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A NOTE ON INDEX SUMMABILITY FACTORS OF FOURIER SERIES

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Author : U.K. Misra, Mahendra Misra and B.P. Padhy Published in : Journal of Computer and Mathematical Sciences ABSTRACT A theorem on index summability factors of Fourier Series has been established
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  A NOTE ON INDEX SUMMABILITY FACTORSOF FOURIER SERIES U.K. Misra 1  , Mahendra Misra  2  and B.P. Padhy  3 1  Department of Mathematics Berhampur University Berhampur-760007, Orissa (India) e-mail: umakanta_misra@yahoo.com 2 Principal Government Science CollegeMlkangiri, Orissa 3  Roland Institute of TechnologyGolanthara-760008, Orissa (India) e-mail: iraady@gmail.com ABSTRACT A theorem on index summability factors of Fourier Series has been established  Key words:  Summability factors, Fourier series. AMS Classification No: 40D25 J. Comp. & Math. Sci. Vol. 1(1), 67-70 (2009).1. INTRODUCTION Let   n a  be a given infinite serieswith the sequence of partial sums   n s . Let   n  p  be a sequence of positive numbers suchthat(1.1)  , 0      Pnas pP inr vn  1,0     i p i The sequence to sequence transformation(1.2)    nvvvnnn  s pPt  0 1 defines the sequence of the   n  p N  ,  -meanof the sequence   n s  generated by the sequenceof coefficients   n  p .The series   n a  is said to be summable 1,,   k  p N  k n , if (1.3)        111 nk nnk nn t t  pP In the case when 1  n  p , for all k and n ,1  k n  p N  ,  summability is same as 1, C   summa- bility. For k n  p N k  ,,1   summability is same  [ 68 ] as n  p N  ,  - summability..Let   n    be any sequence of positivenumbers. The series   n a is said to be summable 1,,,   k  p N  k nn    , if (1.4)     111 nnnk n  t t    .Clearly, k nnn  pP p N  ,,  is same as 1,,   k  p N  k n and 1 1,, n  p N   is n  p N  , .Let    t  f   be a periodic function with period 2  and integrable in the sense of Lebesgue over       ,  . Without loss of generality, we mayassume that the constant term in the Fourier series of  )( t  f   is zero, so that(1.5) )( t  f  ~ )()sincos( 11 t  Ant bnt a nnnnn      2. Known Theorem :  Dealing with 1,,   k  p N  k n , summabilityfactors of Fourier series, Bor  1  proved thefollowing theorem: Theorem-A :  If   n    is a non-negativeand non-increasing sequence such that    nn  p    , where   n  p  is a sequenceof positive numbers such that   nasP n and    nvnvv  Pt  AP 1 )(0)( . Then the factored Fourier series   nnn  Pt  A    )(  is summable 1,,   k  p N  k n .We prove an analogue theorem on k nn  p N     ,,  - summability, 1  k  , in thefollowing form:3. Main Theorem : Theorem:  Let   n  p  is a sequence of  positive numbers such that  n  p pP 21   n  p ....  as  n  and   n    is a non-negative, non-increasing sequence such that    nn  p    . If (3.1) (i).    nvnvv  Pt  AP 1 )(0)( (3.2) (ii).             11111 ,0 mvn vvnvnk n P pP p      mas  and  (3.3) (iii).   vvnvvn  pP        0 1  ,then the series   nnn  Pt  A    )(  is summable 1,,,   k  p N  k nn    . and   n   , a sequence of  positive numbers.4. Required Lemma : We need the following Lemma for the proof of our theorem.  Lemma [1]:  If   n    is a non-negative  [ 69 ] and non-increasing sequence such that    nn  p    , where   n  p  is a sequence of  positive numbers such that   nasP n then   nasP nn )1(0    and     nn P    .5. Proof of the Theorem : Let )(  xt  n  be the n -th   n  p N  ,  meanof the series ,)( 1   nnnn  P x A     then by definitionwe have       nvvr r r r vn nn  P x A p P xt  00 )(1)(         nr vvnr nr r r n  pP x A P   0 )(1   r nr r nr r  n PP x A P      0 )(1 .Then       nr r r r r n nnn  x APP P xt  xt  01 1)()(         nr r r r r n n  x APP P 1011 )(1   )( 111  x APPPPP nr r r r nr nnr n                    nr r r r nr nnvn nn  x APPPPP PP 1111 )(1             11111 1  nr r nr nnr n nn PPPP PP            1 )( r vvv  x AP , using partial summationformula with 0  o  p .         11111 1  nr r r nr nnr n nn PP pP p PP           11211 nr r r nr nnr n  P pPP p    , using(3.1)  4,3,2,1,  nnnn  T T T T    , say..In order to complete the proof of thetheorem, using Minkowski's inequality, it issufficient to show that     1,1 .4,3,2,1, nk ink n  i for T     Now, we have          12121111,1 1 mnmnk nvvvvn k nk nk nk n  P pPT                 12111 1 mnnvk vk vvnnk  P pP              111 1 k nvvnn  pP , using Holder's inequality.          121111 11 mnk vvvv nvvnnk n  PP pPO               1111 1 mvn nvnk nmvvv P pPO           mvvv  pO 1 )1(    (using 3.2)    masO ),1( .          1212111112.1 1 mnmnk nvvvvn k nk nk nk n  P pPT                 1211111 1 mnnvk vk vvnnk n  P pP             11111 1 k nvvnn  pP   11211111 )(1)1(       k vvmnvvnvvnnk n  PP pPO                 mvmvn nvnk vv P pPO 111111 )1(       ,)1(0 1    mvvv  p    using (3.2)    mas ,)1(0 . Now,          121211113,1 1 mnk mnvvnvvnk nk nk nk n  P pPT           1111121111 11              k nvvnnmnk vvnvvnnk n  pPP pP          1121111 11        k vvmnvvnvvnnk n  PP pPO          ,11 121111     mnvvnvvnnk n  P pPO      by the lemma          nvnmvnnnmvvv P pPO 11111 )1(          mvvv  p 1 )1(0    , using (3.2)    m ,)1(0 .Finally,       1212114,1 mnmnk nk nk nk nk nk n PP pT              211 k vvnv nv PP      k mnvvvn nvk nk n  P pP         1211111 1)1(0     ,using (3.3)  1211111 1)1(0      mnvnnvnk n  pP       11111 1        k nvvnnk vv  pPP      vvnvvnnnnk n  P pP        1111121 1)1(0     mas ),1(0 , as aboveThis completes the proof of the theorem. REFERENCE 1.Bor Huseyin: On the absolute summabilityfactors of Fourier Series, Journal of Compu-tational Analysis and Applications, Vol. 8, No. 3, 223-227 (2006). [ 70 ]

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