J. Comp. & Math. Sci. Vol. 1 (6), 758-761 (2010)
On
N, p n
Summability Factors of Fourier Series
k
MAHENDRA MISRA1, U. K. MISRA2 and CHITARANJAN KHADANGA3 Principal, Government Science College, Mlkangiri, Orissa1, Department of Mathematics2, Berhampur University Berhampur – 760007,Orissa, India e-mail:umakanta_misra@yahoo.com M.P.Christian College of Engineering and Technology3, P.O Box No.18, Kailash Nagar Bhilai-490026,(C.G) e-mail: chita_khadanga@redifmail.com ABSTRACT A theorem on index summabilty factors of Fourier Series has been established Key Words: Summability factors, Fourier series AMS Classification No: 40D25
1. INTRODUCTION
∑a
Let
In
be a given infinite series
n
with the sequence of partial sums {s n } . Let
{p n } be a sequence of positive numbers such
that (1.1)
n
Pn =
∑ pv → ∞ , as n → ∞
( P−i
= p− i = 0, i ≥ 1)
r =0
The sequence to sequence transformation
tn =
(1.2)
1 Pn
n
∑ v =0
p n −v s v
defines the sequence of the ( N , p n ) -mean of
the sequence {s n } generated by the sequence of coefficients {p n }. The series
∑a
n
is said to be
summable N , p n k , k ≥ 1 , if
Pn ∑ n =1 p n ∞
(1.3)
k −1
t n − t n −1
k
<∞
the
case
n and x = 1, N , p n C ,1
pn = 1 ,
when
summability.
k
for
all
summability is same as For
k = 1, N , p n
k
summability is same an N , p n -summability.
Let f (t ) be a periodic function with period 2π and integrable in the sense of Lebesgue over (− π , π ) . Without loss of generality, we may assume that the constant term in the Fourier series of f (t ) is zero, so that (1.4) ∞
∞
n=1
n=1
f (t) N ∑(an cos nt + bn sin nt) = ∑ An (t) 2. Know Theorem Dealing with
N , pn k , k ≥ 1 ,
summability factors of Fourier Series, Bor1 proved the following theorem: Theorem-A If {λ n } is a non-negative and non-increasing sequence such that p n λ n < ∞ , where {p n } is a sequence of
∑
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
Mahendra Misra et al., J. Comp. & Math. Sci. Vol. 1 (6), 758-761 (2010)
759
Pn → ∞ as n → ∞ then
positive numbers such that Pn → ∞ as n → ∞ and n
∑P v =1
v
Pn λn = 0(1) as n → ∞ and
Av (t ) = 0( Pn ) . Then the factored
Fourier series
∑ A (t ) P λ n
n
n
is summable
∑P
n
5. PROOF OF THE THEOREM Let t n ( x) be the n-th ( N , p n ) mean
N , p n k , k ≥1 . We prove an analogue theorem on N, p n k -summability, k ≥ 1 , in the following form:
∞
n =1
t n ( x) =
=
and {λ n } is a non-negative, non-increasing
sequence such that
∑p
n
(3.1)
(i).
(3.2)
(ii).
∑P
v
v =1
n = v +1
n
n
n
,
then by
definition we have
Theorem: Let {p n } is a sequence of positive numbers such that Pn = p1 + p 2 + .... + p n → ∞ as n → ∞
∑
∑ A ( x) P λ
of the series
3. MAIN THEOREM
m +1
∆λ n < ∞ .
Pn pn
k −1
n
λn < ∞ . If
Av (t ) = 0( Pn )
p n −v −1 p = 0 v , as m → ∞ Pn −1 Pv
and (3.3) (iii).
Pn −v −1 ∆λv = 0( p n − v λv ) , then
the
∑ A (t ) P
series
n
=
n
λn
is
summable
N , pn k , k ≥ 1.
∑ ∑
positive numbers such that
1 Pn
v
∑ p ∑ A ( x) P λ v =0
n−v
n
∑ Ar ( x) Pr λr r =0 n
∑ A ( x) P r =0
r
r
r
r =0
r
r
n
∑p v =r
n −v
Pn − r λ r .
Then tn ( x) − tn−1 ( x) = 1 Pn
n
∑ Pn−r Pr λr Ar ( x) − r =1
1 Pn−1
n−1
∑P r =1
n−r −1
Pr λr Ar ( x)
Pn− r P − n − r −1 Pr λ r Ar ( x) Pn−1 r =1 Pn 1 = (Pn−v Pn−1 − Pn− r −1 Pn ) Pr λr Ar ( x) Pn Pn−1
=
4. REQUIRED LEMMA We need the following for the proof of our theorem. n Lemma [1]: P If {λ n }Pis a non-negative Pn r and Pn n r n r sequence such that r =1 non-increasing p n λ n < ∞ , where {p n } is a sequence of
1 Pn
n
1 Pn
n
∑
1 n −1 ∑ ∆ {( Pn − r Pn −1 − Pn − r −1 Pn ) λr } Pn Pn−1 r =1 r ∑ Pv A v ( x ) v =1
=
}
with p o = 0
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
r
∑P v =1
v
Av x
2
Mahendra Misra et al., J. Comp. & Math. Sci. Vol. 1 (6), 758-761 (2010)
1 n−1 = ∑ ( pn−r Pn−1 − pn−r −1 Pn ) λr Pr Pn Pn −1 r =1 +
n −1
∑(p r =1
n − r −1
≤
n =2
Pn −1 − Pn − r − 2 p n ) Pr ∆λ r
k
Pn n=2 n
k −1
∑ p
Tn,1
k
P = ∑ n n=2 pn
k −1
m
= 0(1)
∑ r =1
P ≤ ∑ n n =2 p n m +1
Pn ∑ n = 2 pn m +1
n −1
1 ∑ p n −v Pv λv k Pn v =1
1 n −1 ∑ p n −v Pvk λkv Pn v =1
m+1 P = O(1) ∑ n n=2 pn
k −1
1 Pn
n −1
∑p r =1
n −v
m
∑ v =1
k
≤
m +1
∑ n=2
k −1
∑ n=2
m+1
∑ n =2
Pn pn
Pn pn
∑
n = v +1
Pv λv ( Pv λv ) k −1
k −1
p n − v −1 Pn −1
,
=
Pv λv (Pv λv )
k −1
Pn pn
m +1
∑
k −1
1 n −1 ∑ pn−v−1 Pv ∆λv Pnk v=1
1 n −1 k 1 ∑ p n − v −1 (Pv ∆λv ) Pn v =1 Pn
Pn pn
k −1
P = O(1)∑ n n=2 p n
1 Pn
k −1
n −1
∑p v =1
1 Pn
n − v −1
v =1
v =1
n − v −1
Pv ∆λv (Pv ∆λ v )
n −1
∑p
n −1
∑p
k −1
n − v −1
k −1
k −1
Pv ∆λv , b
k −1
P = O (1)∑ Pv ∆λv ∑ n v =1 v =1 p n n −1
m
p n −v Pn
= 0(1)
m
∑p v =1
v
n −1
p n− v −1 P n
∆λv , using (3.2)
= 0(1) , m → ∞ .
k −1 k
Pn pn
m +1
n=2
k −1
n=2
p v λv (using 3.2)
Tn.2
k
Tn,3
= O(1) ∑
k −1
= O (1), as m → ∞ . m +1
v =1
n −v −1
k −1
y the lemma
P = O (1)∑ Pv λv ∑ n v =1 n = v +1 p n = O (1)
p n −v ∑ v =1 n −1
∑p
p v λv , using (3.2)
m +1
m +1
m
1 Pn
n −1
1 Pn −1
v =1
n − v −1
Now,
m +1
k −1
k −n
n −1
∑p
= 0(1) , as m → ∞ .
< ∞, for i = 1,2,3,4.
m +1
1 n−1 1 ∑ p n −v −1 Pvk λkv k Pn −1 v =1 Pn −1
v =1
Now, we have m +1
k −1
= O (1) ∑ Pv λ v
k −1
Tn,i
m
In order to complete the proof of the theorem, using Minkowski’s inequality, it is sufficient to show that
Pn ∑ n =1 p n
Pn pn
m +1 P = O(1)∑ n n= pn
= Tn ,1 + Tn , 2 + Tn ,3 + Tn , 4 , say.
∞
m +1
∑
760
=
m +1
∑ n=2
1 n−1 p Pk λv k ∑ n −v −1 v Pn v=1
k
Pn pn
k −1
n −1
∑p v =1
P
n v Finally, 1 v v
Pn ∑ n=2 pn m +1
k −1
Tn, 4
k
=
m +1
∑ n=2
Pn pn
k −1
p nk n −1 ∑ Pv Pn −v − 2 ∆λv k Pn Pnk−1 v =1
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)
k
Mahendra Misra et al., J. Comp. & Math. Sci. Vol. 1 (6), 758-761 (2010)
761
≤ 0(1)
m+1
∑ n=2
Pn pn
k −1
1 n−1 ∑ Pnk−1 v=1
k
pn−v−1
Pv λv ,
= 0(1), as m → ∞ , as above This completes the proof of the theorem.
using (3.3)
REFERENCE ≤ 0(1)
m+1
Pn n
∑ p n=2
k −1
1 Pn−1
P = 0(1) ∑ n n=2 p n n +1
n−1
∑ v=1
k −1
k
1 pn−v−1 (Pv λv ) Pn−1
1 Pn −1
n −1
∑p v =1
n − v −1
n−1
∑p v=1
n−v−1
Pv λv
k −1
1. Bor Huseyin: On the absolute summability factors of Fourier Series, Journal of Computational Analysis and Applications, Vol.8, No.3, 223-227, (2006).
Journal of Computer and Mathematical Sciences Vol. 1, Issue 6, 31 October, 2010 Pages (636-768)