Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 989865, 8 pages
doi:10.1155/2009/989865
Research Article
On N, p, qE, 1 Summability of Fourier Series
H. K. Nigam and Ajay Sharma
Department of Mathematics, Faculty of Engineering & Technology, Mody Institute of Technology and
Science (Deemed University), Laxmangarh, Sikar (Rajasthan) 332311, India
Correspondence should be addressed to Ajay Sharma, ajaymathematicsanand@gmail.com
Received 7 November 2008; Revised 17 March 2009; Accepted 30 March 2009
Recommended by Hüseyin Bor
A new theorem on N, p, qE, 1 summability of Fourier series has been established.
Copyright q 2009 H. K. Nigam and A. Sharma. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let {pn } and {qn } be the sequences of constants, real or complex, such that
P n p0 p1 p2 · · · pn n
pν −→ ∞,
as n −→ ∞,
ν0
Qn q0 q1 q2 · · · qn n
qν −→ ∞,
as n −→ ∞,
1.1
ν0
Rn p0 qn p1 qn−1 · · · pn q0 n
pν qn−ν −→ ∞,
as n −→ ∞.
ν0
Given two sequences {pn } and {qn } convolution p ∗ q is defined as
n
pn−k qk .
Rn p ∗ q n k0
Let
∞
n0 un
be an infinite series with the sequence of its nth partial sums {sn }.
1.2
2
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We write
p,q
tn n
1 pn−k qk sk .
Rn k0
1.3
If Rn /
0, for all n, the generalized Nörlund transform of the sequence {sn } is the sequence
p,q
{tn }.
p,q
If tn → S, as n → ∞, then the series ∞
n0 un or sequence {sn } is summable to S by
generalized Nörlund method Borwein 1 and is denoted by
Sn −→ S N, p, q .
1.4
The necessary and sufficient conditions for N, p, q method to be regular are
n pn−k qk O|Rn |,
1.5
k0
and pn−k o|Rn |, as n → ∞ for every fixed k ≥ 0, for which qk / 0.
Now
En1
If En1 → s, as n → ∞, then the series
p,q,E
tn
n
n
1
sk .
n
2 k0 k
∞
n0 un
1.6
is said to be E, 1 summable to s Hardy 2:
n
1 pn−k qk Ek1
Rn k0
1.7
n
k
k
1 1
sν .
pn−k qk · k
Rn k0
2 ν0 ν
p,q,E
If tn
→ ∞, as n → ∞, then we say that the series
summable to S by N, p, qE, 1 summability method.
∞
n0 un
or the sequence {sn } is
Particular Cases
1 N, p, qE, 1 mean reduces to N, pn E, 1 summability mean if qn 1, ∀n.
2 N, p, qE, 1 mean reduces to N, 1/n 1E, 1 mean if pn 1/n 1 and qn 1, ∀n.
3 N, p, qE, 1 method reduces to N, qn E, 1 if pn 1, ∀n.
nα−1 4 N, p, qE, 1 method reduces to C, αE, 1 if pn , α > 0, and qn α−1
1, ∀n.
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3
Let ft be a periodic function with period 2π and integrable in the sense of Lebesgue over
the interval −π, π.
Let its Fourier series be given by
ft ∼
∞
1
a0 an cos nt bn sin nt.
2
n1
1.8
We use the following notations:
φt fx t − fx − t − 2fx,
t
Φt φu du,
0
1
1
τ
the integral part of ,
t
t
1
R
Rn Rn,
Rτ ,
t
Kn t 1.9
n
cosk t/2 cosk 1t/2
1 .
pn−k qk
2πRn k0
sin t/2
2. Theorem
A quite good amount of work is known for Fourier series by ordinary summability method.
The purpose of this paper is to study Fourier series by N, p, qE, 1 summability method in
the following form.
Theorem 2.1. Let {pn } and {qn } be positive monotonic, nonincreasing sequences of real numbers
such that
Rn n
pk qn−k −→ ∞,
as n −→ ∞.
2.1
k0
Let αt be a positive, nondecreasing function of t. If
Φt t
φudu o
0
αn −→ ∞,
t
,
α1/t
as t −→ 0,
as n −→ ∞,
2.2
2.3
then a sufficient condition that the Fourier Series 1.8 be summable N, p, qE, 1 to fx at the
point t x is
n
Ru
du ORn ,
uαu
1
as n −→ ∞.
2.4
4
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3. Lemmas
Proof of the theorem needs some lemmas.
Lemma 3.1. For 0 ≤ t ≤ 1/n,
|Kn t| On.
3.1
Proof.
n
cosk t/2 sink 1t/2 1 |Kn t| pn−k qk
2πRn k0
sin t/2
n
n
1 1 k 1|sin t/2|
≤
On
pn−k qk
pn−k qk On.
2πRn k0
Rn k0
|sin t/2|
3.2
Lemma 3.2. If {pn } and {qn } are nonnegative and nonincreasing, then for 0 ≤ a ≤ b < ∞, 0 ≤ t ≤ π,
and any n we have
b
Rτ
cosk t/2 sin k 1t/2 1 .
pn−k qk
O
2πRn ka
sint/2
tRn
3.3
Proof.
b
cosk t/2 sink 1t/2 1 pn−k qk
2πRn ka
sin t/2
b
1 t
t
≤
sink 1 pn−k qk cosk
tπRn ka
2
2
b
1 k t
ik1t/2 e
pn−k qk cos
Im
tπRn 2
ka
b
1 k t
ikt/2 it/2 ≤
e
pn−k qk cos
e tπRn ka
2
b
1 k t
ikt/2 ≤
e
pn−k qk cos
tπRn ka
2
⎧
⎛ ⎞ ⎫
t ⎪
⎪
ikt
⎪
⎪
ik⎝ ⎠ τ−1
b
⎬
1 ⎨
t
t
2
k
k
≤
e 2 pn−k qk cos
e
pn−k qk cos
⎪.
kτ
tπRn ⎪
2
2
⎪
⎪
⎩ka
⎭
3.4
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5
Now considering first term of 3.4, we have
τ−1
τ−1
τ−1
1 1 1 k t
ikt/2 e
pn−k qk eikt/2 ≤
pn−k qk
pn−k qk cos
≤
tπRn ka
tπRn ka
2
tπRn ka
τ−1
1 1
Rτ
≤
pτ−k qk ≤
.
Rτ O
tπRn ka
tπRn
tRn
3.5
Now considering second term of 3.4 and using Abel’s lemma, we have
b
b
1 1 k t
ikt/2 ikt/2 e
pn−k qk cos
≤
pn−k qk e
tπRn kτ
2
tπRn kτ
1 − eik1t/2 2pn−b qτ
≤
max tπRn τ1≤k≤b 1 − eit/2 4pn−b qτ e−it/4
≤
it/4
tπRn e
− e−it/4 τ
2qτ
pn−b
1
≤
where Pτ pτ−k
Pτ tπRn Pτ
sin t/4 k0
1
8qτ
pn−b
≤
Pτ tπRn Pτ
t
≤
8qτ Pτ
tπRn
τ
8Rτ
since Rτ ≤
pτ−k qk ≥ Pτ qτ
tπRn
k0
Rτ
O
.
tRn
3.6
Using 3.5 and 3.6, we get the required result of Lemma 3.2.
4. Proof of Theorem
Following Zygmund 3, the nth partial sum sn x of the series 1.8 at t x is given by
1
sn x fx 2π
π
0
φx t
sinn 1/2t
dt.
sin t/2
4.1
6
International Journal of Mathematics and Mathematical Sciences
So the E, 1 mean of the series 1.8 at t x is given by
En1 x
n
n
1
n
sk x
2 k0 k
fx fx π
1
2n1 π
0
π
1
2n1 π
0
π
n
n
φx t
1
sin k t dt
sin t/2 k0 k
2
n φx t
Im eit/2 1 eit
dt
sin t/2
φx t
n
it/2
Im
e
dt
cos
t
i
sin
t
1
2n1 π 0 sin t/2
π
φx t
1
t n
t
it/2 n
n t
fx n1
Im e 2 cos
dt
cos i sin
2
2
2
2 π 0 sin t/2
π
φx t
nt
1
t
nt
Im eit/2 2n cosn
i sin
fx n1
cos
dt
2
2
2
2 π 0 sin t/2
cosn t/2sinn 1t/2
1 π
dt.
φx t
fx 2π 0
sin t/2
fx 1
4.2
Therefore N, p, q transform of {En1 x} is given by
p,q,E
tn
p,q,E
tn x
n
n
cosk tsink 1t/2
1 1 π 1 pn−k qk Ek1 x fx pn−k qk φx t
Rn k0
2π 0 Rn k0
sin t/2
π
fx Kn tφx tdt,
x − fx ! 1/n
0
π"
Kn tφx tdt I1 I2 I3
δ
0
1/n
say .
δ
4.3
We have
|I1 | ≤
1/n
|Kn t| φx tdt
0
On
1/n
φx tdt
using Lemma 3.1
0
4.4
Ono
1
by 2.2
nαn
1
o
o1 as n −→ ∞
αn
by 2.3 .
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7
Now we consider
|I2 | ≤
δ
|Kn t|φx t dt
where 0 < δ < 1
1/n
δ
O
1/n
R1/t φx t dt
tRn
δ
!
1
O
Rn
1
O
Rn
1/n
1
O
Rn
using Lemma 3.2
R1/t φx t dt
t
R1/t
φx t
t
! R1/t
o
α1/t
δ
"
R1/t
φx t
d
t
1/n
δ
−
1/n
δ
−
δ
1/n
1/n
R1/t
φx td
t
"
by 2.1
! δ
"
1
1
1
R1/tα1/t
o
o
o
φx t d
Rn
αn
Rn
tα1/t
1/n
1
1
1
o
o
o
Rn
αn
Rn
! δ
R1/t
t
dα1/t
×
o
α1/t
tα1/t
1/n
"
1
R1/t
α
d
t
tα1/t
! δ
δ
"
1
1
dα1/t
1
R1/t
o
o
o1
o
td
2
Rn
αn
Rn
tα1/t
1/n {α1/t}
1/n
1
1
1
o
o
o1
Rn
αn
α1/t
1
o
Rn
!
tR1/t
tα1/t
δ
1/n
−
δ 1/n
δ
1/n
"
R1/t
dt
tα1/t
1 1
1
1
R1/t
1
o
o
o
dt
o
Rn
αn
αn
Rn
1/n tα1/t
n 1
1
1
1
Ru
o
o
o
du
∵ u
Rn
αn
Rn
t
1 uαu
1
1
1
o
o
ORn by 2.4
o
Rn
αn
Rn
1
1
o
o1
o
Rn
αn
o1, as n → ∞ by virtue of 2.1 and 2.2 .
4.5
8
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Now by Riemann-Lebesgue theorem and by regularity of the method of summability we
have
I3 π
|kn t| φx tdt
δ
o1,
4.6
as n −→ ∞.
This completes the proof of the theorem.
5. Corollaries
Following corollaries can be derived from our main theorem.
Corollary 5.1. If
Φt o
t
,
log 1/t
as t −→ 0,
5.1
then the Fourier series 1.8 is C, 1E, 1 summable to fx at the point t x.
Corollary 5.2. If
Φt ot,
as t −→ 0,
5.2
then the Fourier series 1.8 is N, pn E, 1 summable to fx at the point t x, provided that
{pn } be a positive, monotonic, and nonincreasing sequence of real numbers such that
pn p0 p1 · · · pn −→ ∞,
as n −→ ∞.
5.3
Acknowledgments
The authors are grateful to Professor Shyam Lal, Department of Mathematics, Harcourt
Butler Technological Institute, Kanpur, India, for his valuable suggestions and guidence
in preparation of this paper. The authors are also thankful to Professor M. P. Jain, Vice
Chancellor, Mody Institute of Technology and Science Deemed University, Lakshmangarh,
Sikar, Rajasthan, India, and to Professor S. N. Puri, Former Dean, Faculty of Engineering,
Technology, Mody Institute of Technology and Science Deemed University, Laxmangarh,
Sikar, Rajasthan.
References
1 D. Borwein, “On product of sequences,” Journal of the London Mathematical Society, vol. 33, pp. 352–357,
1958.
2 G. H. Hardy, Divergent Series, Oxford University Press, Oxford, UK, 1st edition, 1949.
3 A. Zygmund, Trigonometric Series. Vol. I, Cambridge University Press, Cambridge, UK, 2nd edition,
1959.