o ( ) du] α(). p uα(u). p u o { } + o { }. du α(). p α(). p o { } + o { } by (3.) log log Usig secod mea value heorem for he iegral i he secod erm as α() is moooic o() + o() as, o(), as (5.4) By Riema-Lebesgue heorem ad by regulariy codiio of he mehod of Summabily, I 3 () () d o(), as (5.5) Combiig (5.3), (5.4) ad (5.5) we have H E q f(x) o(), as This complees he proof of heorem. 5. Proof of Theorem. Le s ñ (f; x) deoes he parial sum of series (.). The followig Lal[4] ad usig Riema-Lebesgue Theorem, s (f; x) of series (.) is give by s ñ (f; x) f (x) () cos ( + ) si d Therefore usig (.), he (E,q) rasform E q of s ñ (f; x) is give by q Ẽ f (x) ( + q) ψ() { ( v ) cos ( + ) q v si } d 7

8 Mahemaical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol.7, No.5, 7 Now deoig (H, )(E, q) rasform of s ñ (f; x) by H q E we wrie H q E f (x). log [ ( + )( + q) ψ() { ( v ) q v cos ( + ) si } d] ψ() ()d (5.6) I order o prove he Theorem, we have o show ha, uder he hypohesis of heorem For < <, we have We cosider, ψ() ()d ψ() () d o() as [ ψ() + ψ() + ψ() ] ()d J + J + J 3 (Say) (5.7) J ψ() () d O[ ψ() d ] by lemma 3 O( ) [ ψ() d] O() ( ) [o { α().p }] by. o { } α().p Now, J ψ() o { } usig. log o(), as (5.8) () d O[ ψ() d ] by lemma 4 O () [ ψ() d ] O( ) [{ ψ()} O ( ) [o { α( )p } + ψ ()d] + o ( ) d] by (3.3) α( ).p 8

9 Mahemaical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol.7, No.5, 7 Puig u, i secod erm, O ( ) [o { } + o ( α().p uα(u).p ) o { } + o { } α().p α().p o { } + o { } by (3.) log log du]. du Usig secod -mea value heorem for he iegral i he secod erm as α() is moooic o() + o(), as o(), as (5.9) By Riema Lebesgue heorem ad by regulariy codiio of he mehod of Summabiliy CONCLUSION J 3 ψ() () d o(), as (5.) Combiig (5.8), (5.9) ad (5.) we have, H E q - f (x) o(), as This complees he proof of heorem. I he field of Summabiliy heory, various resuls peraiig (H,) ad (E,q), (H,)X ad X(H,) Summabiliy of Fourier series as well as is allied series have bee reviewed.i fuure, he prese wor ca be exeded o esablish ew resuls uder cerai codiios. 9

10 Mahemaical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol.7, No.5, 7 REFERENCES ] Chadra, p. O he E, q summabiliy of a Fourier series ad is cojugae series Riv, Ma, Uiv. Parma (4),3, 65-78(977). ] Tichmarsh, E.C. The Theory of fucios, Oxford (95). 3] Chadra, p. ad Dishi, G.D. O he B ad E, q summabiliy of a Fourier series, is cojugae series ad heir derived series, Idia J. pure applicaios mah.,() 35 36, (98). 4] Hare Krisha Nigam, Kusum Sharma, O (E,) (C,) Summabiliy of Fourier series ad is cojugae series, Deemed uiversiy, , 3 5] H.K. Nigam, O (C,) (E,) produc meas of Fourier series ad is cojugae series,( ), , 3. 6] G.H. Hardy, Diverge series, firs ediio, oxford uiversiy (949). 7] Prasad Kahaiya, o he (N,P) C Summabiliy of a sequece of Fourier series coefficie, Idia J. pure appl. Mah., (7) , (98). 8] Nigam, H.K. ad Sharma, Ajay, o (N,P,q) (E,) Summabiliy of Fourier series, IJMMS, vol. 9, (9). 9] G.H. HARDY ad J.E. LITTLEWOOD, some ew covergece crieria for fourier series, J. Loda Mah. SOC. 7(93), ] Tiwari, Sadeep umar ad Bariwal chadrashehar, degree of approximaio of fucio belogig o he IJMA ( ), -4, (). ] E.C. Tichmarsh, he heory of fucios, oxford uiversiy press (939), ] A. ZYGMUND, Trigoomerical series, Dover, New Yor, ] Zygmud, A. Trigoomerical series, vol. I ad II, waesaw (935).

11 Mahemaical Theory ad Modelig ISSN (Paper) ISSN 5-5 (Olie) Vol.7, No.5, 7 BIOGRAPHY Dr. Kalpaa Saxea received Ph.D. i Mahemaics from Uiversiy of APS Rewa 999. Dr. Kalpaa Saxea is presely posed as a professor a GOVT. M.V.M. College Bhopal. Her mai ieress are Produc Summabiliy of Fourier series ad sequece. Sheela Verma perusig Ph.D. i Mahemaics from Baraullah Uiversiy uder he guidace of Dr. Kalpaa Saxea. I compleed Msc. I mahemaics From Sarojii aidu GOVT. Girl s P.G. College Shivaji Nagar Bhopal.