Absolute Banach Summability of Orthogonal Series

Abstract

In this paper we have proved a theorem on generalized
Nörlund summability of infinite series, which generalizes various
known results. However, the theorem is as follows:
Theorem: Let {W(n)} be positive sequence such that (n)
n
W 
 
 
is a non-increasing sequence and the series
1
1
n n (n)
¥
= W
Σ
converges and
( )
1
log
n
t
u t
du O
u
t
d
 
 
F   =
   
   
   
∫ as t®0,d
being some fixed positive constant then the orthogonal series
1
( ) n n
n
a f x
¥
=
Σ is summable B at t = x , provided
2
1
log( ) ( ( ))
n
k n k O n n
¥
=
Σ + = W .
Definitions and Notations: Let { } n s be the sequence of partial
sums of a series n Σa . Let the sequence { } 1
( ) k k
t n
¥
=
is defined
by
288 Satish Chandra and Devendra Kumar Verma
(1.1)
1
0
1
( ) ,
k
k n v
v
t n s k N
k
−
+
=
= Σ Î
If
(1.2) lim ( ) k
k
t n s
®¥
= , a finite number, uniformly for all
nÎ N , then
n Σu is said to be Banach summable to s 1 .
Further if
1
1
( ) ( ) k k
k
t n t n
¥
+
=
Σ − < ¥ uniformly for all nÎN ,
then the series n Σu is said to be absolute Banach summable or
simply B -summable.
2. Let { } n f be an orthogonal system defined in the interval
(a,b) . We suppose that f (x) belongs to 2 L (a,b) and
0
( ) ( ) n n
n
f x a f x
¥
=
»Σ
We denote by (2) ( ) n E f the best approximation to f (x) in the
metric of
2 L by means of polynomials of
0 1 1 ( ), ( ),........ ( ) n f x f x f x − .
It is well known that
2
(2) ( ) 1
n k 2
k n
E f a
¥
=
 
=  
 
Σ
we write n n n 1 l l l − D = −
,
for any sequence { } n l .
(2.1) ( ) 2
1
2 1 ( )
( , )
( 1) ( )
k
v
v
v t
g k t n v
k k n v t
b
p
−
=
W
= +
+ +
Σ
Absolute Banach Summability of Orthogonal Series
289
(2.2)
1
( , ) ( , )( )
(1 ) u
d
J k u g k t t u dt
F dt
b
b
¥
− = −
−
∫
( w k,u) = uv J (k,u)
[x]= greatest integer not exceeding x
1
U
u
 
=  
 
and
1
t
t
 
=  