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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 || 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. Some Accepts Of Banach Summability of a Factored Conjugate Series
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(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. Some Accepts Of Banach Summability of a Factored Conjugate Series
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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. Some Accepts Of Banach Summability of a Factored Conjugate Series
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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. Some Accepts Of Banach Summability of a Factored Conjugate Series
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   


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. Some Accepts Of Banach Summability of a Factored Conjugate Series
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   



 



 
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 

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    





 



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).