Degree of Approximation of a Class of Function by (C, 1) (E, q) Means of Fourier Series

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1 IAENG Inenaional Jounal of Applied Mahemaic, 4:, IJAM_4 7 Degee of Appoximaion of a Cla of Funcion by C, E, q Mean of Fouie Seie Hae Kihna Nigam and Kuum Shama Abac In hi pape, fo he fi ime, we inoduce he concep of C, E, q ummabiliy mehod and eablih a new heoem on degee of appoximaion of a funcion f Lip, cla by C, E, q poduc ummabiliy mean of Fouie eie. Index Tem Degee of appoximaion, Lip, cla of funcion, C, mean, E, q mean, C, E, q poduc mean, Fouie eie, Lebegue inegal. I. INTRODUCTION Alexi7, Sahney and Goel3, Chanda5, Quehi and Neha, Leindle3 and Rhoade have deemined he degee of appoximaion of a funcion belonging o Lipα cla by Ceào, Nölund and genealied Nölund ingle ummabiliy mehod. Woing in he ame diecion Sahney and Rao4, Khan8, Quehi 9- have udied he degee of appoximaion of a funcion belonging o Lipα, cla by Nölund and genealied Nölund ingle ummabiliy mehod. Theeafe, Lal and Singh6 have udied he eo eimae E n f hough igonomeic Fouie appoximaiontfa of conjugae of a funcion belonging o Lipα, p cla by C, E, poduc ummabiliy mean. Exending he eul of Lal and Singh6, in, Nigam 7 ha udied he eo appoximaion E n f hough igonomeic Fouie appoximaiontfa of conjugae of a funcion belonging o Lip, cla uing C, E, poduc mean. Recenly, Nigam8 ha alo udied Fouie eie and i conjugae eie by C, E, poduc mean. Bu nohing eem o have been done o fa o obain he degee of appoximaion of funcion belonging o Lip, cla by C, E, q poduc ummabiliy mean. The Lip, cla i a genealizaion of Lipα cla and Lipα, cla. The C, E, q mean include a pecial cae of C, E, mean fo q n. Theefoe, in an aemp o mae an advance u in hi diecion, in hi pape, a heoem on degee of appoximaion of a funcion f Lip, cla by C, E, q poduc ummabiliy mean of Fouie eie ha been poved. Ou heoem exend he eul of Lal and Singh6 o Lip, cla uing C, E, q poduc mean. II. DEFINITIONS AND NOTATIONS Le n u n be a given infinie eie wih equence of i n h paial um n. Manucip eceived May 4, ; evied Ocobe,. D. Hae Kihna Nigam i Aian Pofeo in he Depamen of Mahemaic, Faculy of Engineeing and Technology a Mo Iniue of Technology and Science Deemed Univeiy, Laxmangah, Sia Rajahan- 333, India. haeihnan@yahoo.com Kuum Shama Aian Pofeo i Reeach Schola in Depamen of Mahemaic a Mo Iniue of Technology and Science Deemed Univeiy. TheC, anfom i defined a he n h paial um of C, ummabiliy and i given by n n n + n a n n + hen he infinie eie n u n i ummable o he definie numbe by C, mehod. If E, q En q n + q n q n a n hen he infinie eie n u n i aid o be ummable E, q o a definie numbe 6. The C, anfom of E,q anfom define C,E,q anfom and we denoe i by CnE n. q Thu if C ne q n n + E q a n 3 whee Cn denoe he C, anfom of n and En q denoe he E,q anfom of n. Then he eie n u n i aid o be ummable by C, E, q mean o ummable C, E, q o a definie numbe. A. Paicula Cae Following i a paicula cae of C, E, q mean. C, E, q mean educe o C, E, mean when q n. Le f x be peiodic wih peiod and inegable in he ene of Lebegue. The Fouie eie f x i given by f x a + a n co nx + b n in nx n A n x 4 n wih n h paial um n f; x. The conjugae eie of Fouie eie 4 i given by a n co nx b n in nx B n x 5 n n Advance online publicaion: 4 May

2 IAENG Inenaional Jounal of Applied Mahemaic, 4:, IJAM_4 7 L - nom i defined by f f x dx, 6 and le he degee of appoximaion of a funcion be given by Zygmund E n f min n f 7 whee n x i ome n h degee igonomeic polynomial. Thi mehod of appoximaion i called igonomeic Fouie appoximaion TFA. A funcion f Lipα if f x + f x α fo < α 8 f Lip α, if f x + f x dx α, < α, 9 definiion 5.38 of Mc Fadden 4 Given a poiive inceaing funcion and an inege, f Lip, if f x + f x dx If α hen Lip, cla coincide wih he Lip α, cla and if hen Lip α, cla educe o he Lipα cla. We wie φ f x + + f x f x K n n + + q q in + in III. PREVIOUS RESULT Lal and Singh6 ha poved a heoem on he degee of appoximaion of a funcion fx, peiodic wih peiod and belonging o he cla Lip α, p fo < α, p by C, E, mean of conjugae eie of a Fouie eie. He ha poved he following heoem. A. Theoem If f : R R i peiodic and Lip α, p funcion hen he degee of appoximaion of i conjugae funcion f by he C, E, poduc mean of conjugae eie of Fouie eie of f aifie, fo n,,,... M n f Min CE n f p n + α p whee CE n n n+ i i i i C, E, mean of eie 5. IV. MAIN RESULT In hi pape, exending all nown eul of hi line of wo, we pove he following heoem. A. Theoem If a funcion f, peiodic, Lebegue inegal on,, belong o Lip, cla, hen i degee of appoximaion by C, E, q ummabiliy mean of i Fouie eie i given by n En q f n + n + povided ha aifie he following condiion: and n+ n+ φ δ φ d d o n + n + δ 3 whee δ i an abiay poiive numbe uch ha δ >, +,, condiion and hold unifomly in x and CnE n q i C, E, q mean of he Fouie eie 4. V. LEMMAS Fo he poof of ou heoem, following lemma ae equied. A. Lemma K n n +, fo n +. Poof: Fo n+, in n n in K n n + + q q + in in + n + + q q n + n + + ince q + q Advance online publicaion: 4 May

3 IAENG Inenaional Jounal of Applied Mahemaic, 4:, IJAM_4 7 B. Lemma K n, fo n + Poof: Fo n+, by applying Jodan lemma in and in n. K n n + + q q + q n + + q ince q + q n + VI. PROOF OF THEOREM Following Tichmah5 and uing Riemann-Lebegue heoem, n f; x he eie 4 i given by n f; x f x φ in n + in d Theefoe, uing 4, he E, q anfom E q n of n f; x i given by En q f x + q n n q n in φ in + d Now denoing C, E, q anfom of n f; x a CnE n, q we wie CnE n q n φ f x + n + q in q n in + d We conide, n+ + φ K n d n+ I + I ay 4 I n+ φ K n d Uing Hölde inequaliy and he fac ha φ Lip,, I n+ n+ φ d Kn d Kn n + n+ n + n+ n + d d by by Lemma Since i a poiive inceaing funcion and uing econd mean value heoem fo inegal, n+ I n + n + n + n + Now we conide, I d, fo ome < n + + n+ + n + n + n+ Uing Hölde inequaliy, δ φ I n+ Kn n+ δ n + δ n+ n + δ n+ Now puing y, I n + δ φ K n d d d Kn ince + 5 δ d δ n+ y y δ d by by Lemma Since i a poiive inceaing funcion and uing econd y Advance online publicaion: 4 May

4 IAENG Inenaional Jounal of Applied Mahemaic, 4:, IJAM_4 7 mean value heoem fo inegal, I n + δ n+, n + η y δ + fo ome η n + n + δ n+, n + y δ + n + δ n + n + δ n + n + n + fo ome n + y δ δ n + δ n + n + Combining I and I yield, C ne q n f n + n+ ince + 6 n + Now, uing L - nom, we ge n En q f n En q f / dx / n + dx n + n + / dx n + n + n + Thi complee he poof of he heoem. VII. APPLICATIONS Following coollaie can be deived fom ou main heoem: A. Coollay If α, < α, hen he cla Lip,,, educe o he cla Lip α, and he degee of appoximaion of a funcion f Lip α,, < α < i given by n En q f Poof: We have C ne q n f o n + n + n + α n En q f dx CnE n q f dx o Hence O n En q f dx O n + n+ n En q f n + n + fo if no he igh-hand ide will be O, heefoe n En q f α n + n + n + α B. Coollay If in coollay, hen he cla Lip α, educe o he cla f Lipα and he degee of appoximaion of a funcion f Lipα, < α < i given by C. Coollay 3 C ne q n f n + α If α, < α, hen he cla Lip,,, educe o he cla Lip α, and if q hen E, q ummabiliy educe o E, ummabiliy and he degee of appoximaion of a funcion f Lip α,, < α < i given by n En q f D. Coollay 4 n + α If in coollay 3, hen he cla Lip α, educe o he cla f Lipα and he degee of appoximaion of a funcion f Lipα, < α < i given by E. Rema n E q n f n + α Independen poof of above coollaie and 3 can be obained along he ame line of ou heoem. ACKNOWLEDGMENT The fi auho i gaeful o hi beloved paen fo hei encouagemen and uppo o hi wo. We alo expe ou incee han o he efeee fo hi valuable uggeion and commen fo he impovemen of hi pape. Advance online publicaion: 4 May

5 IAENG Inenaional Jounal of Applied Mahemaic, 4:, IJAM_4 7 REFERENCES A. Zygmund, Tigonomeic eie, Cambidge Univ. Pe, 939, Cambidge, New Yo. B. E. Rhaode, On he degee of appoximaion of funcion belonging o Lipchiz cla by Haudoff mean of i Fouie eie, Tamang J. Mah, 34 no. 3, 3, B.N. Sahney and D.S. Goel, On he degee of coninuou funcion, Ranchi Univeiy Mah. Jou., 4 973, B.N. Sahney and V. Rao, Eo bound in he appoximaion of funcion, Bull. Aualian Mah. Soc., 6 97, E.C. Tichmah, The Theoy of funcion, Oxfod Univeiy Pe, G. H. Ha, Divegen eie, Oxfod Univeiy Pe, G. Alexi, Convegence poblem of ohogonal eie, Tanlaed fom Geman by I Földe. Inenaional eie of Monogam in Pue and Applied Mahemaic, Vol H. H. Khan, On degee of appoximaion of funcion belonging o he cla Lipα, p, Indian J. Pue Appl. Mah., 5 No. 974, Anal. Appl., K. Quehi, On he degee of appoximaion of a peiodic funcion f by almo Nölund mean, Tamang J. Mah., No. 98, K. Quehi, On he degee of appoximaion of a funcion belonging o he cla Lipα, Indian J. pue Appl. Mah., 3 No. 8 98, K. Quehi, On he degee of appoximaion of a funcion belonging o weighed W, cla, Indian J. pue Appl. Mah., 3 98, K. Quehi and H. K. Neha, A cla of funcion and hei degee of appoximaion, Gania., 4 No. 99, Lázló Leindle, Tigonomeic appoximaion of funcion in L p nom, J. Mah. Anal. Appl., Leonad McFadden, Abolue Nölund ummabiliy, Due Mah. J., 9 94, Pem Chanda, Tigonomeic appoximaion of funcion in L p nom, J. Mah. Anal. Appl., 75 No., Lal and Singh, Degee of appoximaion of conjugae of Lip α, p funcion by C, E, mean of conjugae eie of a Fouie eie, Tamang Jounal of Mahemaic, Vol. 33, No., H. K. Nigam, Appoximaion of conjugae of a funcion belonging o Lip, cla by C, E, poduc mean of conjugae eie of Fouie eie, Ula Scieni of Phyical Science, Vol., No. M, H. K. Nigam, On C, E, poduc mean of Fouie eie and i conjugae Fouie eie, Ula Scieni of Phyical Science, Vol., No., Advance online publicaion: 4 May

Research Article On Double Summability of Double Conjugate Fourier Series

Research Article On Double Summability of Double Conjugate Fourier Series Inernaional Journal of Mahemaic and Mahemaical Science Volume 22, Aricle ID 4592, 5 page doi:.55/22/4592 Reearch Aricle On Double Summabiliy of Double Conjugae Fourier Serie H. K. Nigam and Kuum Sharma