Microsoft Word - Binod Prasad Dhakal _151-158_.doc Binod Prasad Dhakal / BIBECHANA 9 (2013) 151-158 : BMHSS, p.151 (Online Publication: Nov., 2012) BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Approximation of a generalized Lipschitz class function by Euler - Cesàro means of Fourier series Binod Prasad Dhakal Central Department of Education Mathematics Tribhuvan University, Nepal E-mail: binod_dhakal2004@yahoomail.com Article history: Received 19 November, 2011; Accepted 27 November, 2012 Abstract In this paper, I have taken product of two summability methods, Euler and Cesàro; and establish a new theorem on the degree of approximation of the function f belonging to W(L p , ξ(t)) classes by Euler - Cesàro method. Key words and phrases: Degree of approximation; (E,1) (C,1) Summability; Fourier series. 1. Definitions and Notations A function f(x)∈ Lipα, if ( )α =−+ tO)x(f)tx(f for 0 < α ≤ 1 and f ∈ Lip (α, p), if ( ),tOdx)x(f)tx(f p 1 2 0 p α π =      −+∫ 0 < α ≤ 1, p ≥ 1. Given a positive increasing function ξ(t), p ≥ 1, f (x) ∈ Lip (ξ(t), p), if ))t((Odx)x(f)tx(f p 1 2 0 p ξ=      −+∫ π and f ∈ W (L p , ξ(t)), if ( ) ).0()),t((Odxxsin)x(f)tx(f p 1 2 0 p ≥βξ=      −+∫ π β [1] It is noted that, α →α →ξ→ξ ∞→=ξ=β α Lip)p,(Lip)p),t((Lip))t(,L(W pt)t(0p So, Lip α ⊆Lip (α, p)⊆ Lip (ξ(t), p) ⊆ W (L p , ξ(t)) for 0 < α ≤ 1 and 1p≥ . Binod Prasad Dhakal / BIBECHANA 9 (2013) 151-158 : BMHSS, p.152(Online Publication: Nov., 2012) We define the norm p by 1p,dx)x(ff p 1 p 2 0 p ≥         = ∫ π . The degree of approximation En(f) of function f: R→R is given by pnn ftMin)f(E −= . Where tn is trigonometric polynomial of degree n [2]. Let f be 2π periodic, integrable over (-π,π) in the sense of Lebesgue and belonging to ( ))t(,LW p ξ class, then its “Fourier series” is given by )ntsinbntcosa(a 2 1 )t(f nn 1n o ++= ∑ ∞ = . (1) Let ∑ ∞ =0n nu be the infinite series whose nth partial sum is given by ∑ = = n 0i in .uS The Cesåro means (C, 1) of sequence {Sn} is .S 1n 1 n 0k kn ∑ =+ =σ If ∞→→σ nas,Sn then sequence {Sn} or the infinite series ∑ ∞ =0n nu is said to be summable by Cesåro means method (C,1) to S. It is denoted by n S(C,1), as n [3].σ → → ∞ The Euler means (E, 1) of sequence {Sn} is ∑ =       = n 0k kn 1 n S k n 2 1 E If ∞→→ n,asSE1 n then sequence {Sn} or infinite series ∑ ∞ =0n nu is said to be summable by Euler means method (E, 1) to S. It is denoted by 1 n E S(E,1), as n [4].→ →∞ The 1 nE transformation of {σn} is denoted by ,t 11 C,E n which is (E, 1) (C, 1) transformation of {Sn} and defined as ∑ = σ      = n 0k kn C,E n k n 2 1 t 11 ∑ ∑ = =+      = n 0k r k or n .S 1k 1 k n 2 1 If ∞→→ nas,St 11 C,E n then sequence {Sn} or infinite series ∑ ∞ =0n nu is said to be summable by (E, 1) (C, 1) means method to S. It is denoted by .nas),1,C()1,E(St 11 C,E n ∞→→ We use following notations. )x(f2)tx(f)tx(f)t( −−++=φ (2) 2 t2 2 t2n 0k 1n C,E n sin)1k( )1k(sin k n 2 1 N 11 + +       π = ∑ = + (3) Binod Prasad Dhakal / BIBECHANA 9 (2013) 151-158 : BMHSS, p.153 (Online Publication: Nov., 2012) 2. Main Theorem In present paper, the degree of approximation of a function ( ))t(,LWf p ξ∈ class by (E,1) (C,1) means of a Fourier series has been determined in the following form: Theorem: If f: R → R is 2π periodic, Lebesgue integrable function in ),( ππ− and is W ( ))t(,Lp ξ , then the degree of approximation of function f by (E,1)(C,1) means of Fourier series (1) satisfies, ( )     ξ+=− + +β 1n 1 p C,E n p 111 )1n(Oft , for ..,.........4,3,2,1n = Provided ξ(t) satisfy the following conditions;      ξ t )t( is monotonic decreasing (4)       + =                 ξ φ β∫ + 1n 1 Odttsin )t( )t(t p 1 p p 0 1n 1 , (5) ( )( )δ δ−π +=                 ξ φ ∫ + 1nOdt )t( )t(t p 1 p 1n 1 (6) where δ is an arbitrary number such that q(1-δ)-1> 0, condition (5) and (6) hold uniformly in x. 3. Lemmas We need the following Lemmas for the proof of our theorem. Lemma 1: Let ∑ = + + +       π = n 0k 2 t2 2 t2 1n C,E n , sin)1k( )1k(sin k n 2 1 )t(N 11 then )1n(O)t(N 11 C,E n += , for . 1n 1 t0 + << Proof: ∑ = + + +       π = n 0k 2 t2 2 t2 1n C,E n sin)1k( )1k(sin k n 2 1 )t(N 11 ∑ = + + +       π ≤ n 0k 2 t2 2 t22 1n sin)1k( sin)1k( k n 2 1 ∑ = + +      π = n 0k 1n )1k( k n 2 1 Binod Prasad Dhakal / BIBECHANA 9 (2013) 151-158 : BMHSS, p.154 (Online Publication: Nov., 2012)             +      π = ∑ ∑ = = + n 0k n 0k 1n k n k n k 2 1 [ ]n1n 1n 22n 2 1 + π = − +       π + = 4 2n ( ) π + ≤ 2 1n )1n(O += (7) Lemma 2: Let 11 C,E nN be given as Lemma I, then         + = 2 C,E n t)1n( 1 O)t(N 11 , for π<< + t 1n 1 Proof: ∑ = + + +       π ≤ n 0k 2 t2 2 t2 1n C,E n sin)1k( )1k(sin k n 2 1 )t(N 11 ∑ = + + +−       π = n 0k 2 t21n sin)1k(2 t)1k(cos1 k n 2 1 ∑ = + +       π ≤ n 0k 2 t21n sin)1k( 1 k n 2 1 ∑ = + +      π = n 0k 21n )1k( 1 k n t2       + −π = + + 1n 12 t2 1n 21n       − + π = +1n2 2 1 1 t)1n( 2t)1n( + π ≤ . t)1n( 1 O 2         + = (8) 4. Proof of the Theorem Following Titchmarsh [5], n th partial sum of Fourier series (1) at t = x is given by Binod Prasad Dhakal / BIBECHANA 9 (2013) 151-158 : BMHSS, p.155 (Online Publication: Nov., 2012) ( ) dt sin tnsin )t( 2 1 )x(f)x(S 2 t 2 t 0 n + φ π =− ∫ π . (C,1) transform of Sn i.e. σn is given by ( ) ( ) dttksin sin )t( )1n(2 1 )x(f)x(S 1n 1 2 t n 0k2 t 0 k n 0k + φ π+ =− + ∑∫∑ = π = dt sin )1n(sin )t( )1n(2 1 )x(f)x( 2 t2 2 t2 0 n + φ π+ =−σ ∫ π . Similarly, (E,1) transform of σn i.e. 11 C,E nt is ( ) ( ) dt sin)1k( 1ksin k n )t( 2 1 )x(f)x( k n 2 1 2 t2 2 t2n 0k0 1nk n 0k n + +       φ π =−σ      ∑∫∑ = π + = dt sin)1k( )1k(sin k n 2 1 )t()x(f)x(t 2 t2 2 t2n 0k 1n 0 C,E n 11 + +       π φ=− ∑∫ = + π dt)t(N)t( 11 C,E n 0 φ= ∫ π dt)t(N)t(dt)t(N)t( 11 1n 1 11 1n 1 C,E n C,E n 0 φ+φ= ∫∫ π + + 21 II += , say. (9) For I1, applying Holder inequality and fact that ( ))t(,LW)t( p ξ∈φ , we have p 1 p 0 1 dttsin )t( )t(t I 1n 1                 ξ φ ≤ β∫ + q 1 q C,E n 0 dt)t(N tsint )t( 11 1n 1               ξ β∫ + = ( ) 1 1 +nO q 1 1n 1 dt t )1n)(t( 0 q 1               +ξ ∫ + +β = O ( ))( 1 1 +nξ q 1 1n 1 dtt )1(q         ∫ + ε +β− , by the Mean Value Theorem, where 1n 1 0 + <ε< . ( ) q 1 1n 1 1)1(q t )(O 1)1(q 1n 1               ++β− ξ= + ε ++β− + ( ) ( ){ }q 1 1q)1( 1n 1 1n)(O −+β + +ξ= Binod Prasad Dhakal / BIBECHANA 9 (2013) X151-158: BMHSS, p.156 (Online Publication: Nov., 2012)      ξ+= + −+β )()1n(O 1n 1 1 q 1      ξ+= + +β )()1n(O 1n 1p 1 . (10) For I2, applying Holder’s inequality and taking δ as an arbitrary number such that q (1- δ) -1 > 0, we have p 1 p 2 dttsin )t( )t(t I 1n 1                 ξ φ ≤ β δ−π ∫ + q 1 q C,E n dt tsint )t(N)t( 11 1n 1                     ξ βδ− π ∫ + p 1 p dt )t( )t(t 1n 1                 ξ φ = δ−π ∫ + q 1 q 2 dt )1n(t )t( 1n 1               + ξ +β+δ− π ∫ + ( )1)1n(O −δ+= q 1 2 q 2 y 1 1n y dy y )( 1               −        ξ −β−δ + ∫ π = O             + ξ+ δ 1n 1 )1n(                   −δ−β+ + ∫ π q 1 1 dyyO 2)1(q 1n Using condition ( 4) = O ( ))()1n( 1n 1 + δ ξ+                         −δ−β+ +−δ−β+ π q 1 1 1n 1)1(q 1)1(q y O =O ( ))()1n( 1n 1 + δ ξ+ ( )               −+ −δ−β+ −δ−β+ π −δ−β+ q 1 1)1(q11)1(q )()1n( 1)1(q 1 O = O ( ))()1n( 1n 1 + δ ξ+ ( )( )q 11 1nO −δ−β++ = O      ξ+ + −+β )()1n( 1n 1 1 q 1 = O      ξ+ + +β )()1n( 1n 1p 1 . (11) By (9), (10) and (11), we have      ξ+=− + +β )()1n(ft 1n 1C,E n p 1 11 or p 1 p 111 dx)()1n(Oft p 1n 1 2 0p C,E n            ξ+=− + +β π ∫ Binod Prasad Dhakal / BIBECHANA 9 (2013) 151-158 : BMHSS, p.157 (Online Publication: Nov., 2012) = O      ξ+ + +β )()1n( 1n 1p 1 p 1 dx 2 0       ∫ π = O      ξ+ + +β )()1n( 1n 1p 1 . (12) This completes the proof of theorem. 5. Corollaries Corollary 1: If 10,t)t(ando ≤α<=ξ=β α then the degree of approximation of a π2 periodic function f belonging to class )p,(Lip α is given by         + =− −α p 1 11 )1n( 1 Oft p C,E n Proof: ( )     ξ+=− + +β 1n 1 p C,E n p 111 )1n(Oft = O      ξ+ + )()1n( 1n 1p 1 = O      + α+ )1n( 1p 1 )1n( = O         + −α p 1 )1n( 1 . (13) Corollary 2: If ∞→p in corollary 1 then the degree of approximation of a π2 periodic function f belonging to class )10(Lip <α<α is given by 10for, )1n( 1 Oft 11 C,E n <α<      + =− α ∞ . (14) Remarks: An independent proof of corollary can be developed along the same line as the theorem. Example: Consider the infinite series, ∑ ∞ = −−− 1n 1n)3(41 . (15) The (E,1) (C,1) means of the sequence {Sn} is given by ∑ = σ      = n 0k kn C,E n k n 2 1 t 11 ( )1n)1(1 )1n(2 1 +−− + = . (16) The infinite series (15) is neither (C,1) nor ( E,1) summable. But from (16), it is summable by (E,1) (C,1) method. Therefore product summability (E,1) (C,1) is more powerful than the individual methods (C,1) Binod Prasad Dhakal / BIBECHANA 9 (2013) 151-158 : BMHSS, p.158 (Online Publication: Nov., 2012) and (E,1). Consequently, (E,1) (C,1) means gives the better approximation than individual methods (C,1) and (E,1). References [1] B. E. Rhoades, Journal of Mathematics, 3(2003) 245-247. [2] A .Zygmund: Trigonometric Series, Cambridge University Press (1959). [3] G. H. Hardy: On the Summability of Fourier series, Proc. London Math. Soc.,12(1913) 365-372. [4] G. H. Hardy : Divergent Series, The University Press, Oxford ( 1949). [5] E. C. Titchmarsh.: The Theory of functions, Second Edition, Oxford (1939).