IJMMS 25:6 (2001) 365–372
PII. S0161171201005130
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON ALMOST (N, p, q) SUMMABILITY OF CONJUGATE
FOURIER SERIES
SHYAM LAL and HARE KRISHNA NIGAM
(Received 9 May 2000)
Abstract. A new theorem on almost generalized Nörlund summability of conjugate series
of Fourier series has been established under a very general condition.
2000 Mathematics Subject Classification. Primary 42B05, 42B08.
1. Introduction. Lorentz [3], for the first time in 1948, defined almost convergence
of a bounded sequence. It is easy to see that a convergent sequence is almost convergent [4]. The idea of almost convergence led to the formulation of almost generalized Nörlund summability method. Here, almost generalized Nörlund summability
method is considered. In 1913, Hardy [1] established (c, α), α > 0 summability of
the series. Later on in 1948, harmonic summability which is weaker than the summability (c, α), α > 0 of the series was discussed by Siddiqi [8]. The generalization of
Siddiqi has been obtained by several workers, for example, Singh [9, 10], Iyengar [2],
Pati [5], Tripathi [11], Rajagopal [7] for Nörlund mean. But nothing seems to have been
done so far in the direction of study of conjugate Fourier series by almost generalized
Nörlund summability method. Almost generalized Nörlund summability includes almost Nörlund, Riesz, harmonic and Cesàro as particular cases. In an attempt to make
an advance study in this direction, in the present paper, a theorem on almost generalized Nörlund summability of conjugate Fourier series has been obtained.
2. Definitions and notations. Let an be an infinite series with {Sn } as the sequence of its nth partial sums. Lorentz [3] has given the following definition.
A bounded sequence {Sn } is said to be almost convergent to a limit S, if
n+m
1
Sν = S,
n→∞ n + 1
ν=m
lim
uniformly with respect to m.
(2.1)
Let {pn } and {qn } be the two sequences of non-zero real constants such that
Pn = p0 + p1 + p2 + · · · + pn ,
Q n = q 0 + q 1 + q2 + · · · + q n ,
P−1 = p−1 = 0,
(2.2a)
Q−1 = q−1 = 0.
(2.2b)
Given two sequences {pn }, {qn }, convolution p ∗ q is defined by
Rn = (p ∗ q)n =
n
k=0
pk qn−k .
(2.3)
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S. LAL AND H. K. NIGAM
It is familiar and can be easily verified that the operation of convolution is commutative and associative, and
n
(p ∗ 1)n =
pk .
(2.4)
k=0
The series
an or the sequence {Sn } is said to be almost generalized Nörlund
(N, p, q) (Qureshi [6]) summable to S, if
tn,m =
n
1 pn−ν qν Sν,m
Rn ν=0
(2.5)
tends to S, as n → ∞, uniformly with respect to m, where
Sν,m =
ν+m
1 Sk .
ν + 1 k=m
(2.6)
Particular cases. (a) Almost (N, p, q) method reduces to almost Nörlund method
(N, pn ) if qn = 1 for all n.
(b) Almost (N, p, q) method reduces to almost Riesz method (N, qn) if pn = 1 for all n.
(c) In the special case when pn = n+α−1
, α > 0, the method (N, pn ) reduces to the
α−1
well-known method of summability (C, α).
(d) pn = (n + 1)−1 of the Nörlund mean is known as harmonic mean and is written
as (N, 1/(n + 1)).
Let f(t) be a periodic function with period 2π and integrable in the sense of Lebesgue
over an interval (−π , π ).
Let its Fourier series be given by
f (t) ∼
∞
∞
1
1
a0 +
An (t)
an cos nt + bn sin nt = a0 +
2
2
n=1
n=1
(2.7)
and then the conjugate series of (2.7) is given by
∞
∞
Bn (t).
an sin nt − bn cos nt =
n=1
(2.8)
n=1
We will use the following notations:
φ(t) = f (x + t) + f (x − t) − 2f (x),
ψ(t) = f (x + t) − f (x − t),
Φ(t) =
Ψ (t) =
t
0
t
0
φ(u) du,
ψ(u) du,
1
1
= The integral part of ,
τ=
t
t
(2.9)
367
ON ALMOST (N, p, q) SUMMABILITY OF CONJUGATE FOURIER SERIES
Nn,m (t) =
n
sin(ν + 1)(t/2) cos(ν + 2m + 1)(t/2) − cos(t/2)
1
, (2.10)
pn−ν qν
2π Rn ν=0
(ν + 1) sin2 (t/2)
N n,m (t) =
n
cos(ν + 2m + 1)(t/2) sin(ν + 1)(t/2)
1
.
pn−ν qν
2π Rn ν=0
(ν + 1) sin2 (t/2)
(2.11)
3. Known theorem. Pati [5] has established the following theorem for Nörlund
summability of a Fourier series.
Theorem 3.1. Let (N, pn) be a regular Nörlund method defined by a real nonnegative monotonic non-increasing sequence of coefficients {pn } such that
Pn =
n
pν → ∞,
as n → ∞,
(3.1)
ν=0
and
log n = O Pn ,
then if
Φ(t) =
t
0
φ(u)du = o
as n → ∞,
t
,
Pτ
(3.2)
as t → +0,
(3.3)
the series (2.7) is summable (N, pn ) to f (x) at the point t = x.
4. Main theorem. In this paper, we aim to generalize the above result for almost
(N, p, q) summability of conjugate Fourier series in the following form.
Theorem 4.1. Let {pn } and {qn } be the monotonic non-increasing sequences of real
n
constants such that Rn = ν=0 pν qn−ν → ∞, as n → ∞. If
Ψ (t) =
t
ψ(u) du = o α(1/t)t ,
R(1/t)
0
1/(n+m)δ ψ(t)
1/(n+m)
t2
dt = o(n),
as t → +0,
as n → ∞,
(4.1)
(4.2)
where 0 < δ < 1/2, uniformly with respect to m, and α(t) is a positive monotonic
non-increasing function of t such that
α(n + m) log(n + m) = O Rn+m , as n → ∞,
n Rn
pn−ν qν
=O
,
(ν + 1)
n
ν=0
then the conjugate Fourier series (2.8)
π
−(1/2π ) 0 cot(1/2)t ψ(t)dt at point t = x.
is
almost
(N, p, q)
(4.3)
(4.4)
summable
to
368
S. LAL AND H. K. NIGAM
5. Lemmas. For the proof of Theorem 4.1, the following lemmas are required.
Lemma 5.1. For 0 < t < 1/(n + m), we have
Nn,m (t)
n
sin(ν + 1)(t/2) cos(ν + 2m + 1)(t/2) − cos(t/2) 1 pn−ν qν
=
2π Rn ν=0
(ν + 1) sin2 (t/2)
n
sin(ν+1)(t/2) 2 sin (ν+2m + 2)/2 (t/2) sin (ν+2m)/2 (t/2) 1 =
pn−ν qν
2π Rn ν=0
(ν+1) sin2 (t/2)
n
(ν+1) sin(t/2) 2 sin (ν+2m + 2)/2 (t/2) sin (ν+2m)/2 (t/2) 1 pn−ν qν
≤
2π Rn ν=0
(ν+1) sin2 (t/2)
n
2 (ν + 2m + 2)/2 sin(t/2) sin (ν + 2m)/2 (t/2)
1
≤
pn−ν qν
sin(t/2)
2π Rn ν=0
n
1
=
pn−ν qν (n + 2m + 2)
2π Rn ν=0
= O(n + m)
n
1 pn−ν qν
Rn ν=0
Nn,m (t) = O(n + m)
by (4.4).
(5.1)
Lemma 5.2. For 1/(n + m) < t < π , we have
N n,m (t) =
n
cos m + (ν + 1)/2 t sin (ν + 1)/2 t
1
,
pn−ν qν
2π Rn ν=0
(ν + 1) sin2 (t/2)
n
cos m + (ν + 1)/2 t sin (ν + 1)/2 t
1
pn−ν qν
2π Rn ν=0
(ν + 1) sin2 (t/2)
n
n 1
1 1 pn−ν qν
1
pn−ν qν
≤
=O 2
,
2
2π Rn ν=0 (ν + 1) sin (t/2)
t Rn ν=0 (ν + 1)
N n,m (t) ≤
N n,m (t) = O
1
t2n
by (4.4).
(5.2)
Proof of Theorem 4.1. Let Sk (x) denote the nth partial sum of the series (2.8).
Then we have
1 π cos k + (1/2) t − cos(t/2)
Sk (x) =
ψ(t) dt
2π 0
sin(t/2)
(5.3)
1 π cos k + (1/2) t
1 π
t
=
ψ(t) dt −
ψ(t) dt.
cot
2π 0
sin(t/2)
2π 0
2
Now, by using (2.6) we get
ν+m
1 Sν,m =
ν + 1 k=m
1
2π
π
0
cos k + (1/2) t
1 π
t
ψ(t) dt −
ψ(t) dt ,
cot
sin(t/2)
2π 0
2
(5.4)
ON ALMOST (N, p, q) SUMMABILITY OF CONJUGATE FOURIER SERIES
369
so that by using (2.5) we have
ν+m
n
1 π
1 t
1 π cos k + (1/2) t
ψ(t) dt −
ψ(t) dt
pn−ν qν
cot
tn,m =
Rn ν=0
2π 0 k=m (ν + 1) sin(t/2)
2π 0
2
1 π
t
ψ(t)dt
cot
tn,m − −
2π 0
2
π ν+m
n
cos k + (1/2) t
1
1
=
ψ(t)dt
pn−ν qν
Rn ν=0
2π (ν + 1) 0 k=m
sin(t/2)
=
=
=
π
n
sin(ν + m + 1)t − sin mt
1
ψ(t) dt
pn−ν qν
2π Rn ν=0
2(ν + 1) sin2 (t/2)
0
π 0
π
0
N n,m (t)ψ(t) dt
=
n
cos(ν + 2m + 1)(t/2) sin(ν + 1)(t/2)
1
ψ(t) dt
pn−ν qν
2π Rn ν=0
(ν + 1) sin2 (t/2)
1/(n+m)
0
+
1/(n+m)δ
1/(n+m)
+
π
1/(n+m)δ
N n,m (t)ψ(t) dt = I1 + I2 + I3 .
(5.5)
First we consider,
I1 =
=
=
1/(n+m)
0
1/(n+m)
0
1/(n+m)
0
+
N n,m (t)ψ(t) dt
n
1
cos(ν + 2m + 1)(t/2) sin(ν + 1)(t/2)
ψ(t) dt
pn−ν qν
2π Rn ν=0
(ν + 1) sin2 (t/2)
n
sin(ν+1)(t/2) cos(ν+2m+1)(t/2)−cos(t/2)
1
ψ(t) dt
pn−ν qν
2π Rn ν=0
(ν+1) sin2 (t/2)
1/(n+m)
0
n
1
sin(ν + 1)(t/2) cot(t/2)
ψ(t) dt = I1.1 + I1.2 .
pn−ν qν
2π Rn ν=0
(ν + 1) sin(t/2)
(5.6)
Now
1/(n+m)
I1.1≤
=
0
n
sin(ν+1)(t/2) cos(ν+2m+1)(t/2)−cos(t/2) 1 ψ(t) dt
pn−ν qν
2π Rn ν=0
(ν+1) sin2 (t/2)
1/(n+m)
0
Nn,m (t)ψ(t) dt
= O(n + m)
1/(n+m)
0
ψ(t) dt
by Lemma 5.1
α(n + m)
= O(n + m)o
by (4.1)
(n + m)Rn+m
α(n + m)
1
=o
by (4.3)
=o
Rn+m
log(m + n)
I1.1 = o(1),
as n → ∞, uniformly with respect to m.
(5.7)
370
S. LAL AND H. K. NIGAM
Next, for 0 < t ≤ 1(n + m)
1/(n+m)
n
sin(ν + 1)(t/2) cot(t/2)
1
ψ(t) dt
pn−ν qν
2π Rn ν=0
(ν + 1) sin(t/2)
0
1/(n+m)
n
1
(ν + 1) sin(t/2) cot(t/2)
≤
ψ(t) dt
pn−ν qν
2π Rn ν=0
(ν + 1) sin(t/2)
0
1 1/(n+m)
t
−
ψ(t) dt
cot
2π 0
2
I1.2 ≤
since the conjugate function exists, therefore
−
1
2π
1/(n+m)
0
cot
t
ψ(t) dt = o(1), as n → ∞, uniformly with respect to m.
2
(5.8)
Hence,
I1.2 = o(1),
(5.9)
thus from (5.6), (5.7), and (5.9)
I1 = o(1).
(5.10)
Now, we take
I2 ≤
1/(n+m)δ
1/(n+m)
N n,m (t)ψ(t) dt
1/(n+m)δ ψ(t)
dt by Lemma 5.2
t2n
1/(n+m)δ ψ(t)
1
dt
=O
n 1/(n+m)
t2
1
=O
o(n) by (4.2)
n
I2 = o(1), as n → ∞, uniformly with respect to m.
=O
1/(n+m)
(5.11)
Finally, we have
n
cos(ν + 2m + 1)(t/2) sin(ν + 1)(t/2) 1
ψ(t) dt
pn−ν qν 2
(ν + 1) sin (t/2)
1/(n+m)δ 2π Rn ν=0
π
n
sin(ν + m + 1)t − sin mt 1
ψ(t) dt
=
pn−ν qν 2
2(ν + 1) sin (t/2)
1/(n+m)δ 2π Rn ν=0
n
π
sin(ν + m + 1)t 1
ψ(t) dt
=
pn−ν qν
2
δ
2π Rn ν=0
2(ν
+
1)
sin
(t/2)
1/(n+m)
π
sin mt
ψ(t) dt
+
2
1/(n+m)δ 2(ν + 1) sin (t/2)
I3 ≤
π
= I3.1 + I3.2 .
(5.12)
ON ALMOST (N, p, q) SUMMABILITY OF CONJUGATE FOURIER SERIES
371
Now, by using second mean value theorem, we have
I3.1 ≤
'
n
1
1
pn−ν qν
sin(ν + m + 1)tψ(t) dt,
2
δ
δ
2π Rn ν=0 2(ν + 1) 2 sin 1/2(n + m)
1/(n+m)
1
1
< ' < π, 0 < δ <
(n + m)δ
2
2 '
1
1/2(n + m)δ
ψ(t) dt
=O
(n + m)2δ
δ
δ
n
sin 1/2(n + m)
1/(n+m)
where
I3.1 = o(1),
as n → ∞, uniformly with respect to m.
(5.13)
Now,
I3.2 ≤
≤
1
2π Rn
2 sin
π
n
1/(n+m)δ ν=0
1
1/2(n + m)δ
2
I3.2 = o(1),
pn−ν qν sin mt
ψ(t) dt
2
2(ν + 1) sin (t/2)
'
1/(n+m)δ
ψ(t) dt
as n → ∞, uniformly with respect to m.
(5.14)
Hence,
I3 = o(1),
as n → ∞.
(5.15)
Now, by combining (5.5), (5.10), (5.11), and (5.15), we have
π
0
Nn,m (t)ψ(t) dt = o(1),
as n → ∞, uniformly with respect to m.
(5.16)
Thus, the theorem is established.
6. Applications. In this section, we deduce some corollaries from Theorem 4.1.
Corollary 6.1. If
Ψ (t) =
t
0
ψ(u) du = o
log(n + m) = O Rn+m ,
t
R(1/t)
,
as n → ∞,
(6.1)
(6.2)
conditions (4.2) and (4.4) of the main theorem are satisfied, then the conjugate Fourier
π
series is almost (N, p, q) summable to −(1/2π ) 0 ψ(t) cot(1/2)t dt.
Corollary 6.2. If
Ψ (t) =
t
0
ψ(u) du = o
t
,
log(1/t)
(6.3)
conditions (4.2) and (4.4) of Theorem 4.1 hold, then the conjugate Fourier series is alπ
most (N, p, q) summable to −(1/2π ) 0 ψ(t) cot(1/2)t dt without employing (4.3).
372
S. LAL AND H. K. NIGAM
Acknowledgements. The authors are grateful to L. M. Tripathi, Department of
Mathematics, B.H.U, Varanasi-221005 for reading the manuscript. The authors are also
thankful to A. P. Dwivedi, Head of Department of Mathematics and V. K. Jain, Director,
H.B.T.I, Kanpur for their encouragement.
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Shyam Lal: Department of Mathematics, Harcourt Butler Technological Institute,
Nawab Ganj, Kanpur 208002 (U.P), India
Hare Krishna Nigam: Department of Mathematics, Harcourt Butler Technological
Institute, Nawab Ganj, Kanpur 208002 (U.P), India
E-mail address: harekrishnan@hotmail.com