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• W3207238302 abstract "In this paper we have proved a theorem on generalizedNorlund summability of infinite series, which generalizes variousknown results. However, the theorem is as follows:Theorem: Let {W(n)} be positive sequence such that (n)nW   is a non-increasing sequence and the series11n n (n)¥= WΣconverges and( )1logntu tdu Outd  F   =         ∫ as t®0,dbeing some fixed positive constant then the orthogonal series1( ) n nna f x¥=Σ is summable B at t = x , provided21log( ) ( ( ))nk n k O n n¥=Σ + = W .Definitions and Notations: Let { } n s be the sequence of partialsums of a series n Σa . Let the sequence { } 1( ) k kt n¥=is definedby288 Satish Chandra and Devendra Kumar Verma(1.1)101( ) ,kk n vvt n s k Nk−+== Σ IIf(1.2) lim ( ) kkt n s®¥= , a finite number, uniformly for allnI N , thenn Σu is said to be Banach summable to s 1 .Further if11( ) ( ) k kkt n t n¥+=Σ − < ¥ uniformly for all nIN ,then the series n Σu is said to be absolute Banach summable orsimply B -summable.2. Let { } n f be an orthogonal system defined in the interval(a,b) . We suppose that f (x) belongs to 2 L (a,b) and0( ) ( ) n nnf x a f x¥=»ΣWe denote by (2) ( ) n E f the best approximation to f (x) in themetric of2 L by means of polynomials of0 1 1 ( ), ( ),........ ( ) n f x f x f x − .It is well known that2(2) ( ) 1n k 2k nE f a¥= =   Σwe write n n n 1 l l l − D = −,for any sequence { } n l .(2.1) ( ) 212 1 ( )( , )( 1) ( )kvvv tg k t n vk k n v tbp−=W= ++ +ΣAbsolute Banach Summability of Orthogonal Series289(2.2)1( , ) ( , )( )(1 ) udJ k u g k t t u dtF dtbb¥− = −−∫( w k,u) = uv J (k,u)[x]= greatest integer not exceeding x1Uu =   and1tt =  " @default.
• W3207238302 created "2021-10-25" @default.
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• W3207238302 date "2013-01-01" @default.
• W3207238302 modified "2023-10-17" @default.
• W3207238302 title "Absolute Banach Summability of Orthogonal Series" @default.
• W3207238302 cites W2010863639 @default.
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