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ON PRODUCT SUMMABILITY OF CONJUGATE SERIES OF FOURIER SERIES
1
B. P. Padhy,
2Banitamani Mallik,
3U. K. Misra* and
4Mahendra Misra
1
Department of Mathematics, Roland Institute of Technology, Berhampur, Odisha E-mail: [email protected]
2
Department of Mathematics, JITM, Paralakhemundi, Gajapati, Odisha, India E-mail: [email protected]
3
P. G. Department of Mathematics, Berhampur University, Odisha, India
*E-mail: [email protected]
4
P. G. Department of Mathematics, N. C. College (Autonomous), Jajpur, Odisha, India
E-mail: [email protected]
(Received on: 14-12-11; Accepted on: 05-01-12)
________________________________________________________________________________________________ ABSTRACT
In this paper, a theorem on
A
(
E
,
z
)
product summability of conjugate series of Fourier series is proved.Keywords:
A
mean,A
(
E
,
z
)
product mean,Fourier series, conjugate series.Mathematics subject classification: 42B05, 42B08.
________________________________________________________________________________________________
1. INTRODUCTION:
Let
a
n be a given infinite series with the sequence of partial sums{ }
s
n . LetA
=
(
a
mn)
∞×∞ be a triangular matrix .Then the sequence –to-sequence transformation(1.1)
,
1
,
2
,
0
=
=
=
m
s
a
t
m
m m
υ
υ ν
defines the sequence
{ }
t
m of theA
-mean of the sequence{ }
s
n . If(1.2)
t
m→
s
,
asm
→
∞
,then the series
a
n is said to beA
summable tos
.The conditions for regularity of
A
-summability are easily seen to be[3](i)
a
H
n mn m
<
∞
=0
sup
whereH
is an absolute constant.(ii)
lim
=
0
∞→ mn
m
a
(iii)
lim
1
0
=
∞
= ∞ →
n mn
m
a
Let
(1.3)
(
)
(
)
01
,
,
1
n n
z n
n n
E z
E
z
s
s
z
υ υ υ
υ
−
=
=
=
→
+
asn
→
∞
.Then the series
a
n is said to be summable(
E
,
z
)
to a definite numbers
.________________________________________________________________________________________________ 3
U. K. Misra*
© 2012, IJMA. All Rights Reserved 59
Let
(1.4)
(
)
z
s
s
k
z
a
T
kk n
k
k nk
n
→
+
=
−= =
υ υ υ 0
ν
01
as
n
→
∞
.Then the series
a
n is said to be summable tos
by theA
(
E
,
z
)
method .It is known [1] that
(
E
,
z
)
is regular. It is supposed that the methodA
(
E
,
z
)
is regular through out this paper.Let
f
(
t
)
be a periodic function with period 2π, integrable in the sense of Lebesgue over (-π,π) then(1.5)
(
)
∞
= ∞
=
=
+
+
=
0 1
0
cos
sin
(
)
2
)
(
n n n
n
n
nt
b
nt
A
t
a
a
t
f
where
a
nandb
nare the Euler-Fourier constants, is the Fourier series associated withf
and the conjugate series of the Fourier series (1.5) is(1.6)
(
)
∞
= ∞
=
≡
−
1 1
)
(
cos
sin
n n n
n
n
nx
b
nx
B
x
a
We use the following notation through out this paper
(1.7)
{
(
)
(
)
}
,
2
1
)
(
t
=
f
x
+
t
−
f
x
−
t
ψ
(1.8)
(
)
2 sin
2 1 cos 2 cos
1 1 ) (
0
0 t
t t
z k
z a t
K
k
k n
k
k nk n
+ −
+ =
=
− =
υ
ν
π
νυ .
2. KNOWN THEOREM:
:
Dealing with
(
N
,
p
n)(
E
,
z
)
methodof a Fourier series, Nigam,et.al[2] proved the following theorem:Theorem: 2.1 Let
{ }
p
n be a positive, monotonic, non-increasing sequence of real constants such that∞
→
=
= n
n
p
P
0 υ
υ as
n
→
∞
.If
(2.1) ,
1 )
( ) (
0
= =
Φ
t t O du u t
t
α
φ as
t
→
+
0
and
(2.2)
α
(
n
)
→
∞
asn
→
∞
where
α
(
t
)
be a positive, non-increasing function oft
, then the Fourier series∞
=0
)
(
n n
t
A
is summable(
N
,
p
n)(
E
,
z
)
tof
(
x
)
at the pointt
=
x
.In this paper, we have generalized it to
A
(
E
,
z
)
summability of conjugate series of Fourier series (1.6).3. MAIN THEOREM:
Theorem: 3.1 Let
A
=
(
a
mn)
∞×∞ be a regulartriangularmatrix and(3.1) ,
1 )
( ) (
0
= =
Ψ
t t O du u t
t
α
' $* "* $*" $( #+ , - ( + . / 0 12+3
where
α
(
t
)
is positive, non-increasing function oft
and(3.2)
α
(
n
)
→
∞
asn
→
∞
,then the conjugate Fourier series
∞ =0
)
(
n nt
B
is summableA
(
E
,
z
)
at the pointt
.4. REQUIRED LEMMAS:
We require the following Lemmas to prove the theorem.
Lemma: 4.1 If
K
n(
t
)
is as defined in (1.8), then
≤
≤
+
+
≤
≤
=
π
t
n
t
O
n
t
n
O
t
K
n1
1
,
1
1
1
0
,
)
(
)
(
Proof: For
1
1
0
+
≤
≤
n
t
, we have sinnt n sint then)
(
t
K
n(
)
2
sin
2
1
cos
2
cos
1
1
0 0t
t
t
z
k
z
a
k k n k k nk+
−
+
=
= − =υ
ν
π
ν υ(
)
2
sin
2
sin
.
sin
2
cos
.
cos
2
cos
1
1
0 0t
t
t
t
t
t
z
k
z
a
k k n k k nkυ
υ
ν
π
ν υ+
−
+
≤
= − =(
+
)
+
≤
= − =t
t
t
t
z
k
z
a
k k n k k nkυ
υ
ν
π
ν υsin
2
sin
2
sin
2
2
cos
1
1
2 0 0(
+
)
+
≤
= − =t
t
t
O
z
k
z
a
k k n k k nksin
2
sin
2
sin
2
1
1
0 0υ
υ
υ
ν
π
ν υ(
)
= − =+
≤
k k n k knk
O
k
k
z
z
a
0 0)
(
1
1
ν υν
π
(
)
(
)
k n k k nkz
z
a
k
O
+
+
=
=1
1
)
(
1
0π
=
O
(
n
)
.For
≤
≤
π
+
t
n
1
1
, we have by Jordan’s lemma,
π
t
t
≥
2
sin
, then© 2012, IJMA. All Rights Reserved 61
(
)
2
sin
2
sin
.
2
sin
2
cos
.
2
cos
2
cos
1
1
0 0t
t
t
t
t
t
z
k
z
a
k k n k k nkυ
υ
ν
π
νυ
−
+
+
≤
= − =(
+
)
+
≤
= − =2
sin
2
sin
2
sin
2
2
cos
2
1
1
2 0 0t
t
t
t
z
k
t
z
a
k k n k knk
υ
υ
ν
π
π
ν υ(
)
= − =+
≤
k k n k knk
k
z
z
a
t
01
02
1
ν υν
(
)
(
)
=+
+
=
n k k k nkz
z
a
t
01
1
2
1
.=
t
O
1
.5. PROOF OF THE THEOREM 3.1:
If
s
n(
f
;
x
)
is the n-th partial sum of the conjugate of Fourier series given by (1.6), then by using Riemann-Lebesgue theorem, following Titchmarch [4] we have(
f
x
)
s
n;
-f
(
x
)
=
dt
t
t
t
t
+
−
πυ
ψ
π
02
sin
2
2
1
sin
2
cos
)
(
2
Thus, the
(
E
,
z
)
transformE
nz ofs
n is given by(
)
k
t
dt
t
z
k
t
t
z
x
f
E
n k k n n zn
−
+
+
=
−
= − 0 02
1
sin
2
cos
2
sin
2
)
(
1
2
)
(
υ
ψ
π
πIf
T
n denote theA
(
E
,
z
)
transform ofs
n , we then havedt
t
t
z
k
t
t
z
a
x
f
T
k k n k k nkn
−
+
+
=
−
= − −=0 0 0
2
1
sin
2
cos
2
sin
2
)
(
)
1
(
2
)
(
υ υ πυ
υ
ψ
π
=
t
K
n( )
t
dt
πψ
0
)
(
In order to prove the theorem, under an assumption, it is sufficient to show that
=
πψ
0)
1
(
)
(
)
(
t
K
nt
dt
O
asn
→
∞
For
0
<
δ
<
π
, we have=
−
πψ
0)
(
)
(
)
(
x
t
K
t
dt
f
' $* "* $*" $( #+ , - ( + . / 0 12+3
t
K
nt
dt
n
n
)
(
)
(
1 1
0 1 1
ψ
π
+
=
+
+
=
I
1+
I
2,
say
Now
+
=
1 / 1
0
1
(
)
(
)
n
n
t
dt
K
t
I
ψ
(
)
(
)
.
1 / 1
0
dt
t
K
t
nn+
≤
ψ
(
)
(
)
,
1 / 1
0
dt
t
n
O
n+
≤
ψ
Using Lemma -1
,
)
(
1
)
(
=
n
n
O
n
O
α
using (3.1).
,
)
(
1
=
n
O
α
asn
→
∞
.=O
( )
1 , asn
→
∞
, using (3.2). Next
+
≤
πψ
1 1
2
(
)
(
)
n
n
t
dt
K
t
I
=
+
π
ψ
1 / 1
)
(
ndt
t
t
O
, using lemma -2
=
Ψ
+
Ψ
+ +
π π
1 / 1
2 1
/ 1
)
(
)
(
n n
dt
t
t
t
t
O
.
,
)
(
1
1
1
1 / 1
1 / 1
+
=
+
+
π π
α
α
nn
du
u
u
O
t
O
O
where u=1/t
+
+
=
n
n
du
n
n
O
n
O
1 / 1
)
(
1
)
(
1
α
α
, using second mean-Value theorem for the integral in the 2nd term as
α
(
n
)
is monotonic.
=
O
(
1
)
+
O
(
1
)
, asn
→
∞
, using (3.2)
=
O
(
1
)
, asn
→
∞
. Thus,© 2012, IJMA. All Rights Reserved 63
REFERENCES:
[1] Hardy, G.H.: “Divergent series” clarendon press , Oxford university press, (1949).
[2] Nigam,H.K and Kusum sharma : A study on
(
N
,
p
n)(
E
,
q
)
product summability of Fourier series, ultra scientist,vol.22(3), m 927-932, (2010).[3] T eoplitz,O.: “Uber allge-meire Lineare mittelbildungen”, Prace Matimatyczno-Fizyczne, 113-119, 22(1911).
[4] Titechmalch, E. C: The theory of functions, Oxford University Press, 402- 403(1939).
