In this paper a theorem on degree of approximation of a function f&nbsp;Lip&nbsp;?&nbsp;by product Summability(E,r)(N,p,q)of conjugate series of Fourier series associated with f, has been established

Misra, Mahendra

Misra, U. K.

Nayak, Santosh Kumar

Padhy, B. P.

English

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ISSN 2347-1921                                                           1955 | P a g e                                                           J u l y  2 4 ,  2 0 1 4  Degree of approximation of Conjugate Series of a Fourier Series by ),,)(,( qpNrE  Means B. P. Padhy 1, U. K. Misra2, Mahendra Misra3  and Santosh Kumar Nayak4 1Department of Mathematics, Roland Institute of Technology Golanthara-761008, Odisha, India 2Department of Mathematics, National Institute of Science and Technology PallurHills-761008, Odisha, India 3Department of Mathematics, Binayak College,Berhampur Odisha, India 4Department of Mathematics, Jeevan Jyoti Mahavidyalaya, Raikia Khandamal, Odisha, India Abstract: In this paper a theorem on degree of approximation of a function Lipf    by product Summability  ),,)(,( qpNrE  of conjugate series of Fourier series associated with f , has been established. Keywords: Degree of Approximation; Lip  class of function;  rE,  mean; ),,( qpN   mean; ),,)(,( qpNrE  product mean; conjugate Fourier series; Lebesgue integral.  2010-Mathematics subject classification: 42B05, 42B08.                        Council for Innovative Research Peer Review Research Publishing System Journal: Journal of Advances in Mathematics Vol 9, No 2 editor@cirjam.org www.cirjam.com, www.cirworld.com  ISSN 2347-1921                                                           1956 | P a g e                                                           J u l y  2 4 ,  2 0 1 4  1. Introduction:         Let  na  be a given infinite series with sequence of partial sums  ns . Let  nt  denote the sequence of ),,( qpN   mean of the sequence ns . Then  nt is defined as follows: (1.1)                      nnnn sqprt01 , where                                               )0(0110   qpqpqpr nnnn                                               0111   rqp  If (1.2)                               ,stn  as n  ,  then the series  na  is said to be ),,( qpN  summable  to s  . The necessary and sufficient conditions for the regularity of ),,( qpN method are: (1.3)               (i) 0nnrqp  , as n for each integer 0  and (1.4)   (ii) nnn rHqp 0  , where H is a positive number independent of n .The sequence –to-sequence transformation [1],   (1.5)         nT    srnrknnn  011  , defines the sequence  nT   of the  rE,   mean of the sequence   ns . If (1.6)                      sTn   , as n ,    then the series  na  is said to be  rE,  summable to s .Clearly   rE,   method is regular[1].  Further, the  rE,  transform of the ),,( qpN  transform of  ns   is defined by  n   kknnknTrknr  011 (1.7)                           sqprqknrkkkknnkn00111 If  (1.8)                     sn    , as n , then  na  is said to be   qpNrE ,,, -summable to s .              Let )(tf   be a periodic function with period 2 , L-integrable over (-,), The Fourier series associated with f  at any point x is defined by  ISSN 2347-1921                                                           1957 | P a g e                                                           J u l y  2 4 ,  2 0 1 4  (1.9)    010 )(sincos2~)(nnnnn xAnxbnxaaxf  and its conjugate series is  (1.10)   1 1cos sin ( )n n nn nb nx a nx B x       Let  xfsn ;  be the n-th partial sum of the series given by (1.10).The L -norm of a function  RRf :  is defined by  (1.11)                        Rxxff :)(sup  and the L -norm is defined by  (1.12)                   120  xff   , 1 . The degree of approximation of a function RRf :  by a trigonometric polynomial )(xPn  of degree n under norm  .  is defined by  (1.13)     RxxfxpfP nn   :)()(sup     and the degree of approximation  )( fEn  of a function  Lf   is given by  (1.14)    fPfE nPnn min)( .  This method of approximation is called Trigonometric Fourier approximation.  A function  Lipf    if  (1.15)      10,)()(  tOxftxf .              We use the following notation throughout this paper: (1.16)          ,)()(21)( txftxft   and             2sin21cos2cos1121)(00tttqprrknrtKkknknnknn . Further, the method   qpNrE ,,,  is assumed to be regular and this case is supposed throughout the paper. 2. Known Theorems:          Dealing with the degree of approximation by the product    1,, CqE -mean of Fourier series, Nigam et al [3] proved the following theorem. Theorem 2.1: ISSN 2347-1921                                                           1958 | P a g e                                                           J u l y  2 4 ,  2 0 1 4  If  a function  2f is  - periodic  and of  class Lip , then its degree of approximation by    1,, CqE  summability  mean  on its Fourier series 0)(nn tA  is given  by  10,111 nOfCE nqn ,where  1nqnCE    represents the   qE,   transform of  1,C  transform of  xfsn ; . Subsequently Misra et al [2] have proved the following theorem on degree of approximation by the product mean    , , nE q N p   of the conjugate series  1.10 of the Fourier series  1.9 . Theorem 2.2: If  f  is a  2  Periodic function of class Lip , then degree of approximation by the product   , , nE q N p  summability  means on the conjugate series of its Fourier series (defined above) is given by  10,11nOfn  ,   where  n  as defined in (1.7) . 3. Main theorem:  In this paper, we have proved a theorem on degree of approximation by the product mean    qpNrE ,,,   of the Fourier series of a function of class Lip  .  We prove:  Theorem -3.1:  If  f  is a  2  Periodic function of the class  rLip , , then degree of approximation by the product   qpNrE ,,,  summability means on its Fourier series (1.9) is given by,  10,11)( nOxfn  , where n  is as defined in (1.7). 4. Required Lemmas:     We require the following Lemma for the proof the theorem. Lemma -4.1:                                  110,)()(ntnOtKn . Proof of Lemma-4.1:  For  110nt , we have   tnnt sinsin   then    2sin21cos2cos1121)(00tttqprrknrtKkkkknnknn  ISSN 2347-1921                                                           1959 | P a g e                                                           J u l y  2 4 ,  2 0 1 4                  2sin2sin.sin2cos.cos2cos112100ttttttqprrknrkkkknnkn   ttttqprrknrkkkknnkn sin2sin2sin22cos1121200        OOqprrknrkkkknnkn001121   qprkOrknrkkkknnkn 00121              ).(nO  This proves the lemma.  Lemma-4.2:                             tnfortOtKn11,1)( . Proof of Lemma-4.2: For tn 11  , we have by Jordan’s lemma, 1sin,2sin nttt.  Then                 2sin21cos2cos1121)(00tttqprrknrtKkkkknnknn     2sin2sin.2sin2cos.2cos2cos112100ttttttqprrknrkkkknnkn    2sin2sin.2sin2sin22cos1121200tttttqprrknrkkkknnkn   ISSN 2347-1921                                                           1960 | P a g e                                                           J u l y  2 4 ,  2 0 1 4                          kkkknnkn tqprrknr 00 21121              kkkknnknqprrkntr 001141 .  knnknrkntq 0141                      tO1. This proves the lemma. 5. Proof of Theorem 3.1:              Using Riemann –Lebesgue theorem, for the n-th partial sum  xfsn ;  of the Fourier series (1.9) of )(xf  and following Titchmarch [4], we have   dtKtxfxfs nn 0)(21)(;. Using (1.1),  the  qpN ,,  transform  of  xfsn ;  is given by  001cos cos1 2 2( ) ( )2sin2nn n k kkntn tt f x t p q dttr         . Denoting the    qpNrE ,,,  transform of  xfsn ;  by n , we have       fn   0 001cos cos1 1 2 2( )2 1 sin2n kn kknk kttnt r p q dttk rr                                        0( ) ( )nt K t dt                  dttKt nnn)()(11011    (5.1)                    sayII ,21   Now  ISSN 2347-1921                                                           1961 | P a g e                                                           J u l y  2 4 ,  2 0 1 4              1I  110 001cos cos1 1 2 2( )2 1 sin2n n kn kknk kttnt r p q dttk rr                                            110( ) , using lemma-4.1nO n t dt                          110)(ndttnO    ,           11011)(ntnO   111)(nnO   111)2.5(nO  Next  112 )()(nn dttKtI  2111( ) , using lemma-4.2nI t O dtt    ,              111ndttOt 111ndtt )1(1)3.5(nO Then  from (5.2) and (5.3) , we have  ISSN 2347-1921                                                           1962 | P a g e                                                           J u l y  2 4 ,  2 0 1 4                                              10,11)(  nOxfn    10,11)(sup)(  nOxfxf nxn            This completes the proof of the theorem.                                                                           References [1] G.H. Hardy:  Divergent series, First edition, Oxford University press 70(19). [2] Misra, U.K. Misra,  M. Misra, B.P. Padhy and S.K. Buxi: On degree of approximation by product mean   , , nE q N p  of the conjugate series of the  Fourier series”, International Journal of Math. and                 Sciences and Engineering Applications, ISSN 0973 – 9424, Vol.6, No.122 (2012), pp 363-370.   [3] H.K. Nigam  and Ajay Sharma: On degree of Approximation by product means, Ultra  Scientist of Physical Sciences, Vol.22 (3) M, 889-894, (2010). [4] E.C. Titchmarch: The theory of functions, oxford university press, p.p402-403(1939).  [5] A . Zygmund : Trigonometric Series , second Edition ,Vol.I , Cambridge University  press ,  Cambridge , (1959). 

Degree of approximation of Conjugate Series  of a Fourier Series by (E,r)(N,p,q) Means

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Degree of approximation of Conjugate Series of a Fourier Series by (E,r)(N,p,q) Means

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