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F02402047053

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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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F02402047053

  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 || April. 2014 || PP-47-53 www.ijmsi.org 47 | P a g e Some Accepts Of Banach Summability of a Factored Conjugate Series Chitaranjan Khadanga1 , Garima Swarnakar2 Rcet, Bhilai, Department of Mathematics, ABSTRACT: An elementary proposition states that an absolutely convergent series is convergent, i.e. that if /s0-s1/+/s1-s2/+…+/sn-sn-1/< ∞ This is the analogue for series of the theorem on functions that if a function f(x) is of bounded variation in an interval, the limits exist at every point. Consider the function f(x) = ∑an xn ,lthen the series being supposed convergent in ( 0 < x < 1 ) . Summability theory has historically been concerned with the notion of assigning a limit to a linear space-valued sequences, especially if the sequence is divergent. In this paper we have been proved a theorem on Banach summability of a factored conjugate series. KEY WORDS: Summability theory, Absolute Banach summability, Conjugate series, infinite 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 (1.2)   n v vvn n n sp P t 0 1 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 the case when 1np , for all knpNkandn ,,1 summability is same as 1,C summability . For knpNk ,,1 summability is same an npN, -summability. Now , Let    1n n xB be the conjugate Fourier series of a 2 -periodic function  tf and L-integrable on  , . Then
  2. 2. Some Accepts Of Banach Summability of a Factored Conjugate Series www.ijmsi.org 48 | P a g e (1.4 )     ,...3,2,1,sin 2 0   ndttxBn    where (1.5 )     txftxft  2 1 )( Dealing with B -summability of a conjugate Fourier Series, We have the following results: Known Results: Theorem-2.1: Let )(tf be a 2 -periodic, L-integrable function on  , . Then the conjugate Fourier Series  )(xBn of )(tf is B -integrable if (i)   ,0)( BVt  and (ii)    dt t t   0 Theorem-2.2: If       0 dt t t , then the factored Fourier series   xBnn is B -summable for  n to be a non-negative convex sequence such that  n n . we wish to generalize the above two results to absolute Riesz-Banach summability. We prove Theorem-2.3: Let  np be a positive non-decreasing sequence of numbers such that   n v vn nonpP 1 , . Let (i)     ,0BVt  (ii)       0 dt t t , and (iii)    nasPOnp nn , . Then the conjugate Fourier Series   xBn is absolutely Riesz-Banach summable i.e.    BpN n, summable.
  3. 3. Some Accepts Of Banach Summability of a Factored Conjugate Series www.ijmsi.org 49 | P a g e Theorem-2.4: Let  np be a sequence of positive numbers such that   n v vn pP 1 , as n . Let (i)    dt t t   0 , and (ii)  nasPnp nn ),(0 . Then the factored conjugate Fourier series  xBnn is absolutely Riesz-Banach summable i.e.   BpN n , -summable for  n to be a non-negative convex sequence such that   n n . We need the following Lemmas for the proof of the above theorems. Lemma-2.3.1 Let  np be a positive non-decreasing sequence of numbers. Let      t 1  , then {tn} is a monotonically decreasing sequence. Lemma-2.3.2 Let  np be a sequence of positive non-decreasing, then        kn Pk is monotonically increasing in k . Proof. We have       1 1 1 11        knkn PknPkn kn P kn P kkkk              11 121        knkn ppppppnp knkn Ppkn kkkkkkk   npm,0 is non-decreasing. This proves the lemma. Lemma-2.3.3 If  n is a positive convex sequence such that   n n , then  n is a monotonically decreasing sequence. Proof of the theorem - 2.3 If  nTk is the k-th element of the Riesz-Banach transformation of the conjugate Fourier Series   xBn , then
  4. 4. Some Accepts Of Banach Summability of a Factored Conjugate Series www.ijmsi.org 50 | P a g e 1 1 1 )(    vn k v v k k sp P nT     1 11 )( 1 vn r i k v v k xBp P            1 1 1 nk ni inik k n i i xBPP P xB . Now       k v vnv kk k kk xBP PP p nTnT 11 1 1 )()()( For the series  )(xBn , we have              1 1 11 1 1 )()()( k k k v vnv kk k kk xBP PP p nTnT         1 1 01 1 )(sin)( 2 k k v v kk k dttvntP PP p           1 101 1 )(sin)( 2 k k v v kk k dttvntP PP p     21 1 21 , 2                    t where k k , say We have                   1 1 0 11 1 1 0 11 1 )( sinsin      k k v v kk k k k v v kk k dt t t tvnPt PP p dttvntP PP p Since          0 1 1 , k dt t t , if        1 11 1 sin k k v v kk k tvnPt PP p , uniformly for  t0 Now       12 1 11 1 )sin(  k k v v kk k tknP PP p t       1 11 1 )(0 k k v v kk k P PP p t     )(0),1(0)(0)(0 1 )(01)(0 1 1 1 1 1 1 n k k n k k k kk k Pnpast P pk tPk PP P t              Next,
  5. 5. Some Accepts Of Banach Summability of a Factored Conjugate Series www.ijmsi.org 51 | P a g e       2 0 11 1 )()(sin    k k v v kk k dtttvnp PP p Since 0)()0(   , we have   )( )cos( )sin( 00 td vn tvn dttvnt       Then,          2 0 11 1 )()cos(    k k v v kk k tdtvn vn P PP p     ),0(,cos)1(0 11 1   BVtastvn vn P PP p k k v v kk n        by results,      0 td        k k kn k kk k tvn kn P PP p ,)(cos)1(0 1 1        k k k Pkn p 1 1 )( )(0 , by Lemma-2.3.1,    nn k Pnpas nkn 0, 1)( 1 )(0        )1(00)(0 1    . Then      1 1 )()( k kk nTnT , uniformly for Nn . Hence  )(xBn is absolutely Riesz-Banach summable. This completes the proof of the theorem. Proof of the Theorem - 2.4 Let )(nTk be the k-th element of the Riesz – Banach transformation of the factored conjugate Fourier series  )(xBnn . Then         n i ii nk ni nik k iik nBPP P xBnT 1 1 1 )()(  Then  xBP PP p nTnT vn k v vnv kk k kk        11 1 1 )()(  …series  xBnn , we have
  6. 6. Some Accepts Of Banach Summability of a Factored Conjugate Series www.ijmsi.org 52 | P a g e               1 1 11 1 1 )()( k k k v vnvnv nk n nk xBP PP p nTnT               1 1 01 1 sin 2 n k v vnv nk n dtvn t t tP PP p     Since            0 1 1, )( k nk nTnTdt t t if            1 11 1 sin k k v vnv nk k tvnPt PP p  , uniformly for  t0 . Now                k v vnv kk k k k tvnPt PP p 111 sin     1 2 , say. We have          1 1 11 1 sin   k k v vnv kk k tvnPt PP p             1 11 1 sin0 k k v vnv kk k tvnP PP p t       1 11 1 )(0 k k v v kk k P PP p t       1 1 1 1)(0 k k kk k Pk PP p t          1 1 1 0),1(0)(0)(0 )1( )(0 k nn k k Pnpont P pk t . Next          2 11 1 sin   k k v vnv kk n tvnP PP p t . By Abel’s partial summation formula                    k v k v v p k p knkvnvvnv tpnPtpnPtvnP 1 1 1 1 1 sinsinsin 
  7. 7. Some Accepts Of Banach Summability of a Factored Conjugate Series www.ijmsi.org 53 | P a g e                1 1 110 k v knkvnvvnv PPp  Thus           2 1 1 11 1 1 )1(0   k k v knkvnvvnv kk k PPp PP p                      k k nn knk vnvvnv kk k PP P pp PP p 1 11 1 1 )1(0                 k kk k v nvvnv PP p Pp 1 1 1 111)1(0                    vk n kn k k nn n v vnvvnv P p PP p Pp 2 1 1 1 1 11                            1 11 11 1 1 )1(0 v v k kn v vnvvnv vnvvnv kP Pp Pp P                                            k k v v v vnvvnv v v vnvvnv kP P P P P p P p 1 )1(0 1 1 11 1 11                     1 1 1 1 1 1 )1(0 v v k n vn v vnv kP p                      1 11 )1(0 v v vn vn v                    1 11 )1(0 v v vn vn v   )1(0 , on n is decreasing and   n n , Hence   , uniformly for Nn . Then      BpNisnB nnn , -summable. This proves the theorem. REFERENCES: [1]. CHAMPMAN, S : On non-integral orders of summability of series and integrals, Proceedings of the London Mathematical Society, (2), 9(1910) 369-409. [2]. CHOW, A.C : Note on convergence and summability factors, Journal of the London Mathematical Society, 29(19054),459 – 476. [3]. FEKETE, M : A Szettarto vegtelen sorak elmeletchez, Mathematikal as Termesz Ertesito, 29 (1911), 719-726. [4]. GREGORY, J : Vera Circuli et hyperbalse quardrature, (1638-1675), published in 1967. [5]. HARDY, G.H : Divergent Series, Clarendow Press, Oxford, (1949).

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