1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 4 (Jul. - Aug. 2013), PP 38-42
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www.iosrjournals.org 38 | Page
On Absolute Weighted Mean | , |kA  -Summability Of Orthogonal
Series
Aditya Kumar Raghuvanshi, Riperndra Kumar & B.K. Singh
Department of Mathematics IFTM University Moradabad (U.P.) India, 244001
Generalizing the theorem of Kransniqi [On absolute weighted mean summability of orthogonal series, Slecuk J.
Appl. Math. Vol. 12 (2011) pp 63-70], we have proved the following theorem which gives some interesting and
new results.
If the series
1 22 1
2 2
, ,
1 0
ˆ| | | | | | .
k
n
k
n n n j j
n j
a a c

    
 
 
  
 
  
 
Converges for 1 2k  then the orthogonal series
1
c ( )n n
n
x



is summable | , |kA  almost every where,
Abstract: In this paper we prove the theorems on absolute weighted mean | , |kA  -summability of orthogonal
series. These theorems are generalize results of Kransniqi [1].
Keywords: Orthogonal Series, Nörlund Matrix, Summability.
I. Introduction
Let
0
n
n
a


 be a given infinite series with its partial sums { }ns and Let ,( )n vA a be a normal matrix,
that is lower-semi matrix with non-zero entries. By ( )nA s we denote the A -transform of the sequence
{ }ns s , i.e.
0
( ) .n nv v
v
A s a s


 
The series
0
n
n
a


 is said to be summable | A |k , 1k  (Sarigöl [4]) if
1
1
0
| | | ( ) ( )|k k
nn n n
n
a A s A s




  
And the series
0
n
n
a


 is said to be summable | , |kA  1, 0k   if
1
0
| | | ( ) |k k k
nn n
n
a A s

  

  
where 1( ) ( ) ( )n n nA s A s A s  
In the special case when A is a generalized Nörlund matrix | A |k summability is the same as
| , , |kN p q summability (Sarigöl [5]).
By a generalized Nörlund matrix we mean one such that
n v
nv
n
p vq
a
R

 for 0 v n 
nva O for v n

2. On Absolute Weighted Mean | , |kA  -Summability Of Orthogonal Series
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where for given sequences of positive real constant { }np p and { }nq q , the convolution ( )n nR p q  is
defined by
0 0
( * )n v n v v v v
v v
p q p q p q
 
 
 
  
where ( * ) 0np q  for all n, the generalized Nörlund transform of the sequence { }ns is the sequence
,
{ ( )}p q
nt s defined by
,
0
1
( )p q
n n m m m
mn
t s p q s
R



 
and | |kA summability reduces to the following definition:
The infinite series
0
n
n
a


 is absolutely summable | N,p,q | 1k k  if the series
1
, ,
1
0
| ( ) ( ) |
k
p q p q kn
n n
n n
R
t s t s
q




 
   
 

and we write
0
| ,p,q |n k
n
a N



Let { ( )}x be an orthonormal system defined in the interval ( , )a b . We assume that ( )f x belongs to
2
( , )L a b and
0
( ) ~ ( )n n
n
f z c f x


 (1.1)
where ( ) ( ) , 0,1,2...
b
n na
c f x x dx n 
we write 1 0
, 0,j n
n n v v n n n
v j
R p q R R R




  
and
0
( *1)n n v
v
P p p


   and
0
(1* )n n v
v
Q q q


  
Regarding to 1| , , | | , , |N p q N p q , summability of orthogonal series (1.1) the following two theorems are
proved.
Theorem 1.1 (Okuyama [3]) If the series
1/22
21
0 1 1
| |
j jn
n n
j
n j n n
R R
c
R R


  
   
    
   
 
then the orthogonal series ( )n nc x is summable | , , |N p q almost every where.
Theorem 1.2 (Okuyama [3]) Let { ( )}n be a positive sequence such that { (n) / n} is a non increasing
sequence and the series
1
1
( ( ))
n
n n



 converges. Let { }np and { }nq be non negative. If the series
2 (1)
1
| | ( ) ( )n
n
c n w n


 converges then the orthogonal series
0
( ) | , , |j j
j
c x N p q


 almost everywhere, where
(1)
( )nw is defined by
2
(1) 1 2 1
( )
1
.
j j
n n
j
n j n n
R R
w j n
R R

 
 
 
  
 

The main purpose of this paper is studying of the | , |kA  summability of the orthogonal series (1.1),
for 1 2k  . Before starting the main result we introduce some further notations.

3. On Absolute Weighted Mean | , |kA  -Summability Of Orthogonal Series
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Given a normal matrix nvA a , we associates two lower semi matrices ,n vA a and ,
ˆ ˆn vA a as follows
, , , 0,1,2n v ni
i v
a a n i


 
and 1,
ˆnv nv n va a a   , 00 00
ˆ , 1,2,...a a n  .
It ma be noted that A and ˆA are the well-known matrices of series-to-sequence and series-to-series
transformations resp.
Throughout this paper we denote by Ka constant that depends only on k and may be different in different
relations.
II. Main Results
We prove the following theorems:
Theorem 2.1 If the series
/2
1
2 1
2 2
, ,
1 0
ˆ| | | | | |
k
n
k
n n n j j
n j
a a c

    
 
 
  
 
  
 
Converges for 1 2k  , then the orthogonal series
0
( )n n
n
c x



is summable | , |kA  almost every where.
Proof. For the matrix transform (s)(x)nA of the partial sums of the orthogonal series
0
( )n n
n
c x


 we have
0
,
0 0
0 0
,
0
( )( ) ( )
( )
( )
( )
n
n nv v
v
n v
n v j n
v j
n n
j n nv
j v
n
n j j n
j
A s x a s x
a c x
c x a
a c x




 
 






 
 

where
0
( )
v
j n
j
c x

 is the partial sum of order v of the series (1.1)
Hence
1
, 1,
0 0
1
,n , 1,
0
1
,n ,
0
,
0
( )( ) ( ) ( )
( ) ( ) ( )
ˆ ˆ( ) ( )
ˆ ( )
n
n n j j j n j j j
j j
n
n j n n j n j j j
j
n
n j n n j j j
j
n
n j j j
j
A s x a c x a c x
a c x a a c x
a c x a c x
a c x
 
 
 

 

 






  
  
 

 



Using the Hölder’s inequality and orthogonality to the latter equality we have

4. On Absolute Weighted Mean | , |kA  -Summability Of Orthogonal Series
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 1 222
1
2 2
1
2
,
0
| ( )( ) | ( ) | ( )( ) ( )( ) |
ˆ( ) ( )
Kk
b b
k
n n na a
K
k nb
n j j ja
j
A s x dx b a A s x A s x dx
b a a c x




   
 
  
 
 
 

Thus the series
1
1
| | | ( )( )|
b
k k k
nn na
n
a A s x dx

 

 
/22
2 2
2 2
,
1 0
ˆ| | | | | |
k
n
k
nn n j j
n j
k a a c
  
 
 
  
 
  (2.1)
Converges by the assumption.
From this fact and since the function | ( )( )|nA s x are non negative and by Lemma of Beppo-Levi [2] we have
1
1
| | | ( )( )|k k k
nn n
n
a A s x

 


Converges almost every where.
This completes the proof of theorem.
If we put
2 2
2 2
( ) 2
,2
1
1
ˆ( ; , ) | | | |k k k
nn n j
n jk
H A j n na a
j


  
 
  (2.2)
then the following theorem holds true.
Theorem 2.2: Let 1 2k  and { ( )}n be a positive sequence such that
( )n
n
 
 
 
is a non decreasing
sequence and the series
1
1
( )n n n

 
 converges. If the following series
2
1
2
1
| | ( ) ( , , )kk
n
n
c n H A n
 


converges, then the orthogonal series
0
( ) | , |n n k
n
c x A 


 almost every where, where ( , , )k
H A n is defined
by (2.2)
Proof. Applying Hölder’s inequality to inequality (2.1) we get
/2
1 1 2 2
,
1 1 1
/22 2
2 2 1
2 2
,2
1 02
2
2 22 2 2 1
1 1,
ˆ| | ( )( ) | | | | | |
1
ˆ| | ( ( )) | | | |
( ( ))
1
| | ( ( ))
( )
k
nb kk k k k
nn n nn n j ja
n n j
k
n
k k
nn n j jk
n j
k
k k
nn
n nn n
a A s x dx a a c
k a n n a c
n n
k a n n
a n
 


 
   
  
   

 

    
 
 
   
 
 
  
 
 
    
  
 
 
/2
2 2
,
0
/22 2
2 2 1
2 2
, ,
1
/22
1 2 2
2
2 2
,
1
/22
1
2
1
ˆ| | | |
ˆ| | | | ( ( ) | |
( )
ˆ| | | | | |
| | ( , , )
k
n
n j j
j
k
k k
j n n n j
j n j
k
k
k k
j nn n j
j n j
k
kk
j
j
a c
k c a n n a
j
k c n na a
j
k c j H A j

 


    
 

    
 
 

 
 
 
 
  
 
 
  
   
  
 
 
  
 

 
 

which is finite by virtue of the hypothesis of the theorem and completes the proof of the theorem.

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References:
[1] Kransniqi, Xhevat Z.; On absolute weighted mean summability of orthogonal series, Slecuk J. App. Math. Vol. 12 (2011) pp 63-70.
[2] Nantason, I.P.; Theory of functions of a real variable (2 vols), Frederick Ungar, New York 1955, 1961.
[3] Okuyama, Y.; On the absolute generalized, Nörlund summability of orthogonal series, Tamkang J. Math. Vol. 33, No. 2, (2002) pp.
161-165.
[4] Sarigöl, M.A.; On absolute weighted mean summability methods, Proc. Amer. Math. Soc. Vol. 115 (1992).
[5] Sarigöl, M.A.; On some absolute summability methods, Proc. Amer. Math. Soc. Vol. 83 (1991), 421-426.