International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.or...
Norlund Index Summability…
www.ijmsi.org 43 | P a g e
Theorem-2.1
If  n is a non-negative and non-increasing sequence s...
Norlund Index Summability…
www.ijmsi.org 44 | P a g e
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Norlund Index Summability…
www.ijmsi.org 45 | P a g e
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Norlund Index Summability…
www.ijmsi.org 46 | P a g e
 mas),1(0 , as above
This completes the proof of the theorem.
REF...
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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.

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F024042046

  1. 1. International Journal of Mathematics and Statistics Invention (IJMSI) E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759 www.ijmsi.org Volume 2 Issue 4 ǁ April.2014 ǁ PP-42-46 www.ijmsi.org 42 | P a g e Norlund Index Summability of a Factored Fourier series Chitaranjan Khadanga1 Shubhra Sharma2 Lituranjan Khadanga3 Professor Dept .of Mathematics R C E T Bhilai(c.g) Chitakhadanga@gmail.com Shubhrasharma13@gmail.com Asst.Professor Dept. of Mathematics RCET Bhilai(c.g) Asst.professor Dept of Mathematics CEC Bhilai(c.g) ABSTRACT :- In this paper we have been proved an analogue theorem on knpN, -summability, 1k . KEY WORDS:- Summability Theory., Infinite Series , Fourier series . I. INTRODUCTION:- Let  na be a given infinite series with the sequence of partial sums  ns . Let  np be a sequence of positive numbers such that (1.1)  1,0, 0     ipPnaspP ii n r vn The sequence to sequence transformation is defined by (1.2)   n v vvn n n sp P t 0 1 and which defines the sequence of the  npN, -mean of the sequence  ns generated by the sequence of coefficients  np . The series  na is said to be summable 1,, kpN kn , if (1.3)            1 1 1 n k nn k n n tt p P In this case, when 1np , for all knpNkandn ,,1 summability is same as 1,C summability. Let  tf be a periodic function with period 2 and integrable in the sense of Lebesgue over  , . Without loss of generality, we may assume that the constant term in the Fourier series of )(tf is zero, so that (1.4)         1 1 )(sincos)( n n nnn tAntbntatf II. Dealing with 1,, kpN kn , summability factors of Fourier series, Bor has been proved the following theorem:
  2. 2. Norlund Index Summability… www.ijmsi.org 43 | P a g e Theorem-2.1 If  n is a non-negative and non-increasing sequence such that  nnp  , where  np is a sequence of positive numbers such that  nasPn and   n v nvv PtAP 1 )(0)( . Then the factored Fourier series  nnn PtA )( is summable 1,, kpN kn . III. MAIN THEOREM Let  np is a sequence of positive numbers such that  nn pppP ....21 as n and n is a non-negative, non-increasing sequence such that  nnp  . If (2.1) (i).   n v nvv PtAP 1 )(0)( (2.2) (ii).                        1 1 1 1 1 ,0 m vn v v n vn k n n mas P p P p p P and (2.3) (iii).  vvnvvn pP    01 , then the series  nnn PtA )( is summable 1,, kpN kn . 3. We need the following Lemma for the proof of our theorem. Lemma-3.1 If  n is a non-negative and non-increasing sequence such that  nnp  , where  np is a sequence of positive numbers such that  nasPn then  nasP nn )1(0 and   nnP  . PROOF OF MAIN THEOREM Let )(xtn be the n-th  npN, mean of the series ,)( 1   n nnn PxA  then by definition we have     n v v r rrrvn n n PxAp P xt 0 0 )( 1 )(       n rv vnr n r rr n pPxA P  0 )( 1 r n r rnrr n PPxA P   0 )( 1 . Then          n r n r rrrrn n rrrrn n nn xAPP P xAPP P xtxt 0 1 0 1 1 1 )( 11 )()(  )( 1 1 1 xAP P P P Pn r rrr n rn n rn           
  3. 3. Norlund Index Summability… www.ijmsi.org 44 | P a g e   )( 1 11 1 11 xAPPPPP PP rrrnrnnvn n rnn                             1 1 1 11 1 )( 1 n r r v vvrnrnnrn nn xAPPPPP PP  , using partial summation formula with 0op           1 1 11 1 1 n r rrnrnnrn nn PPpPp PP            1 1 211 n r rrnrnnrn PpPPp  , using (2.1) 4,3,2,1, nnnn TTTT  , say. In order to complete the proof of the theorem, using Minkowski’s inequality, it is sufficient to show that           1 , 1 .4,3,2,1, n k in k n n iforT p P Now, we have                             1 2 1 2 1 1 1 1, 1 1m n m n kn v vvvnk n k n nk n k n n Pp Pp P T p P                                 1 2 1 1 1 1 1 1 11m n k n v vn n n v k v k vvn n k n n p P Pp Pp P  , using Holder’s inequality                    1 2 1 1 1 1 1 1 m n k vvvv n v vn n k n n PPp Pp P O                 1 1 1 1 1 m vn n vn k n n m v vv P p p P PO    m v vvpO 1 )1(  (using 2.2)  masO ),1( .                              1 2 1 2 1 1 1 1 1 2. 1 1m n m n kn v vvvnk n k n nk n k n n Pp Pp P T p P                                   1 2 1 1 1 1 1 1 1 1 1 1 11m n k n v vn n n v k v k vvn n k n n p P Pp Pp P 
  4. 4. Norlund Index Summability… www.ijmsi.org 45 | P a g e 1 1 2 1 1 1 1 )( 1 )1(                 k vv m n vv n v vn n nk n n PPp Pp P O                      m v m vn n vn k n n vv P p p P PO 1 1 1 1 1 1 )1(  , ,)1(0 1   m v vvp  using (2.2)  mas,)1(0 . Now,                             1 2 1 2 1 1 1 1 3, 1 1m n k m n vv n v vnk n k n nk n k n n Pp Pp P T p P    1 1 1 1 1 2 1 1 1 1 11                                k n v vn n m n k vv n v vn n k n n p P Pp Pp P      1 1 2 1 1 1 1 1 1                 k vv m n vv n v vn n k n n PPp Pp P O    , 1 1 1 2 1 1 1 1                m n vv n v vn n k n n Pp Pp P O  by the lemma (3.1)                    n vn m vn n n n m v vv P p p P PO 1 1 1 1 1 )1(    m v vvp 1 )1(0  , using (2.2)  m,)1(0 . Finally,                             1 2 1 2 2 1 11 1 4, 1 m n k m n vvnv n v k n k n k n k n nk n k n n PP PP p p P T p P  k m n vvvn n v k n k n n Pp Pp P                      1 2 1 1 11 1 1 )1(0  , using (2.3)   1 1 1 1 1 1 2 1 1 11 1 11 )1(0                           k n v vn n k m n vvvn n vn k n n p P Pp Pp P  vv n v vn n n n k n n Pp Pp P                1 1 1 1 1 2 1 1 )1(0
  5. 5. Norlund Index Summability… www.ijmsi.org 46 | P a g e  mas),1(0 , as above This completes the proof of the theorem. REFERENCES: [1] ABEL, N. H : Üntersuchungen über die Reihe    2 2 )1( 1 x mm mx ,Journal fur reine and angewants mathematik (crelle) I(1826), 311-339. [2] Bor, H : On the local property of summability of factored Fourier series, Journal of mathematical Analysis and applications,163,220-226(1992). [3] HARDY, G.H : Divergent Series, Clarendow Press, Oxford, (1949). [4] MISRA, M., MISRA, M., RAUTO, K. : Absolute Banach Summabilty of Fourier Series, International Journal of Mathematical Sciences, June 2006, Vol. 1, No.1, 39 – 45. [5] PAIKARAY, S.K. : Ph.D. thesis submitted to Berhampur University, (2010) [6] PETERSEN, M : Regular matrix transformations, McGraw-Hill Publishing company limited (1996).

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