162
ON THE LOCAL PROPERTY OF
N
,
p
n,
n kSUMMABILITY OF A
FACTORED FOURIER SERIES
U.K. Misra1,Mahendra Misra2, B.P. Padhy3 & S.K. Buxi4
1
Department of Mathematics,Berhampur University,Berhampur – 760007,Orissa, India,
2
Principal,Government Science College,Mlkangiri, Orissa, India.
3
Roland Institute of Technology,Golanthara-760008,Orissa, India.
4
Department of Mathematics,Gunupur Science College,Gunupur,Odisha, India. Email: 1umakanta_misra@yahoo.com , 2mahendramisra@2007.gmail.com , 3iraady@gmail.com
ABSTRACT In this paper we have established a theorem on the local property of
k n n
p
N
,
,
summability of factored Fourier series.Keywords:
k n
p
N
,
-summability,N
,
p
n,
n k-summability,k n
p
N,
-summability and Fourier series. AMS CLASSIFICATION NO: 40D251. INTRODUCTION
Let
a
n be a given infinite series with sequence of partial sums
s
n .Let
p
n be a sequence of positive realconstants such that
(1.1)
n i in
p
as
n
P
p
i
P
0
)
1
,
0
(
The sequence –to-sequence transformation
(1.2)
n nn
n
p
s
P
t
0
1
defines
N
,
p
n -means of the sequence
s
n generated by the sequence of coefficients
p
n .The series
a
n issaid to be summable
k n
p
N
,
,k
1
,
if(1.3)
1
1 1
n
k n n k
n n
t
t
p
P
.
For k=1,
k n
p
N
,
-summability is same asN
,
p
n -summability. Whenp
n
1
for
all
n
and
k
1
,k n
p
N
,
-summability is same asC
,
1
-summability.Let
n be any sequence of positive numbers. The series
a
n is said to be summableN
,
p
,
,
k
1
k n
n
, if(1.4)
kn n n
k
n
t
t
1 11
.A sequence
n is said to be convex if0
2
n for every positive integer n.Let
f
t
be a periodic function with period2
and integrable in the sense of Lebesgue over(
,
)
.Without loss of generality we may assume that the constant term in the Fourier series off
t
is zero, so that(1.5)
1 1
)
sin
cos
(
~
n n n
n
n
nt
b
nt
A
t
a
t
163 2. KNOWN THEOREMS
Dealing with the
k n
p
N
,
-
summability of an infinite series Bor[1] proved the following theorem: THEOREM:Let
k
1
and let the sequences
p
nand
n be such that(2.1)
n
O
X
n1
,
(2.2)
1
1 1
n
k n k n k n
n
X
,
and
(2.3)
1
)
1
(
n
n k
n
X
,
where
.
n n n
np
P
X
Then the summabilityk n
p
N
,
of the factored Fourier series
1
)
(
n
n n n
t
X
A
at a point canbe ensured by the local property.
In this paper we have proved a theorem on the local property of
k n n
p
N
,
,
summability of factored Fourier series:3. MAIN THEOREM
Let
k
1
.
Suppose
n be a convex sequence such that
n
1
nis convergent and
p
n be a sequence such that(3.1)
n
O
X
n1
,
(3.2)
r r n
r n n
r n
p
P
P
p
O
P
P
1 1 1
,
(3.3)
r r n
r n k m
r n
n
P
p
O
P
p
1 1
1
,(3.4)
1 1
n
k n k n
n
X
,and
(3.5)
1 1
n
k n k n
n
X
,
where
.
n n n
np
P
X
Then the summabilityN
,
p
,
,
k
1
k n
n
of the factored Fourier series
1
)
(
n
n n n
t
X
A
at a
164 4. LEMMA
Let
k
1
.
Suppose
n be a convex sequence such that
n
1
n is convergent and
p
n be a sequence suchthat the conditions (3.1)-(3.5)are satisfied. If
s
n is bounded then the series
1
n
n n n
X
a
isk n n
p
N
,
,
-summable, where
n is a sequence of positive numbers . 5. PROOF OF THE LEMMALet
T
n denote theN
,
p
n-
mean of the series
1
n
n n n
X
a
.
Then by definition we have
nr
r r r n
n
n
p
a
X
P
T
0 0
1
nr n n
r
r r r n
p
X
a
P
0
1
nr n n
r
r r r n
p
X
a
P
0
1
nr
r r r n r n
X
P
a
P
01
Hence
11 1 1
1 1
1
1
nr
r r r r n n
n
r
r r r r n n n
n
P
a
X
P
X
a
P
P
T
T
r r r
n
r n
r n n
r
n
a
X
P
P
P
P
1 1
1
r r rn
r
n r n n r n n
n
X
a
P
P
P
P
P
P
1
1 1
1
1
n rr
r r n r n n r n n
n
a
X
P
P
P
P
P
P
11
1
1 1
1
1
1
1
1 2
1 1 1
1
2 1
1 1
1
1 1
1 1
n
r
r r r n r n n r n n
r
r r r n r n n r n n
r
r r r n r n n r n n
n
s X P
P P P
s X P
P P P
s X P p P p P
P
(by Abel’s transformation)
T
n,1
T
n,2
T
n,3
T
n,4
T
n,5
T
n,6(
say
)
.
To complete the theorem by using Minokowski’s inequality, it is sufficient to show that
1
,
2
,
3
,
4
,
5
,
6
.
1
,
1
i
for
T
n
k i n k n
Now, we have
1
2
1 , 1
m
n
k n k n
T
12
1
1
1 1
1
1
m
n
k n
r
r r r n r n n
n k
n
p
P
X
s
P
P
165
1 1
1 1
2
1
1
1
1
1
k n
r r n n m
n
n
r
k r k r k r r n n
k
n
p
P
X
s
p
P
11 1
1
)
1
(
m
r
n n
r n k n m
r
k r k r
P
p
X
O
r r m
r
k r k r
P
p
X
O
1
)
1
(
,by(3.3)
(
1
)
,
1
1
r r r r m
r
k r k r
rp
P
P
p
X
O
asn n n
np
P
X
r
X
O
k r m
r k r
1 1
)
1
(
O
(
1
)
as
m
,
by (3.4). Next,
1
2
2 , 1
m
n
k n k n
T
12
1
1 1 1
1
1
m
n
k n
r
r r r n r n n
n k
n
p
P
X
s
P
P
1 1
1 1 1
1
2
1
1 1 1
1
1
1
k n
r r n n
m
n
n
r
k r k r k r r n n
k
n
p
P
X
s
p
P
11 1
1 1
1
)
1
(
m
r
n n
r n k n m
r
k r k r
P
p
X
O
r r m
r
k r k r
P
p
X
O
1
)
1
(
,by(3.3)
(
1
)
,
1
1
r r r r m
r
k r k r
rp
P
P
p
X
O
asn n n
np
P
X
r
X
O
k r m
r k r
1 1
)
1
(
O
(
1
)
as
m
,
by (3.4). Further,
1
2
3 , 1
m
n
k n k n
T
12
1
1
1 1 1
1
1
m
n
k n
r
r r r n r n n
n k
n
P
P
X
s
P
P
1 1
1 1 1
2
1
1 1
1
1
1
k
r n
r r n n m
n
n
r
k r k r k r r n n
k
n
P
P
X
s
P
P
11
1 1
1
)
1
(
m
r
n n
r n k n m
r
k r k r
P
P
X
O
1
1 1
1
1
(
1
)
1
nr r n
r
r r n n
O
P
P
Since
r r m
r
k r k r
P
p
X
O
1
)
1
166
(
1
)
,
1
1
r r r r m
r
k r k r
rp
P
P
p
X
O
asn n n
np
P
X
r
X
O
k r m
r k r
1 1
)
1
(
O
(
1
)
as
m
,
by (3.5). Now,
1
2
4 , 1
m
n
k n k n
T
12
1
1 2 1
1
1
m
n
k n
r
r r r n r n n
n k
n
P
P
X
s
P
P
1 11 2 1
1
2
1
1 2 1
1
1
1
k
r n
r r n n m
n
n
r
k r k r k r r n n
k
n
P
P
X
s
P
P
11 1
2 1
1
)
1
(
m
r
n n
r n k n m
r
k r k r
P
P
X
O
,(as above)
r r m
r
k r k r
P
p
X
O
1
)
1
(
, by (3.3)
(
1
)
,
1
1
r r r r m
r
k r k r
rp
P
P
p
X
O
asn n n
np
P
X
r
X
O
k r m
r k r
1 1
)
1
(
O
(
1
)
as
m
,
by (3.5). Again
1
2
5 , 1
m
n
k n k n
T
12
1
1
1 1 1 1
1
1
m
n
k n
r
r r r n r n n
n k
n
P
P
X
s
P
P
12
1
1
1 1 1
m
n
k n
r
r r r n r n k
n
X
s
P
P
12
1
1
1 1
1 1
m
n
k n
r
r r r r r n
r n k
n
X
s
p
P
P
p
,by(3.2)
12
1
1
1 1
1
1
1
m
n
k n
r
r r r r n
r n k
n
r
s
p
P
P
p
, by(3.1)
12
1
1
1 1
1 1
m
n
k n
r r
r r r r r r n
r n k
n
P
p
X
s
p
P
P
p
,asn n n
np
P
X
1 1
1 1
1 1
2
1
1
1 1
1 1
k n
r n r n m
n
n
r
k r k r k r n
r n k
n
P
p
X
s
P
p
11 1
1 1
1 1
)
1
(
m
r
n n
r n k n m
r
k r k r
P
p
X
167
(
1
)
,
1
1 1
r r r r m
r
k r k r
rp
P
P
p
X
O
asn n n
np
P
X
and by(3.3)
(
1
)
,
1
1 1
mr
k r k r
X
r
O
O
(
1
)
as
m
,
by (3.4). Finally,
1
2
6 , 1
m
n
k n k n
T
12
1
1
1 2 1
1
1
m
n
k n
r
r r r n r n n
n k
n
P
P
X
s
P
P
12
1
1
1 1
2 1
m
n
k n
r
r r r n
r n k
n
X
s
P
P
12
1
1
1 2
2 1
m
n
k n
r
r r r r r n
r n k
n
X
s
p
P
P
p
,by(3.2)
12
1
1
1 2
2
1
1
m
n
k n
r
r r r r n
r n k
n
r
s
p
P
P
p
, by(3.1)
12
1
1
1 2
2 1
m
n
k n
r r
r r r r r r n
r n k
n
P
p
X
s
p
P
P
p
,asn n n
np
P
X
1 1
1 2
2 1
2
1
1
1 2
2 1
k n
r n r n m
n
n
r
k r k r k r n
r n k
n
P
p
X
s
P
p
11 2
2 1
1 1
)
1
(
m
r
n n
r n k n m
r
k r k r
P
p
X
O
(
1
)
,
1
1 1
r r r r m
r
k r k r
rp
P
P
p
X
O
asn n n
np
P
X
and by(3.3)
(
1
)
,
1
1 1
mr
k r k r
X
r
O
O
(
1
)
as
m
,
by (3.4). This completes the proof of the Lemma.5. PROOF OF THE THEOREM
Since the behavior of the Fourier series, as far as convergence is concerned , for a particular value of x depends on the behavior of the function in the immediate neighborhood of this point only, hence the truth of the theorem is necessary consequence of the Lemma.
REFERENCES
[1]. H.Bor., 1992, On the Local Property of
k n
