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. e-mail: 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
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
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
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
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, 45-47. 3 B.N. Sahney and D.S. Goel, On he degee of coninuou funcion, Ranchi Univeiy Mah. Jou., 4 973, 5-53. 4 B.N. Sahney and V. Rao, Eo bound in he appoximaion of funcion, Bull. Aualian Mah. Soc., 6 97, -8. 5 E.C. Tichmah, The Theoy of funcion, Oxfod Univeiy Pe, 939. 6 G. H. Ha, Divegen eie, Oxfod Univeiy Pe, 949. 7 G. Alexi, Convegence poblem of ohogonal eie, Tanlaed fom Geman by I Földe. Inenaional eie of Monogam in Pue and Applied Mahemaic, Vol. 96. 8 H. H. Khan, On degee of appoximaion of funcion belonging o he cla Lipα, p, Indian J. Pue Appl. Mah., 5 No. 974, 3-36. Anal. Appl., 35. 9 K. Quehi, On he degee of appoximaion of a peiodic funcion f by almo Nölund mean, Tamang J. Mah., No. 98, 35-38. K. Quehi, On he degee of appoximaion of a funcion belonging o he cla Lipα, Indian J. pue Appl. Mah., 3 No. 8 98, 898-93. K. Quehi, On he degee of appoximaion of a funcion belonging o weighed W, cla, Indian J. pue Appl. Mah., 3 98, 898-93. K. Quehi and H. K. Neha, A cla of funcion and hei degee of appoximaion, Gania., 4 No. 99, 37-4. 3 Lázló Leindle, Tigonomeic appoximaion of funcion in L p nom, J. Mah. Anal. Appl., 3 5. 4 Leonad McFadden, Abolue Nölund ummabiliy, Due Mah. J., 9 94, 68-7. 5 Pem Chanda, Tigonomeic appoximaion of funcion in L p nom, J. Mah. Anal. Appl., 75 No., 3-6. 6 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., 69-74. 7 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, 95-3. 8 H. K. Nigam, On C, E, poduc mean of Fouie eie and i conjugae Fouie eie, Ula Scieni of Phyical Science, Vol., No., 49-48. Advance online publicaion: 4 May