BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics

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1 Biod Prasad Dhaal / BIBCHANA 9 ( : BMHSS,.5 (Olie Publicaio: Nov., BIBCHANA A Mulidisciliary Joural of Sciece, Techology ad Mahemaics ISSN 9-76 (olie Joural homeage: h://ejol.ifo/idex.h/bibchana Aroximaio of a geeralized Lischiz class fucio by uler - Cesàro meas of Fourier series Biod Prasad Dhaal Ceral Dearme of ducaio Mahemaics Tribhuva Uiversiy, Neal -mail: biod_dhaal4@yahoomail.com Aricle hisory: Received 9 November, ; Acceed 7 November, Absrac I his aer, I have ae roduc of wo summabiliy mehods, uler ad Cesàro; ad esablish a ew heorem o he degree of aroximaio of he fucio f belogig o W(L, classes by uler - Cesàro mehod. Key words ad hrases: Degree of aroximaio; (, (C, Summabiliy; Fourier series.. Defiiios ad Noaios A fucio f(x Li, if ( f (x f (x O for < ad f Li (,, if f (x f (x dx O(, <,. Give a osiive icreasig fucio,, f (x Li (,, if f (x f (x dx O( ad f W (L,, if β ( f (x f (x si x dx O (, ( β. β I is oed ha, W(L, Li(, Li(, Li So, Li Li (, Li (, W (L, for < ad. []

2 Biod Prasad Dhaal / BIBCHANA 9 ( : BMHSS,.5(Olie Publicaio: Nov., We defie he orm by f f (x dx,. The degree of aroximaio (f of fucio f: R R is give by (f Mi f. Where is rigoomeric olyomial of degree []. Le f be eriodic, iegrable over (-, i he sese of Lebesgue ad belogig o ( L, ( he is Fourier series is give by f ( a o (a cos b Le si W ξ class,. ( u be he ifiie series whose h arial sum is give by S u i. i The Cesåro meas (C, of seuece {S } is σ S. If σ S, as he seuece {S } or he ifiie series u is said o be summable by Cesåro meas mehod (C, o S. I is deoed by σ S(C,, as [3]. The uler meas (, of seuece {S } is S If S as, he seuece {S } or ifiie series u meas mehod (, o S. I is deoed by S(,, as [4]., C is said o be summable by uler The rasformaio of {σ } is deoed by, which is (, (C, rasformaio of {S } ad,c defied as σ Sr. r o, C If S, as he seuece {S } or ifiie series u (C, meas mehod o S. I is deoed by is said o be summable by (,, C S(, (C,, as. We use followig oaios. φ ( f (x f (x f (x (,C N si ( ( si (3

3 Biod Prasad Dhaal / BIBCHANA 9 ( : BMHSS,.53 (Olie Publicaio: Nov.,. Mai Theorem class by (, (C, meas of a Fourier series has bee deermied i he followig form: Theorem: If f: R R is eriodic, Lebesgue iegrable fucio i (, ad is W( L, ξ (, he he degree of aroximaio of fucio f by (,(C, meas of Fourier series ( saisfies, I rese aer, he degree of aroximaio of a fucio f W ( L,,C β f O (, for,,3,4,..... Provided saisfy he followig codiios; is moooic decreasig (4 φ β ( si d O, (5 δ φ( d O( δ where δ is a arbirary umber such ha (-δ->, codiio (5 ad (6 hold uiformly i x. (6 3. Lemmas We eed he followig Lemmas for he roof of our heorem.,c si ( Lemma : Le N (, ( si he Proof:, C N ( O(, for < <. si ( ( ( si,c N ( si ( si (

4 Biod Prasad Dhaal / BIBCHANA 9 ( : BMHSS,.54 (Olie Publicaio: Nov., [ ] 4 ( ( O (7 Lemma : Le C, N be give as Lemma I, he,c ( O ( N, for < < Proof:,C si ( ( si ( N si ( ( cos si ( ( ( (. ( O (8 4. Proof of he Theorem Followig Tichmarsh [5], h arial sum of Fourier series ( a x is give by

5 Biod Prasad Dhaal / BIBCHANA 9 ( : BMHSS,.55 (Olie Publicaio: Nov., S ( x f ( x φ si ( ( d si (C, rasform of S i.e. σ is give by φ( ( si σ. ( S ( x f ( x si( d ( x f ( x (, C φ( Similarly, (, rasform of σ i.e. is,c ( σ (x f (x (x f (x φ(, C φ( N ( si φ( d si si ( ( si d. ( si ( si (,C,C φ( N ( d φ(n d ( d I I, say. (9 φ ( W L,, we have For I, alyig Holder ieualiy ad fac ha ( φ( β I si d ( ξ O( ( β d ξ N β si (, C ( d d O( ξ ( O O ( ξ ( ε ( β ( ( d ( β ( β ( β { }, by he Mea Value Theorem, where ε < ε <.

6 Biod Prasad Dhaal / BIBCHANA 9 (3 X5-58: BMHSS,.56 (Olie Publicaio: Nov., β O ( β O (. ( For I, alyig Holder s ieualiy ad aig δ as a arbirary umber such ha (- δ - >, we have δ φ( β si ( ξ I d O ( δ φ( d ( ξ δ ( y y δ β, C ( N ( d ξ δ β si d ( δβ dy y O ξ ( δ (β δ O y dy (β δ O( ( δ y O (β δ O( ( δ ( O ( (β δ Usig codiio ( 4 β δ (β δ ( ( O( O( δ ( β δ ( O β ( β O (. ( By (9, ( ad (, we have or,c β f (,C f O β ( dx

7 Biod Prasad Dhaal / BIBCHANA 9 ( : BMHSS,.57 (Olie Publicaio: Nov., β O ( dx β O (. ( This comlees he roof of heorem. 5. Corollaries Corollary : If β o ad, < he he degree of aroximaio of a fucio f belogig o class Li(, is give by Proof:,C f O ( O (,C β f eriodic O ( ξ ( O ( ( O. (3 ( Corollary : If i corollary he he degree of aroximaio of a eriodic fucio f belogig o class Li (<< is give by, C f O, for << (. (4 Remars: A ideede roof of corollary ca be develoed alog he same lie as he heorem. xamle: Cosider he ifiie series, The (, (C, meas of he seuece {S } is give by,c σ 4 ( 3. (5 ( (. (6 ( The ifiie series (5 is eiher (C, or (, summable. Bu from (6, i is summable by (, (C, mehod. Therefore roduc summabiliy (, (C, is more owerful ha he idividual mehods (C,

8 Biod Prasad Dhaal / BIBCHANA 9 ( : BMHSS,.58 (Olie Publicaio: Nov., ad (,. Coseuely, (, (C, meas gives he beer aroximaio ha idividual mehods (C, ad (,. Refereces [] B.. Rhoades, Joural of Mahemaics, 3( [] A.Zygmud: Trigoomeric Series, Cambridge Uiversiy Press (959. [3] G. H. Hardy: O he Summabiliy of Fourier series, Proc. Lodo Mah. Soc.,( [4] G. H. Hardy : Diverge Series, The Uiversiy Press, Oxford ( 949. [5]. C. Tichmarsh.: The Theory of fucios, Secod diio, Oxford (939.

Prakash Chandra Rautaray 1, Ellipse 2

Prakash Chandra Rautaray 1, Ellipse 2 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: 48-96 www.ijera.com Vol. 3, Issue, Jauar -Februar 3,.36-337 Degree Of Aroimaio Of Fucios B Modified Parial