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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. Norlund Index Summability…
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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. Norlund Index Summability…
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  )(
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. Norlund Index Summability…
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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. Norlund Index Summability…
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 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).