J. Comp. & Math. Sci. Vol. 1(1), 67-70 (2009).
A NOTE ON INDEX SUMMABILITY FACTORS OF FOURIER SERIES U.K. Misra1, Mahendra Misra2 and B.P. Padhy3 1
2 3
Department of Mathematics Berhampur University Berhampur-760007, Orissa (India) e-mail: umakanta_misra@yahoo.com
Principal Government Science CollegeMlkangiri, Orissa
Roland Institute of TechnologyGolanthara-760008, Orissa (India) e-mail: iraady@gmail.com ABSTRACT A theorem on index summability factors of Fourier Series has been established Key words: Summability factors, Fourier series. AMS Classification No: 40D25
1. INTRODUCTION Let
a
be a given infinite series
n
defines the sequence of the N, p n -mean of the sequence s n generated by the sequence
with the sequence of partial sums s n . Let
of coefficients p n .
p n be a sequence of positive numbers such
The series
that
n
is said to be summable
N , p n k , k 1, if n
(1.1) Pn
p
v
, as n Pi
r 0
p i 0, i 1 The sequence to sequence transformation (1.2)
a
1 tn Pn
v 0
Pn (1.3) n 1 p n
k 1
t n t n 1
k
In the case when p n 1 , for all n and k 1,
N, p n k summability is same as C ,1 summa-
n
p n v sv
bility. For k 1, N , pn k summability is same
[ 68 ] as N, p n - summability.. Let n be any sequence of positive numbers. The series a n is said to be summable
N , p n , n k , k 1, if
k 1 n n
t
n
n
n
is summable
N , p n , k 1. k
We prove an analogue theorem on
N , pn , n
- summability, k 1 , in the following form:
(1.4)
A (t ) P
Fourier series
t n 1 .
n 1
k
3. Main Theorem :
Clearly, N , p n ,
Pn is same as N , pn k , k 1 pn k
Theorem: Let p n is a sequence of
and N , p n ,1 1 is N, pn .
positive numbers such that Pn p1 p 2
Let f t be a periodic function with period 2 and integrable in the sense of Lebesgue over
negative, non-increasing sequence such that
.... p n as n and n is a non-
p , . Without loss of generality, we may n n
assume that the constant term in the Fourier series of f (t ) is zero, so that (1.5)
f (t ) ~ (a n cos nt bn sin nt ) An (t ) n 1
Dealing with N , p n k , k 1 , summability factors of Fourier series, Bor 1 proved the following theorem: Theorem-A : If n is a non-negative and non-increasing sequence such that n
n , where p n is a sequence
of positive numbers such that Pn as n n
and
P
v
v 1
n
(3.1) (i).
P
v
Av (t ) 0( Pn )
v 1
m 1
(3.2) (ii).
n
k 1
n v 1
n 1
2. Known Theorem :
p
. If
Av (t ) 0( Pn ) . Then the factored
p n v 1 p 0 v , Pn1 Pv
as m
and
(3.3) (iii). Pn v 1 v 0 p n v v , then the series
A (t ) P n
n
n is summable
N , pn , n k , k 1 . and n , a sequence of positive numbers. 4. Required Lemma : We need the following Lemma for the proof of our theorem. Lemma [1]: If n is a non-negative
[ 69 ] and non-increasing sequence such that
r
pn n , where pn is a sequence of positive numbers such that Pn as n
P
then Pn n 0(1) as n and
n
Av ( x ) , using partial summation
Pv v 1
formula with p o 0 .
n .
1 n 1 p n r Pn 1 pn r 1 Pn r Pr Pn Pn 1 r 1
p
5. Proof of the Theorem :
n 1
Let t n (x ) be the n-th N, p n mean
Pn 1 Pn r 2 p n Pr r , using (3.1)
n r 1
r 1
of the series
A ( x) P n
n
n
, then by definition
Tn ,1 Tn, 2 Tn, 3 Tn, 4 , say..
n 1
we have n
1 t n ( x) Pn
v
p A ( x) P n v
v0
r
r
r
r 0
In order to complete the proof of the theorem, using Minkowski's inequality, it is sufficient to show that
n
1 Pn
A ( x) P p r
r
r
r 0
n v
k 1
Tn ,i
n
A ( x) P P r
r
n r
r .
r 0
, for i 1,2,3, 4.
n 1
Now, we have m 1
Then
k 1 n
Tn,1
k
m 1
n 2
t n ( x) t n 1 ( x )
k
vr
n
1 Pn
n
1 Pn
n
P
n r
Pr r Ar x
r 0
k 1 n
n 2
m 1
k 1
n 2
1 Pn1
1 Pn Pn 1
P
n r 1
Pr r Ar ( x)
r 0
Pn r P n r 1 Pr r Ar ( x) Pn 1 r 1 Pn
n
Pnv
k
1 n 1 p n v Pvk kv Pn v 1
n 1
n
1 n1 p n v Pv v k Pn v 1
1 Pn
r 1
1 n 1 Pn r Pn 1 Pn r 1 Pn r Pn Pn 1 r 1
p n v v 1
k 1
, using Holder's
inequality. m 1
Pn 1 Pn r 1 Pn Pr r Ar ( x )
n 1
O1 n n2
m
k 1
1 Pn
O1 Pv v v 1
n 1
p
n v
Pv v Pv v
v 1
m 1
n v 1
n
k 1
p n v Pn
k 1
[ 70 ] m
O(1)
p
O(1) Pv v
v 1
m 1
k
n k 1 Tn.2
n 2
n
k 1
n 2
m 1
O (1) n
1 Pn 1
n2
O (1) Pv v v 1
m 1
n
v , using (3.2)
k 1
Tn, 4
k
m 1
n v 1
Pv v ( Pv v ) k 1
0(1)
p v v , using (3.2)
p n v 1 Pv v ,
k 1
n
1 Pn 1
k 1
n2
k
k
1 n1 Pnk1 v 1 n 2 using (3.3)
n
m 1
p nk Pnk Pnk1
k 1
n 2
m 1
0(1)
p k 1 nv1 Pn1
n
n1 Pv Pnv 2 v v 1
v 1
n v 1
v
Finally,
n 1
p
n 2
m
0(1)
0(1)
0(1) , m .
k 1
p
m 1
m
n v 1
v 1
1 p n v 1 Pvk kv Pn 1 v 1
p n v 1 v 1 k 1
k
n 1
n 1
1 Pn 1
p n v 1 P n
n 1 n
m
1 n 1 p n v1 Pv v Pnk1 v 1
n k 1
n2
m 1
v 1
O(1), as m . m 1
m 1
m
v (using 3.2)
v
n 1
p n v 1
v 1
v 1 k
0(1) , as m .
Pv
v
Now, m 1
n
k 1
Tn ,3
k
m 1
n2
k 1
n2
m 1
n
k 1 n
n 2
1 n 1 p n v 1 Pv v Pnk v 1
1 n1 k 1 pn v 1 Pv v Pn v 1 Pn
m 1
O 1 n
k 1
n 2
m 1
O1 n n 2
1 Pn k 1
n 1
p
n v 1
n 1
p v 1
n v 1
Pv v Pv v
by the lemma
n 1
0(1)
n n 2
k 1
1 Pn1
n 1
p n v 1 v 1
k 1
n 1
p
n v 1
Pv v
v 1
k 1
0(1), as m , as above This completes the proof of the theorem.
k 1
REFERENCE
v 1
1 Pn
k
1 Pn1
n 1
p v 1
n v 1
Pv v ,
1. Bor Huseyin: On the absolute summability factors of Fourier Series, Journal of Computational Analysis and Applications, Vol. 8, No. 3, 223-227 (2006).