Bulletin of Mathematical analysis and Applications
ISSN:1821-1291,URL: http://www.bmathaa.org
Volume 1, Issue 1, (2009) Pages 49-54
On the local properties of factored Fourier series
∗
Mehmet Ali Sarigöl
Abstract
In this paper we have improved the result of
Bor [Bull.
Math. Anal.
Appl.1, (2009), 15-21] on local property of N , pn , θn k summability of
factored Fourier series by proving under weaker conditions.
1
Introduction
P
Let
an be a given series with partial sums (sn ), and let (pn ) be a sequence of
positive numbers such that Pn = p0 + p1 + · · · + pn → ∞ as n → ∞. The
sequence- to- sequence transformation
n
1 X
pv sv
Pn v=0
defines the sequence (Tn ) of the N , pn means of the sequence (sn ), generated
P
by the sequence coefficients (pn ). The series
an is summable N , pn , θn k
summability, k ≥ 1, if (see [4])
Tn =
∞
X
k
θnk−1 | Tn − Tn−1 | < ∞.
(1.1)
n=1
Let f be a function with period 2π, integrable (L) over (−π, π) . Without any
loss of generality we may assume that the constant term in the Fourier series of
is zero, so that
Zπ
f (t)dt = 0
−π
and
f (t) ∼
∞
X
(an cos nt + bn sin nt) ≡
n=1
∞
X
Cn (t).
(1.2)
n=1
It is well known [5] that convergence of a Fourier series at any point t = x is a
local property of f, i.e., for arbitrarily small δ > 0, the behaviour of (sn (t)), the
∗ Mathematics Subject Classifications: 40G, 42A24, 42B24.
Key words: Absolute summabilitiy; Fourier series; Local property.
c
2009
Universiteti i Prishtines, Prishtine, Kosovë.
Submitted November, 2009. Published March, 2009.
49
50
On the local properties of factored Fourier series
n-th partial sum of the series (1.2), depends only the natura of f in the interval
(x − δ, x + δ) and is not affected by the values it takes outside the interval.A
sequence (λn ) is said to be convex if ∆2 λn ≥ 0 for every positive integers n,
where ∆2 λn = ∆ (∆λn ) and ∆λn = λn − λn+1 .
Lemma 1 ([3]). If the sequence (pn ) satisfies the conditions
Pn = O (npn )
(1.3)
Pn ∆pn = O (pn pn+1 ) ,
(1.4)
and
then
∆ (Pn /npn ) = O(1/n).
(1.5)
Lemma 2 ([2]). 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
2
Known result
Theorem A. Let k ≥ 1 and (pn ) be a sequence such that the conditions
P −1 (1.3)
and (1.4) are satisfied. Let (λn ) is a convex sequence such that
n λn is
convergent. If (θn ) is any sequence of positive constants such that
k−1
∞ X
1 k
θ v pv
λn < ∞
P
v
v
v=1
k−1
∞ X
θ v pv
v=1
Pv
∆λv < ∞
k−1
∞ X
θ v pv
1 k
λv+1 < ∞
P
v
v
v=1
(2.1)
(2.2)
(2.3)
and
!
k−1
k−1
∞ X
θ n pn
pn
θ v pv
1
=O
Pn
Pn Pn−1
Pv
Pv
n=v+1
then the summability of N , pn , θn k of the series
∞
X
Cn (t) λn Pn /npn
n=1
at a point can be ensured by local property of f.
(2.4)
(2.5)
Mehmet Ali Sarigöl
3
51
The main result
The purpose of this paper is to improve Theorem A by proving under weaker
conditions. Now, we give the following theorem.
Theorem. Let k ≥ 1 and (pn ) be a sequence such that
condition (1.5) is
P the
satisfied. Also let (λn ) is a convex sequence such that
n−1 λn is convergent.
If (θn ) is any sequence of positive constants such that
∞
X
θvk−1
v=1
∞
X
θvk−1
v=1
λv
v
k
< ∞
(3.1)
Pv
∆λv < ∞
v k pv
(3.2)
and
(
)
k−1
k−1
∞ X
pn
θ v pv
1
θ n pn
=O
Pn
Pn Pn−1
Pv
Pv
n=v+1
then the summability of N , pn , θn k of the series
∞
X
Cn (t) λn Pn /npn
(3.3)
(3.4)
n=1
at a point can be ensured by local property of f.
It may be remarked that (1.3) and (1.4) ⇒ (1.5) by Lemma 1. It is obvious that
(1.3) and (2.1) ⇒(3.1), and also (1.3) and (2.2) ⇒ (3.2). Furthermore, since
(λn ) monotonic decreasing, the conditions (2.1) and (2.3) are the same.
Proof of the Theorem. As mentioned in the beginning, the convergence
of Fourier series at a point is a local property. Therefore in order to prove
the theorem it is sufficient to prove P
that if (sn ) is bounded, then
under the
conditions of our theorem, the series
λn an Pn /n pn is summable N , pn ; θn k .
Now, let (Tn ) denote the N , pn means of this series. Then we have
Tn − Tn−1 =
n
X
pn
Pv−1 Xv λv av ,
Pn Pn−1 v=1
Xn = Pn /npn
Applying Abel’s transformation to this sum we get
Tn − Tn−1
n
X
pn
pn sn Pn−1 Xn λn
=
sn ∆ (Pv−1 Xv λv ) +
Pn Pn−1 v=1
Pn Pn−1
=
n
n
X
X
pn
pn
sn λn
sv λv+1 ∆ (Pv−1 Xv ) +
sv Pv−1 Xv ∆λv +
Pn Pn−1 v=1
Pn Pn−1 v=1
n
= T1 + T2 + T3 ,
say.
52
On the local properties of factored Fourier series
For the proof of the lemma, by Minkowski’s inequality, it suffices to show that
∞
X
k
θnk−1 | Tr | < ∞,
r = 1, 2, 3..
n=1
Now, since sn = O(1), It follows that
m+1
X
θnk−1
k
| T1 |
= O(1)
n=2
m+1
X
θnk−1
n=2
pn
Pn Pn−1
k (n−1
X
)k
λv+1 |∆ (Pv−1 Xv )|
.
v=1
On the other hand, in view of
∆ (Pv−1 Xv ) = −pv Xv + Pv ∆Xv = −
Pv
1
+ Pv ∆Xv = Pv − + ∆Xv ,
v
v
it is clear
that the condition ∆Xv = O(1/v) is equivalent to ∆ (Pv−1 Xv ) =
O Pvv . Therefore, making use of Hölder’s inequality and lemma 2, we get
m+1
X
θnk−1
k
| T1 | = O(1)
n=2
m+1
X
θnk−1
pn
Pn Pn−1
k (n−1
X
)k
pn
Pn Pn−1
k (n−1
X
)k
n=2
= O(1)
m+1
X
θnk−1
n=2
Pv
λv+1
v
v=1
λ v X v pv
v=1
) (n−1 )k−1
k (n−1
X
X
pn
k k
λ v X v pv
pv
= O(1)
Pn Pn−1
v=1
v=1
n=2
m
m+1
X
X θn pn k−1
pn
= O(1)
λkv Xvk pv
Pn
Pn Pn−1
v=1
n=v+1
m
k−1
X
1
θ v pv
= O(1)
Xvk |λv |k pv
P
P
v
v
v=1
m
k
X
λv
= O(1), as m → ∞,
= O(1)
θvk−1
v
v=1
m+1
X
θnk−1
by virtue of (3.1). Again, since
n−1
P
v=1
lemma 2, we get
Pv−1 ∆λv ≤ Pn−1
n−1
P
v=1
∆λv = O(Pn−1 ) by
Mehmet Ali Sarigöl
m+1
X
θnk−1
k
| T2 |
= O(1)
n=2
m+1
X
θnk−1
n=2
53
pn
Pn Pn−1
k (n−1
X
)k
Pv−1 Xv ∆λv
v=1
( n−1
)k−1
k n−1
X
X
p
n
= O(1)
θnk−1
Pv−1 Xvk ∆λv
Pv−1 ∆λv
.
P
P
n
n−1
n=2
v=1
v=1
k
n−1
m+1
X
1 X
pn
Pv−1 Xvk ∆λv
= O(1)
θnk−1
P
P
n
n−1
v=1
n=2
k−1
m
m+1
X
X
pn
θ n pn
k
= O(1)
Pv−1 Xv ∆λv
Pn
Pn Pn−1
v=1
n=v+1
m+1
X
= O(1)
m
X
θvk−1
v=1
Pv ∆λv
v k pv
= O(1), as m → ∞,
by virtue of (3.2).
Finally, it is clear that
m
X
k
θnk−1 | T3 | = O(1)
n=1
m
X
n=1
θnk−1
λn
n
k
= O(1) as m → ∞,
by virtue of (3.1). This completes the proof.
References
[1] H. Bor, On the local properties of factored Fourier Series, Bull. Math.
Anal. Appl.1,(2009), 15-21.
[2] H. C. Chow, On the summability factors of infinite series, J. London Math.
Soc. 16,(1941), 215-220.
[3] K. N. Mishra, Multipliers for N , pn summability of Fourier series, Bull.
Inst. Math. Acad. Sinica 14, (1986), 431-438.
[4] W.T. Sulaiman, On some summability factors of infinite series, Proc.
Amer. Math. Soc. 115, (1992), 313-317.
[5] E. C. Tichmarsh, The Theory of Functions, Oxford Univ. Press, London
1961.
54
On the local properties of factored Fourier series
MEHMET AL SARIGÖL
Department of Mathematics,
Pamukkale University,
20017 Denizli, Turkey
e-mail : msarigol@pau.edu.tr