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IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 4 (Jul. - Aug. 2013), PP 38-42 www.iosrjournals.org On Absolute Weighted Mean | A, |k -Summability Of Orthogonal Series Aditya Kumar Raghuvanshi, Riperndra Kumar & B.K. Singh Department of Mathematics IFTM University Moradabad (U.P.) India, 244001 Generalizing the theorem of Kransniqi [On absolute weighted mean summability of orthogonal series, Slecuk J. Appl. Math. Vol. 12 (2011) pp 63-70], we have proved the following theorem which gives some interesting and new results. If the series 1 2 1 k | an,n | n 1 k 2 2 2 ˆ | an , j | | c j | . j 0 n Converges for 1 k 2 then the orthogonal series c n 1 n n ( x) is summable | A, |k almost every where, Abstract: In this paper we prove the theorems on absolute weighted mean | A, |k -summability of orthogonal series. These theorems are generalize results of Kransniqi [1]. Keywords: Orthogonal Series, Nörlund Matrix, Summability. I. Let a n 0 Introduction be a given infinite series with its partial sums {sn } and Let A (an,v ) be a normal matrix, n that is lower-semi matrix with non-zero entries. By An ( s) we denote the A -transform of the sequence s {sn } , i.e. An ( s) anv sv . v 0 The series a n 0 n is said to be summable | A |k , k 1 (Sarigöl [4]) if | a n0 And the series a n 0 n |1 k | An ( s) An 1 ( s) |k nn is said to be summable | A, |k k 1, 0 if | a nn n0 | k 1 k | An ( s) |k where An (s) An ( s) An1 ( s) In the special case when A is a generalized Nörlund matrix | A |k summability is the same as | N , p, q | k summability (Sarigöl [5]). By a generalized Nörlund matrix we mean one such that anv pn vqv for 0 v n Rn anv O for v n www.iosrjournals.org 38 | Page On Absolute Weighted Mean | A, |k -Summability Of Orthogonal Series where for given sequences of positive real constant p { pn } and q {qn } , the convolution Rn ( p q)n is defined by v 0 v 0 ( p * q)n pv qn v pv v qv where ( p * q)n 0 for all n , the generalized Nörlund transform of the sequence {sn } is the sequence {tnp ,q ( s)} defined by 1 Rn tnp ,q ( s) p m 0 q s nm m m and | A |k summability reduces to the following definition: The infinite series a n 0 n is absolutely summable | N,p,q |k k 1 if the series Rn n 0 qn k 1 | tnp ,q ( s) tnp,1q ( s) |k and we write a n 0 n | N , p,q |k Let { ( x)} be an orthonormal system defined in the interval (a, b) . We assume that f ( x) belongs to L2 (a, b) and f ( z ) ~ cn n f ( x) (1.1) n0 where cn b a we write Rnj f ( x) n ( x)dx, n 0,1,2... p v j and Pn ( p *1) n q , Rnn 1 0, Rn0 Rn n v v v 0 v 0 pv and Qn (1* q)n qv Regarding to | N , p, q |1 | N , p, q | , summability of orthogonal series (1.1) the following two theorems are proved. Theorem 1.1 (Okuyama [3]) If the series 1/ 2 n R j R j 2 n n 1 | c j |2 Rn 1 n 0 j 1 Rn then the orthogonal series cn n ( x) is summable | N , p, q | almost every where. Theorem 1.2 (Okuyama [3]) Let {(n)} be a positive sequence such that {(n) / n} is a non increasing sequence and the series (n(n)) n 1 1 converges. Let { pn } and {qn } be non negative. If the series | cn |2 (n)w(1) (n) converges then the orthogonal series n 1 c j 0 j j ( x) | N , p, q | almost everywhere, where w((1)n ) is defined by 2 Rj Rj w j n n n 1 . n j Rn Rn 1 The main purpose of this paper is studying of the | A, |k summability of the orthogonal series (1.1), for 1 k 2 . Before starting the main result we introduce some further notations. (1) ( j) 1 2 www.iosrjournals.org 39 | Page On Absolute Weighted Mean | A, |k -Summability Of Orthogonal Series ˆ aˆ as follows Given a normal matrix A anv , we associates two lower semi matrices A an,v and A n ,v an,v ani , n, i 0,1, 2 i v and aˆnv anv an 1,v , aˆ00 a00 , n 1,2,... . It ma be noted that A and Â are the well-known matrices of series-to-sequence and series-to-series transformations resp. Throughout this paper we denote by Ka constant that depends only on k and may be different in different relations. II. Main Results We prove the following theorems: Theorem 2.1 If the series 1 2 1 k | a | n,n n 1 | aˆn , j | | c j | j 0 n 2 k /2 2 Converges for 1 k 2 , then the orthogonal series c n n0 n ( x) is summable | A, |k almost every where. Proof. For the matrix transform An (s)(x) of the partial sums of the orthogonal series c n0 n n ( x) we have n An ( s )( x) anv sv ( x ) v 0 n v v 0 j 0 an ,v c j n ( x) n n j 0 v 0 c j n ( x) anv n an , j c j n ( x) j 0 v where c j 0 j n ( x) is the partial sum of order v of the series (1.1) Hence n 1 An ( s )( x) an , j c j j ( x) an 1, j c j j ( x ) j 0 j 0 n 1 an ,n c j n ( x) (an , j an 1, j )c j j ( x) j 0 n 1 aˆn ,n c j n ( x) aˆn , j c j j ( x) j 0 n aˆn , j c j j ( x) j 0 Using the Hölder’s inequality and orthogonality to the latter equality we have www.iosrjournals.org 40 | Page On Absolute Weighted Mean b a 1 | An ( s)( x) | dx (b a) k k 2 | A, |k -Summability Of Orthogonal Series | A (s)( x) A b n 1 n a 2 ( s)( x) | dx K 2 K k 1 2 (b a) 2 2 b n aˆn , j c j j ( x) a j 0 Thus the series | a b nn n 1 |1 k k | An ( s)( x) |k dx a 2 2 2 k k | ann | n 1 | aˆn, j |2 | c j |2 j 0 n k /2 (2.1) Converges by the assumption. From this fact and since the function | An ( s)( x) | are non negative and by Lemma of Beppo-Levi [2] we have | a nn n 1 |1 k k | An ( s)( x) |k Converges almost every where. This completes the proof of theorem. If we put 1 H ( k ) ( A; , j ) j n 2 1 n j k 2 k 2 | nann | k 2 2 | aˆn, j |2 (2.2) then the following theorem holds true. ( n ) is a non decreasing n Theorem 2.2: Let 1 k 2 and {(n)} be a positive sequence such that sequence and the series 1 n(n) converges. If the following series n 1 n 1 converges, then the orthogonal series | c c n0 n n n 2 1 |2 k (n) H k ( A, , n) ( x) | A, |k almost every where, where H k ( A, , n) is defined by (2.2) Proof. Applying Hölder’s inequality to inequality (2.1) we get b n k | ann |1 k k An ( s )( x) dx | ann |1 k k | aˆn , j |2 | c j |2 a n 1 n 1 j 1 k n 1 1 ( n( n)) 2k 2 1 k n 1 a ( n) n,n 2 2 2 2 1 n k ( n( n)) k | aˆn , j |2 | c j |2 | ann | j 0 2k 2 k /2 k /2 2 2 2 2 1 n k ( n( n)) k | aˆn , j |2 | c j |2 | ann | j 0 n 1 2 2 2 2 1 k | c j |2 |an , n | k ( n( n) k | aˆn , j |2 n j j 1 k /2 k /2 2 1 2 2 k 2 2 ( j ) 2 k k ˆ k | c j | n | na | | a | nn n, j j n j j 1 k /2 k /2 2 1 k | c j |2 k j H k ( A, , j ) j 1 which is finite by virtue of the hypothesis of the theorem and completes the proof of the theorem. www.iosrjournals.org 41 | Page On Absolute Weighted Mean | A, |k -Summability Of Orthogonal Series References: [1] [2] [3] [4] [5] Kransniqi, Xhevat Z.; On absolute weighted mean summability of orthogonal series, Slecuk J. App. Math. Vol. 12 (2011) pp 63-70. Nantason, I.P.; Theory of functions of a real variable (2 vols), Frederick Ungar, New York 1955, 1961. Okuyama, Y.; On the absolute generalized, Nörlund summability of orthogonal series, Tamkang J. Math. Vol. 33, No. 2, (2002) pp. 161-165. Sarigöl, M.A.; On absolute weighted mean summability methods, Proc. Amer. Math. Soc. Vol. 115 (1992). Sarigöl, M.A.; On some absolute summability methods, Proc. Amer. Math. Soc. Vol. 83 (1991), 421-426. www.iosrjournals.org 42 | Page