Prakash Chandra Rautaray 1, Ellipse 2

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1 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, Degree Of Aroimaio Of Fucios B Modified Parial Sum Of Their Cojugae Fourier Series B Geeralized Mari Mea Prakash Chadra Rauara, Ellise Dearme of Mahemaics, KIIT Uiversi, Bhubaeswar-754, Odisha, Idia Dearme of Mahemaics, Maharishi College of Naural Law, Bhubaeswar -757, Odisha, Idia Absrac The aer sudies he degree of aroimaio of cojugae of a -eriodic Lebesgue iegrable fucio f b usig modified arial sum of is cojugae Fourier series b geeralized mari mea i geeralized Holder meric. Keword. Baach Sace, geeralized Holder meric ad regular geeralized mari.. Defiiio ad Noaio The followig defiiios will be used hroughou he aer (see Zgmud [8].6,4, [5].,49 ad [3] ). (i) The sace L, icludes he sace of all -eriodic Lebesgue iegrable coiuous fucios defied i, wih -orm give b su f ; f f d ; f d ;. (ii) w w, f su f h f h whe, is called he modulus of coiui, su w w f f h f h is called he iegral modulus of coiui. c or su f h f h whe. (iv) K (+ve cosa ) The Holder meric sace H is defied b H f C : f f K ; K, wih Holder meric iduced b he orm f f f f su f, su f su c, where f,. f f ad for (v) A ormed liear sace which is comlee i he meric defied b is orm is called Baach Sace. (vi) The geeralized Holder meric sace H, is defied b, : H f L f h f K h where K > (cosa), < ad. Also he meric give b f h f f f su f h, f su, h h h w w, f h su f h f h f is called he iegral modulus of smoohess. (iii) The Lischiz codiio is give b su f h f h c K (+ve cosa) whe f ad o, Holder meric. (vii) H, f for, is called geeralized is a comlee ormed liear sace ad hece a Baach sace for. Also H,. H 36 P a g e

2 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, (viii)a geeralized mari M m i is said o be regular if k ad mk i M su m i i, k (i) Le k as uiforml i i. k f L, ;, be a -eriodic Lebesgue iegrable fucio. The he cojugae Fourier series of f a, k k k is give b f a si k b cos k ; () ak f k d where cos bk f si k d. ad The followig oaios shall be used hroughou he aer. cos D is called Dirichle s modified cojugae kerel. a f f i m i m i, k, k k ; K i mk i k ; H i mk i k cos k a cos k a f d is called cojugae fucio of f. a f, f, d a ; (4) is ver small osiive umber. where S D d (5) is called modified h arial sum of cojugae Fourier series of k k k f give b (). M S m i S (6) uiforml i i, rovided he series eiss for each, which is called he mari rasformaio of S. (7) l i; M S f, () (3) 37 P a g e

3 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, Iroducio Chadra [] ad Sahe have deermied he degree of aroimaio of a fucio C,, C, ad belogig o Li b N, meas. I 98, Quereshi discussed he degree of aroimaio of cojugae of a fucio belogig o Li ad Li, N, b meas of cojugae series of a Fourier series. I, Sham Lal [4] deermied he degree of aroimaio of cojugae of fucio belogig o weighed class W L, b mari meas of cojugae series of Fourier series. Also i, G.Das, R.N.Das ad B.K.Ra[3] sudied he degree of aroimaio i same direcio usig ifiie mari mea i geeralized Holder meric. The objecive of he rese aer is o sud more comrehesivel he resul of G.Das,R.N.Das & B.K.Ra[3] b geeralized mari mea. 3. Resul I his aer we have sudied he degree of aroimaio of cojugae fucio of f b modified arial sum of is cojugae Fourier series b geeralized mari mea i geeralized Holder meric i.e. l i; m k is k f,, k, uiforml i i. Lemma. Le. The followig lemma will be required for esablishig he heorem. The (a) w, f ad (b) K f f Proof. (a) For ad b Mikowski s iequali, we have ;where K > (cosa). f f d f f d f f d ad for, we have b modified Mikowski s iequali f f d f f d f f d su f f su f f su f f su f f f f su su f f w, f (b) Now = f f f f = f f f f B Mikowski s iequali for ad searael, we ge f f f f 38 P a g e

4 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, Theorem. which esablishes he lemma. If for, K f f ;where K > is a cosa ad for osiive icreasig sequeces ad ad such ha k (sace of regular marices) such ha k mk i O ad f H, M m i l i; O uiforml i i. Proof., The equaio (7) ca be wrie as l i; M S f, = mk i S k f, k = k k k k k, he log ; i log log ; cosk m i d m i d a a = ( m i k k as ) cosk mk i d mk i d k k a a (rovided he chage of order of summaio is ermied) cosk cosk mk i d mk i d k k a a = K i; d H i; d (8) 39 P a g e

5 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, Now K i; m i k ; cos k k a mk i cosk k a mk i a for k M O K i O (9) Also for, ad f H,, f f O B lemma(a), Su f f O O () ( b ()) () Agai (b Mikowski s iequali ) O O O ( b ()) () Cosider l i; l i; = K i; d H i; d K i ; d H i ; d I = I (sa) (3) I K i d where ; O O d (b () ad (9)) = O d = 33 P a g e

6 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, = O = O Also b Abel s rasformaio for, m icosk lim S m, i m, i S m, i ; k k k k k k where S cosk k si cos (see[],[8]) si k k si cos m, k i m, k i as lim m, i k si k k si cos mk i cos k m, k i m, k i k si si si k = O m, k i m, k i k = O i k for k m icosk O i I H i d ad ; = cos k O mk i d ( b ()) k a = cos (4) (5) O O mk i k d a for k O O O i d (b (5) = O i d = 33 P a g e

7 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, ; log ; = O i ; = O i log ; Furher order esimaes for I ad I ca be obaied as follows I K i ; d K i ; d I = I (sa) (7) Also K i; m i cosk k k a k k si si m i mk i a a k k k k k 4a 4a mk i mk i si k k mk i mk i k a k k K i; OO = O m i k m i k k k k = OO O ( m k i k k k ad b ecessar codiio of heorem) (8) Agai b lemma (b), O Now ; (9) I K i d O O O d = ( b (8),(9) ) = O O d = O O (6) 33 P a g e

8 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, = O O O = O () I K i d ad ; cosk O mk i d ( b (9)) k a = si k O O mk i d a k = k O O mk i d si k = = O M O d = O d = O log O log I I I O log Combiig (4) ad (), we ge for Hece I I I = O log O () () O log Also, I K i d (3) 333 P a g e

9 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, O O mk i cos k d ( b (9) ad a = k ( b (5) ) = O O O i d O = i d O i d = = O i ) = i O (4) Combiig (6), (4), we have for I I I = io = O i O ; i i O log ; ; log ; l i; l i; I I Hece ; = O log O i log ; l log ; ; i l i su O, ; i log ; l i; Furher K i ; d H i ; d (5) (6) cos k O O d O mk i d k a ( usig () i boh iegrals & (9) i s iegral ) = 334 P a g e

10 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, O d O O m i cos k d a k = k O O O i d = O O i d = ( b (5) ) = O O i ; log ; ; = O O i log ; ; l i; O i (7) log ; Addig (6) ad (7), we have l i; l i; ; ; su l i l i,, ; O log i log log ; (8) Hece he resul follows. This comlees he roof of heorem. 4. Corollaries Usig he above heorem he followig wo corollaries ca be esablished. Corollar,, f H, M m i is a lower riagular mari of o-egaive real If for ad umbers wih moooic icreasig i k such ha k m k k i as uiforml i i, he ; l i; O m, i, log log ; Proof.,, f H,. Le Le M m i such ha k be a lower riagular mari of o-egaive real umbers wih moooic icreasig i k k m k i as uiforml i i i.e.,,,..., ad m i m i m i m i k k, k, k k k as uiforml i i. 335 P a g e

11 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, Choose. k k k The k m i k m i k =... = O (9) Also i m i m i m i m i, k, k, k, k k k = m i m i m i m i k, k, k,, = k m, k i m, k i m, i m, i m, i m, i m, i m, i... m, i m, i m, i = = m, i m, i m, i = m, i m, i m, i, i O m i (3) Clearl all codiios of heorem are hold. Hece ; l i; O O m, i, log log ; ; O m, i (3) log log ; This esablishes corollar. Corollar,, f H, M m i is a lower riagular mari of o-egaive real If for ad umbers wih moooic decreasig i k such ha k m k k i as uiforml i i, he ; li; O m, i, log log ; Proof.,, f H,. Le Le M m i such ha k be a lower riagular mari of o-egaive real umbers wih moooic decreasig i k k m k i as uiforml i i i.e.,,,..., ad m i m i m i m i k k, k, k Choose. k k k The k m i k m i k =... k k as uiforml i i. 336 P a g e

12 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: Vol. 3, Issue, Jauar -Februar 3, O (3) = Also i m i m i, k, k k k k,, k, k m i m i, k, k m i m i = = m, i m, i m, i m, i m, i m, i m, i m, i... i m i (33) Clearl all codiios of heorem are hold. Hece ; li; O m, i, log log ; ; O m, i log log ; This esablishes corollar. (34) Remark The above resul imroves he resul of G. Das, R. N. Das ad B. K. Ra [3] akig modified arial sum S of cojugae Fourier series of f i lace of S (see [8]). Refereces [] Prem Chadra, O degree of aroimaio of fucios belogig o he Lischiz class, Naa mah. 8(975), [] G.Das, S.Paaaak, Fudameals of mahemaical aalsis (987), Taa MCGraw-Hill Publishig coma limied, New Delhi [3] G. Das, R.N. Das &B.K. Ra, Degree of aroimaio of fucios b heir cojugae Fourier series i geeralized Holder meric, Joural of he Orissa mahemaical socie, vol.7-, 998-,.6-74 [4] Sham Lal, O degree of aroimaio of cojugae of a fucio belogig o weighed W L, class b mari sumabili meas of cojugae series of a Fourier series, Tamkag Joural of mahemaics () [5] G.G.Lorez(SrocuseUiversi), Aroimaio of fucios (948), (Ahera series, Edwi Hewill, Edior) [6] Kubuddi Qureshi, O he degree of aroimaio of fucios belogig o he class Li, b meas of cojugae series, Idia J. Pure Alied, 4(98) [7] E.C Tichmarsh (Uiversi of Oford), A heor of fucios (939), (Oford Uiversi Press, New York) [8] A. Zgmud,Trigoomeric series (977) (Cambridge Uiversi Press, New York) vol.i 337 P a g e