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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On Absolute Weighted Mean | A, |k -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 

2   1

 k 
| an,n |

n 1 


k
2
2
2
ˆ
| an , j | | c j |  .

j 0


n
Converges for 1  k  2 then the orthogonal series

c 
n 1
n
n
( x)
is summable | A,  |k almost every where,
Abstract: In this paper we prove the theorems on absolute weighted mean | A,  |k -summability of orthogonal
series. These theorems are generalize results of Kransniqi [1].
Keywords: Orthogonal Series, Nörlund Matrix, Summability.
I.

Let
a
n 0
Introduction
be a given infinite series with its partial sums {sn } and Let A  (an,v ) be a normal matrix,
n
that is lower-semi matrix with non-zero entries. By An ( s) we denote the A -transform of the sequence
s  {sn } , i.e.

An ( s)   anv sv .
v 0

The series
a
n 0
n
is said to be summable | A |k , k  1 (Sarigöl [4]) if

| a
n0

And the series
a
n 0
n
|1 k | An ( s)  An 1 ( s) |k  
nn
is said to be summable | A,  |k k  1,   0 if

| a
nn
n0
| k 1 k | An ( s) |k  
where An (s)  An ( s)  An1 ( s)
In the special case when A is a generalized Nörlund matrix | A |k summability is the same as
| N , p, q | k summability (Sarigöl [5]).
By a generalized Nörlund matrix we mean one such that
anv 
pn  vqv
for 0  v  n
Rn
anv  O for v  n
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On Absolute Weighted Mean
| A, |k -Summability Of Orthogonal Series
where for given sequences of positive real constant p  { pn } and q  {qn } , the convolution Rn  ( p  q)n is
defined by


v 0
v 0
( p * q)n   pv qn v   pv v qv
where ( p * q)n  0 for all n , the generalized Nörlund transform of the sequence {sn } is the sequence
{tnp ,q ( s)} defined by

1
Rn
tnp ,q ( s) 
p
m 0
q s
nm m m
and | A |k summability reduces to the following definition:

The infinite series
a
n 0
n
is absolutely summable | N,p,q |k k  1 if the series
 Rn 
 

n  0  qn 

k 1
| tnp ,q ( s)  tnp,1q ( s) |k  
and we write

a
n 0
n
| N , p,q |k
Let { ( x)} be an orthonormal system defined in the interval (a, b) . We assume that f ( x) belongs to
L2 (a, b) and

f ( z ) ~  cn n f ( x)
(1.1)
n0
where cn 

b
a
we write Rnj 
f ( x) n ( x)dx, n  0,1,2...

p
v j
and Pn  ( p *1) n 
q , Rnn 1  0, Rn0  Rn
n v v


v 0
v 0
 pv and Qn  (1* q)n   qv
Regarding to | N , p, q |1 | N , p, q | , summability of orthogonal series (1.1) the following two theorems are
proved.
Theorem 1.1 (Okuyama [3]) If the series
1/ 2
 n  R j R j 2

n
 n 1  | c j |2 
 

Rn 1 
n  0  j 1  Rn




then the orthogonal series cn n ( 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
 (n(n))
n 1
1
converges. Let { pn } and {qn } be non negative. If the series

| cn |2 (n)w(1) (n) converges then the orthogonal series
n 1

c 
j 0
j
j
( x) | N , p, q | almost everywhere, where
w((1)n ) is defined by
2
 Rj Rj 
w  j  n  n  n 1  .
n j
 Rn Rn 1 
The main purpose of this paper is studying of the | A,  |k summability of the orthogonal series (1.1),
for 1  k  2 . Before starting the main result we introduce some further notations.
(1)
( j)
1

2
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39 | Page
On Absolute Weighted Mean
| A, |k -Summability Of Orthogonal Series
ˆ  aˆ as follows
Given a normal matrix A  anv , we associates two lower semi matrices A  an,v and A
n ,v

an,v   ani , n, i  0,1, 2
i v
and aˆnv  anv  an 1,v , aˆ00  a00 , n  1,2,... .
It ma be noted that A and  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
 1 

2   1

 k 
|
a
|
 n,n

n 1 




| aˆn , j | | c j | 

j 0


n
2
k /2
2
Converges for 1  k  2 , then the orthogonal series

c 
n
n0
n
( x)
is summable | A,  |k almost every where.

Proof. For the matrix transform An (s)(x) of the partial sums of the orthogonal series
c 
n0
n
n
( x) we have
n
An ( s )( x)   anv sv ( x )
v 0
n
v
v 0
j 0
  an ,v  c j n ( x)
n
n
j 0
v 0
  c j n ( x) anv
n
  an , j c j n ( x)
j 0
v
where
c 
j 0
j
n
( x) is the partial sum of order v of the series (1.1)
Hence

n 1
An ( s )( x)   an , j c j j ( x)   an 1, j c j j ( x )
j 0
j 0
n 1
 an ,n c j n ( x)   (an , j  an 1, j )c j j ( x)
j 0
n 1
 aˆn ,n c j n ( x)   aˆn , j c j j ( x)
j 0
n
  aˆn , j c j j ( x)
j 0
Using the Hölder’s inequality and orthogonality to the latter equality we have
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On Absolute Weighted Mean

b
a
1
| An ( s)( x) | dx  (b  a)
k
k
2
| A, |k -Summability Of Orthogonal Series
  | A (s)( x)  A
b
n 1
n
a
2
( s)( x) | dx

K
2
K
k
1
2
 (b  a)
2 2
 b n

   aˆn , j c j j ( x) 
 a j 0



Thus the series

| a
b
nn
n 1
|1 k  k  | An ( s)( x) |k dx
a
2
 2  2

k
 k  | ann |
n 1 

| aˆn, j |2 | c j |2 

j 0


n
k /2
(2.1)
Converges by the assumption.
From this fact and since the function | An ( s)( x) | are non negative and by Lemma of Beppo-Levi [2] we have

| a
nn
n 1
|1 k  k | An ( s)( x) |k
Converges almost every where.
This completes the proof of theorem.
If we put
1
H ( k ) ( A;  , j ) 
j

n
2
1 n  j
k
2
k
2
| nann | k
 2  2
| aˆn, j |2
(2.2)
then the following theorem holds true.
 ( n ) 
 is a non decreasing
 n 
Theorem 2.2: Let 1  k  2 and {(n)} be a positive sequence such that 

sequence and the series

1
 n(n)
converges. If the following series
n 1
n 1

converges, then the orthogonal series
| c
c 
n0
n
n
n
2
1
|2  k (n) H k ( A,  , n)
( x) | A,  |k almost every where, where H k ( A,  , n) is defined
by (2.2)
Proof. Applying Hölder’s inequality to inequality (2.1) we get

b
 n

k
| ann |1 k  k  An ( s )( x) dx  | ann |1 k  k   | aˆn , j |2 | c j |2 

a
n 1
n 1
 j 1



 k
n 1
1
( n( n))
2k
2
 

1
 k
 n 1 a ( n) 
n,n


2
2
 2  2
1 n


k
( n( n)) k  | aˆn , j |2 | c j |2 
| ann |
j 0


2k
2
k /2
k /2
2
2
 2  2
1 n
 

k
( n( n)) k  | aˆn , j |2 | c j |2 
  | ann |
j 0
 n 1

2
2

 2  2
1
 

 k  | c j |2  |an , n | k
( n( n) k | aˆn , j |2 
n j
 j 1

k /2
k /2
2
1
 

2
2

k

 2 

2  ( j ) 
2
k
k
ˆ
 k  | c j | 
n
|
na
|
|
a
|

 
nn
n, j
 j  n j
 j 1



k /2
k /2
2
1
 

 k  | c j |2  k j H k ( A,  , j ) 
 j 1

which is finite by virtue of the hypothesis of the theorem and completes the proof of the theorem.
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On Absolute Weighted Mean
| A, |k -Summability Of Orthogonal Series
References:
[1]
[2]
[3]
[4]
[5]
Kransniqi, Xhevat Z.; On absolute weighted mean summability of orthogonal series, Slecuk J. App. Math. Vol. 12 (2011) pp 63-70.
Nantason, I.P.; Theory of functions of a real variable (2 vols), Frederick Ungar, New York 1955, 1961.
Okuyama, Y.; On the absolute generalized, Nörlund summability of orthogonal series, Tamkang J. Math. Vol. 33, No. 2, (2002) pp.
161-165.
Sarigöl, M.A.; On absolute weighted mean summability methods, Proc. Amer. Math. Soc. Vol. 115 (1992).
Sarigöl, M.A.; On some absolute summability methods, Proc. Amer. Math. Soc. Vol. 83 (1991), 421-426.
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