Degree of Approximation of Fourier Series

Transcription

1 Ieaioal Mahemaical Foum Vol. 9 4 o HIARI Ld h://d.doi.og/.988/im.4.49 Degee o Aoimaio o Fouie Seies by N E Meas B. P. Padhy U.. Misa Maheda Misa 3 ad Saosh uma Naya 4 Deame o Mahemaics Rolad Isiue o Techology Golahaa-768 disha Idia Deame o Mahemaics Naioal Isiue o Sciece ad Techology PalluHills-768 disha Idia 3 Deame o Mahemaics Biaya CollegeBehamu disha Idia 4 Deame o Mahemaics Jeeva Jyoi Mahavidyalaya Raiia hadamal disha Idia Coyigh 4 B. P. Padhy e al. This is a oe access aicle disibued ude he Ceaive Commos Aibuio Licese which emis uesiced use disibuio ad eoducio i ay medium ovided he oigial wo is oely cied. Absac I his ae a heoem o degee o aoimaio o a ucio Li by oduc summabiliy E N o Fouie seies associaed wih has bee esablished. Mahemaics Subjec Classiicaio: 4B5 4B8 eywods: Degee o Aoimaio Li class o ucio E N oduc mea Fouie seies Lebesgue iegal E mea N mea

2 4 B. P. Padhy U.. Misa Maheda Misa ad Saosh uma Naya. Ioducio Le a be a give iiie seies wih seuece o umbes such ha s o is h aial sum.le { } N mea o he seuece { s }. The N s a. is deied as ollows: Whee s I. s as he he seies a is said o be N summable o s. The ecessay ad suicie codiios o egulaiy o N mehod ae: ν ν.3 i o each iege ν as ad deoes he asom o osiive eal.4 ii ν ν H whee H is a osiive umbe ideede o. ν The seuece o-seuece asomaio [].5 T ν ν deies he seuece { } T o he E mea o he seuece { }.6 T s as s ν s. I he he seies a is said o be E summable o s.clealy Fuhe he E asom o he N asom o { } τ T E mehod is egula[]. s is deied by

3 Degee o aoimaio o Fouie seies 4.7 I.8 τ s as he a is said o be E N -summable o s. Le be a eiodic ucio wih eiod L-iegable ove - The Fouie seies associaed wih a ay oi is deied by a A..9 ~ a cos b si Le by s ; be he -h aial sum o.9.the L -om o a ucio. su { R } ad he L -om is deied by :. The degee o aoimaio o a ucio degee ude om. is deied by.. P su { : R } s : R R is deied : R R by a igoomeic olyomial P o ad he degee o aoimaio E o a ucio L is give by.3 E mi P. ν This mehod o aoimaio is called Tigoomeic Fouie aoimaio. A ucio Li i.4 P. We use he ollowig oaio houghou his ae :.5

4 4 B. P. Padhy U.. Misa Maheda Misa ad Saosh uma Naya ad Fuhe he mehod E N ou he ae.. ow Theoems si si ν. is assumed o be egula ad his case is suosed hough Dealig wih he degee o aoimaio by he oduc E C seies Nigam e al [3] oved he ollowig heoem. Theoem.: I a ucio is - eiodic ad o class by C E E summabiliy mea o is Fouie seies C C asom o s ;. whee C -mea o Fouie Li he is degee o aoimaio E eeses he A is give by E asom o Subseuely Misa e al [] have oved he ollowig heoem o degee o E N o Fouie seies: aoimaio by he oduc mea Theoem.: I is a oduc E N Peiodic ucio o class Li he degee o aoimaio by he summabiliy meas o is Fouie seies deied above is give by τ whee τ as deied i Mai heoem I his ae we have oved a heoem o degee o aoimaio by he oduc mea E N o he Fouie seies o a ucio o class Li. We ove:

5 Degee o aoimaio o Fouie seies 43 Theoem -3.: I is a Peiodic ucio o he class Li he degee o aoimaio by he oduc N E summabiliy meas o is Fouie seies.9 is give by τ whee τ is as deied i Reuied Lemmas: We euie he ollowig Lemma o he oo he heoem. Lemma -4.:. Poo o Lemma-4.: Fo we have si si he si si si si. This oves he lemma.

6 44 B. P. Padhy U.. Misa Maheda Misa ad Saosh uma Naya Lemma-4.: o. Poo o Lemma-4.: Fo we have by Joda s lemma si si. The si si.. This oves he lemma. 5. Poo o Theoem 3. Usig Riema Lebesgue heoem o he -h aial sum s ; o he Fouie seies.9 o ad ollowig Tichmach [4] we have d s si si ;.

7 Degee o aoimaio o Fouie seies 45 Usig. he N asom o s ; is give by d si si. Deoig he N E asom o s ; by τ we have τ d si si d d 5. say I I Now I d si si 4. si lemma g u d d

8 46 B. P. Padhy U.. Misa Maheda Misa ad Saosh uma Naya 5. Ne d I 4. si lemma g u d I d d 5.3 The om 5. ad 5.3 we have τ su τ τ This comlees he oo o he heoem.

9 Degee o aoimaio o Fouie seies 47 Reeeces. G.H. Hady: Divege seies Fis ediio od Uivesiy ess 79.. Misa U.. Misa M. Misa B.P. Padhy ad M.. Muduli: degee o aoimaio by oduc mea E N o Fouie seies Ge. Mah. Noes ISSN Vol.6 No. 3. H.. Nigam ad Ajay Shama: degee o Aoimaio by oduc meas Ula Scieis o Physical Scieces Vol. 3 M E.C. Tichmach: The heoy o ucios ood uivesiy ess A. Zygmud : Tigoomeic Seies secod Ediio Vol.I Cambidge Uivesiy ess Cambidge 959. Received: Jauay 7 4