id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
bibechana-7190	Dhakal, Binod Prasad	Approximation of a generalized Lipschitz class function by Euler - Cesàro means of Fourier series	2012	8	.pdf	application/pdf	2608	72	72	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.             +      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 ≤ .	cache/bibechana-7190.pdf	txt/bibechana-7190.txt