Bulletin of Mathematical analysis and Applications
ISSN:1821-1291,URL: http://www.bmathaa.org
Volume 1, Issue 1, (2009) Pages 15-21
On the local properties of factored Fourier series
∗
Hüseyin Bor
Abstract
In the present paper, a theorem on local property of | N̄ , pn , θn |k summability of factored Fourier series which generalizes a result of Bor [3] has
been proved.
1
Introduction
P
Let
an be a given infinite series with partial sums
P (sn ). We denote by tn the
an is said to be summable
n-th (C,1) mean of the sequence (nan ). A series
| C, 1 |k , k ≥ 1 , if (see [6],[8])
∞
X
1
| tn |k < ∞.
n
n=1
(1.1)
Let (pn ) be a sequence of positive numbers such that
Pn =
n
X
pv → ∞ as n → ∞,
(P−i = p−i = 0, i ≥ 1).
(1.2)
v=0
The sequence-to-sequence transformation
σn =
n
1 X
pv sv
Pn v=0
(1.3)
defines the sequence (σn ) of the Riesz mean or simply the (N̄ , pn ) mean of the
sequence (sn ), generated by the sequence of coefficients (pn ) (see [7]). The series
P
an is said to be summable | N̄ , pn |k , k ≥ 1, if (see [2])
∞
X
(Pn /pn )k−1 | ∆σn−1 |k < ∞,
(1.4)
n
X
pn
Pv−1 av ,
Pn Pn−1 v=1
(1.5)
n=1
where
∆σn−1 = −
n ≥ 1.
∗ Mathematics Subject Classifications: 40G99, 42A24, 42B24.
Key words: Absolute summability, infinite series, local property, Fourier series,
c
2009
Universiteti i Prishtines, Prishtine, Kosovë.
Submitted November 2008, Published January 2009.
15
16
On the local properties of factored Fourier series
In the special case pn = 1 for all values of n, | N̄ , pn |k summability is the
same as | C, 1 |k summability. Also,if we take k = 1 and pn = 1/(n + 1), then
summability | N̄ , pn |k is equivalent to the summability
| R, log n, 1 |. Let (θn )
P
be any sequence of positive constants. The series
an is said to be summable
| N̄ , pn , θn |k , k ≥ 1, if (see [12])
∞
X
θnk−1 | ∆σn−1 |k < ∞.
(1.6)
n=1
If we take θn = Ppnn , then | N̄ , pn , θn |k summability reduces to | N̄ , pn |k
summability. Also, if we take θn = n and pn = 1 for all values of n, then we
get | C, 1 |k summability. Furthermore, if we take θn = n, then | N̄ , pn , θn |k
summability reduces to | R, pn |k (see [4]) summability. A sequence (λn ) is said
to be convex if ∆2 λn ≥ 0 for every positive integer n, where ∆2 λn = ∆(∆λn )
and ∆λn = λn − λn+1 .
Let f (t) be a periodic function with period 2π, and integrable (L) over (−π, π).
Without any loss of generality we may assume that the constant term in the
Fourier series of f (t) is zero, so that
Z π
f (t)dt = 0
(1.7)
−π
and
f (t) ∼
∞
X
(an cos nt + bn sin nt) =
n=1
2
∞
X
An (t).
(1.8)
n=1
Known result
Mohanty [11] has demonstrated that the summability | R, log n, 1 | of
X
An (t)/ log(n + 1),
(2.1)
P
at t = x, is a local property of the generating function of
An (t). Later on
Matsumoto [9] improved this result by replacing the series (2.1) by
X
An (t)/ log log(n + 1)1+ , > 0.
(2.2)
Generalizing the above result Bhatt [1] proved the following theorem.
P −1
Theorem A. If (λn ) is a convex sequence suchPthat
n λn is convergent,
then the summability | R, log n, 1 | of the series
An (t)λn log n at a point can
be ensured by a local property.
Also, Mishra [10] has proved the following most general theorem on this matter.
Theorem B. If (pn ) is a sequence such that
Pn = O(npn )
(2.3)
Hüseyin Bor
17
Pn ∆pn = O(pn pn+1 ),
(2.4)
then the summability | N̄ , pn | of the series
∞
X
An (t)λn Pn /npn
(2.5)
n=1
at a point can be ensured by local property, where (λn ) is as in Theorem A.
On the other hand Bor [3] has generalized Theorem B for | N̄ , pn |k summability
in the following form.
Theorem C. Let k ≥ 1 and (pn ) be a sequence such that the conditions (2.3)
and (2.4) of Theorem B are satisfied. Then the summability | N̄ , pn |k of the
series (2.5) at a point can be ensured by local property , where (λn ) is as in
Theorem A.
3
Main result
The aim of this paper is to generalize Theorem C for | N̄ , pn , θn |k summability.
We shall prove the following theorem.
Theorem. Let k ≥ 1 and (pn ) be a sequence such that the conditions (2.3)(2.4) of Theorem B are satisfied. If (θn ) is any sequence of positive constants
such that
k−1
m X
1
θ v pv
(λv )k = O(1)
(3.1)
P
v
v
v=1
k−1
m X
θ v pv
∆λv = O(1)
(3.2)
Pv
v=1
k−1
m X
θ v pv
1
(λv+1 )k = O(1)
(3.3)
P
v
v
v=1
and
m+1
X
n=v+1
θ n pn
Pn
k−1
pn
=O
Pn Pn−1
(
θ v pv
Pv
k−1
1
Pv
)
,
(3.4)
then the summability | N̄ , pn , θn |k of the series (2.5) at a point can be ensured
by local property, where (λn ) is as in Theorem A.
It should be noted that if we take θn = Ppnn , then we get Theorem C. In this case
the conditions (3.1)-(3.3) are obvious and the condition (3.4) reduces to
m+1
X
1
pn
=O
,
P
P
P
v
n=v+1 n n−1
18
On the local properties of factored Fourier series
which always holds.
We need the following lemmas for the proof of our theorem.
Lemma 1 ([10]). If the sequence (pn ) is such that the conditions (2.3) and
(2.4) of Theorem B are satisfied, then
∆(Pn /npn ) = O(1/n).
(3.5)
Lemma 2 ([5]). If (λn ) is a convex sequence such that
n−1 λn is convergent,
then (λn ) is non-negative and decreasing, and n∆λn → 0 as n → ∞.
P
Lemma 3. Let k ≥ 1 .If (sn ) is bounded and all conditions of the Theorem are
satisfied, then the series
∞
X
An λn Pn /npn
(3.6)
n=1
is summable | N̄ , pn , θn |k , where (λn ) is as in Theorem A.
k
Remark. Since
P (λn ) iska convex sequence, therefore (λn ) is also convex sequence and (1/n)(λn ) < ∞.
Proof of Lemma 3. Let (Tn ) denotes the (N̄ , pn ) mean of the series (3.6).
Then, by definition, we have
Tn =
n
v
n
1 X
1 X X
pv
ar λr Pr /rpr =
(Pn − Pv−1 )av λv Pv /vpv .
Pn v=0 r=0
Pn v=0
Then
Tn − Tn−1 =
n
X
pn
av λv
Pv−1 Pv
,
Pn Pn−1 v=1
vpv
n ≥ 1,
(P−1 = 0).
By Abel’s transformation, we have
Tn − Tn−1
n−1
X
1
pn
pv Pv sv λv
= −
Pn Pn−1 v=1
vpv
+
n−1
X
pn
1
Pv sv Pv ∆λv
Pn Pn−1 v=1
vpv
+
n−1
X
pn
1
Pv λv+1 ∆(Pv /vpv )sv + sn λn
Pn Pn−1 v=1
n
= Tn,1 + Tn,2 + Tn,3 + Tn,4 ,
say.
To prove the lemma, by Minkowski’s inequality, it is sufficient to show that
∞
X
n=1
θnk−1 | Tn,r |k < ∞,
f or
r = 1, 2, 3, 4.
(3.7)
Hüseyin Bor
19
Now, applying Hölder’s inequality, we have that
m+1
X
θnk−1
k
| Tn,1 |
≤
n=2
=
=
=
=
)k−1
k
k (
n−1
n−1
pn
1 X
λv Pv
1 X
k
×
| sv | pv
pv
Pn
Pn−1 v=1
vpv
Pn−1 v=1
n=2
k
m+1
m
X θn pn k−1
X
Pv
pn
k 1
O(1)
(λv ) k
pv
p
v
P
P
v
n
n Pn−1
n=v+1
v=1
k−1
k−1
m X
Pv
1 θ v pv
O(1)
(λv )k k
pv
v
Pv
v=1
m
k−1
X
1 θ v pv
O(1)
v k−1 (λv )k k
v
Pv
v=1
m
k−1
X
1
θ v pv
O(1)
(λv )k = O(1) as m → ∞,
P
v
v
v=1
m+1
X
θnk−1
by virtue of the hypotheses of the Theorem. Since
n−1
X
Pv ∆λv ≤ Pn−1
v=1
n−1
X
∆λv ⇒
v=1
1
n−1
X
Pn−1
v=1
Pv ∆λv ≤
n−1
X
∆λv = O(1),
v=1
by Lemma 2, we have that
m+1
X
θnk−1
k
| Tn,2 |
≤
n=2
m+1
X
θnk−1
n=2
(
×
pn
Pn
1
n−1
X
Pn−1
v=1
k
1
Pn−1
n−1
X
v=1
Pv
vpv
k
Pv ∆λv | sv |k
)k−1
pv
k
m m+1
X
X θn pn k−1
Pv
1
pn
= O(1)
P ∆λv
k v
vp
v
P
P
v
n
n Pn−1
v=1
n=v+1
k
k−1
m X
Pv
θ v pv
1
= O(1)
∆λ
v
vpv
vk
Pv
v=1
m
k−1
X
θ v pv
= O(1)
∆λv
Pv
v=1
= O(1) as m → ∞,
in view of the hypotheses of the Theorem and Lemma 2.
20
On the local properties of factored Fourier series
Using the fact that ∆(Pv /vpv ) = O(1/v) by Lemma 1, we have that
k
m+1
m
n−1
X
X
X
pn
1
θnk−1 | Tn,3 |k =
θnk−1
|
Pv λv+1 ∆(Pv /vpv )sv |k
k
P
P
n
n−1 v=1
n=2
n=1
)k
(n−1
k
m+1
X
X Pv
p
1
1
n
= O(1)
θnk−1
pv λv+1
k
Pn
p
v
Pn−1
n=2
v=1 v
k
k
m+1
n−1 X
pn
1
1 X Pv
= O(1)
θnk−1
pv (λv+1 )k k
P
P
p
v
n
n−1
v
n=2
v=1
(
)k−1
n−1
1 X
×
pv
Pn−1 v=1
k
k−1
m+1 m X
1 X
θ n pn
Pv
pn
= O(1)
pv (λv+1 )k k
pv
v n=v+1 Pn
Pn Pn−1
v=1
m
k−1
k−1
X
Pv
1 θ v pv
= O(1)
(λv+1 )k k
pv
v
Pv
v=1
k−1
m
X
1 θ v pv
v k−1 (λv+1 )k k
= O(1)
v
Pv
v=1
m
k−1
X θ v pv
1
(λv+1 )k = O(1) as m → ∞,
= O(1)
P
v
v
v=1
by virtue of the hypotheses of the Theorem. Finally, we have that
m
X
n=1
θnk−1
k
| Tn,4 |
=
m
X
θnk−1 (λn )k
n=1
= O(1)
= O(1)
m
X
1
nk
θnk−1 (λn )k | sn |k
n=1
m X
n=1
θ n pn
Pn
k−1
1
nk−1
1
n
1
(λn )k
n
= O(1) as m → ∞,
in view of the hypotheses of the Theorem. Therefore we get that
m
X
θnk−1 | Tn,r |k = O(1) as m → ∞,
f or
r = 1, 2, 3, 4.
n=1
which completes the proof of the Lemma 3.
Remark. If we take k = 1, then we get a result due to Mishra [10].
Hüseyin Bor
4
21
Proof of the Theorem
Since the behaviour of the Fourier series, as far as convergence is concerned, for a
particular value of x depends on the behaviour of the function in the immediate
neighborhood of this point only, hence the truth of the Theorem is necessary
consequence of Lemma 3.
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HÜSEYİN BOR
Department of Mathematics,
Erciyes University,
38039 Kayseri, Turkey
e-mail: bor@erciyes.edu.tr