Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 2, Issue 3(2010), Pages 27-31.
ON LOCAL PROPERTY OF FACTORED FOURIER SERIES
WAAD SULAIMAN
Abstract. In this paper generalization as well as improvement to the Sarigöl’s
result concerning local property of factored Fourier series has been achieved.
1. Introduction
∑
Let
𝑎𝑛 be a given series with partial sums (𝑠𝑛 ), and let (𝑝𝑛 ) be a sequence of
positive numbers such that
𝑃𝑛 = 𝑝0 + ... + 𝑝𝑛 → ∞ as 𝑛 → ∞.
The sequence to sequence transformation
𝑇𝑛 =
𝑛
1 ∑
𝑝 𝑣 𝑠𝑣
𝑃𝑛 𝑣=0
defines the sequence (𝑇𝑛 ) of the (𝑁 , 𝑝𝑛 ) means
of the sequence (𝑠𝑛 ) generated
by
∑
the sequence of coefficients (𝑝𝑛 ).The series 𝑎𝑛 is said to be summable 𝑁 , 𝑝𝑛 , 𝜃𝑘 ,
𝑘 ≥ 1 if (see [8])
∞
∑
𝑘
𝜃𝑛𝑘−1 ∣𝑇𝑛 − 𝑇𝑛−1 ∣ < ∞.
(1.1)
𝑛=1
In the special case when 𝜃𝑛 is equal to 𝑃𝑛 /𝑝𝑛 , 𝑛,we obtain 𝑁 , 𝑝𝑛 𝑘 , ∣𝑅, 𝑝𝑛 ∣𝑘 summabilities respectively.
Let 𝑓 be a function with period 2𝜋, integrable (𝐿) over (−𝜋, 𝜋). Without loss of
generality, we may assume that the constant term of the Fourier series of 𝑓 is zero,
that is
∫𝜋
𝑓 (𝑡)𝑑𝑡 = 0,
−𝜋
𝑓 (𝑡) ≈
∞
∑
(𝑎𝑛 cos 𝑛𝑡 + 𝑏𝑛 sin 𝑛𝑡) ≡
𝑛=1
∞
∑
𝐶𝑛 (𝑡).
(1.2)
𝑛=1
The sequence (𝜆𝑛 ) is said to be convex if Δ2 𝜆𝑛 ≥ 0 for every positive integer 𝑛,
where Δ𝜆𝑛 = 𝜆𝑛 − 𝜆𝑛+1 .
2000 Mathematics Subject Classification. 40G99, 43A24, 42B24.
Key words and phrases. Absolute summability; Fourier series; Local property.
c
⃝2010
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted April, 2010. Published Jun, 2010.
27
28
W.T. SULAIMAN
Generalizing the results( [1], [2], [5], [6]), Bor [3] has proved the following result
Theorem 1.1. Let 𝑘 ≥ 1 and (𝑝𝑛 ) be a sequence satisfying the conditions
𝑃𝑛 = 𝑂(𝑛𝑝𝑛 ),
(1.3)
𝑃𝑛 Δ𝑝𝑛 = 𝑂(𝑝𝑛 𝑝𝑛+1 ).
(1.4)
If (𝜃𝑛 ) is any sequence of positive constants such that
)𝑘−1
𝑚 (
∑
𝜃𝑣 𝑝 𝑣
1
(𝜆𝑣 )𝑘 = 𝑂(1),
𝑃
𝑣
𝑣
𝑣=1
)𝑘−1
𝑚 (
∑
𝜃𝑣 𝑝 𝑣
𝑃𝑣
𝑣=1
(1.5)
Δ𝜆𝑣 = 𝑂(1),
(1.6)
)𝑘−1
𝑚 (
∑
𝜃𝑣 𝑝 𝑣
1
(𝜆𝑣+1 )𝑘 = 𝑂(1),
𝑃
𝑣
𝑣
𝑣=1
and
𝑚+1
∑
𝑛=𝑣+1
(
𝜃𝑛 𝑝 𝑛
𝑃𝑛
)𝑘−1
𝑝𝑛
=𝑂
𝑃𝑛 𝑃𝑛−1
((
𝜃𝑣 𝑝 𝑣
𝑃𝑣
)𝑘−1
(1.7)
1
𝑃𝑣
)
,
(1.8)
∞
∑
then the summability 𝑁 , 𝑝𝑛 , 𝜃𝑛 𝑘 of the series
𝐶𝑛 (𝑡)𝜆𝑛 𝑃𝑛 /𝑛𝑝𝑛 at a point can
𝑛=1
∑ −1
be ensured by local property, where (𝜆𝑛 ) is convex sequence such that
𝑛 𝜆𝑛 is
convergent.
In his roll, Sarıgöl [7] generalized the above Bor’s result by giving the following
Theorem 1.2. Let 𝑘 ≥ 1 and (𝑝𝑛 ) be a sequence satisfying the conditions
Δ (𝑃𝑛 /𝑛𝑝𝑛 ) = 𝑂(1/𝑛).
(1.9)
∑ −1
Let (𝜆𝑛 ) is a convex sequence such that
𝑛 𝜆𝑛 is convergent. If (𝜃𝑛 ) is any
sequence of positive constants such that
𝑚
∑
𝜃𝑣𝑘−1
𝑣=1
𝑚
∑
𝑣=1
and
𝑚+1
∑
𝑛=𝑣+1
(
𝜃𝑛 𝑝 𝑛
𝑃𝑛
)𝑘−1
𝑃𝑣
Δ𝜆𝑣 < ∞,
𝑣 𝑘 𝑝𝑣
𝜃𝑣𝑘−1
(
𝜆𝑣
𝑣
)𝑘
(1.10)
< ∞,
𝑝𝑛
=𝑂
𝑃𝑛 𝑃𝑛−1
((
𝜃𝑣 𝑝 𝑣
𝑃𝑣
(1.11)
)𝑘−1
1
𝑃𝑣
)
,
(1.12)
∞
∑
then the summability 𝑁 , 𝑝𝑛 , 𝜃𝑛 𝑘 of the series
𝐶𝑛 (𝑡)𝜆𝑛 𝑃𝑛 /𝑛𝑝𝑛 at a point can
be ensured by local property of 𝑓.
𝑛=1
The following Lemmas are needed for our aim
ON LOCAL PROPERTY OF FACTORED FOURIER SERIES
29
Lemma 1.3. [5]. If the sequence (𝑝𝑛 ) satisfies the conditions
𝑃𝑛 = 𝑂(𝑛𝑝𝑛 ),
(1.13)
𝑃𝑛 Δ𝑝𝑛 = 𝑂(𝑝𝑛 𝑝𝑛+1 )
(1.14)
then
Δ (𝑃𝑛 /𝑛𝑝𝑛 ) = 𝑂(1/𝑛).
(1.15)
∑
−1
Lemma 1.4. [4]. If (𝜆𝑛 ) is a convex sequence such that
𝑛 𝜆𝑛 is convergent,
then (𝜆𝑛 ) is non-negative and decreasing and Δ𝜆𝑛 → 0 as 𝑛 → ∞.
2. Main Results
The coming result covers all the results mentioned in the references
Theorem 2.1. Let 𝑘 ≥ 1, and let the sequences (𝑝𝑛 ), (𝜃𝑛 ), (𝜆𝑛 ) and (𝜑𝑛 ) where
𝜃𝑛 > 0, are all satisfying
∣𝜆𝑛+1 ∣ = 𝑂 (∣𝜆𝑛 ∣) ,
(2.1)
(
)
∞
𝑘
∑
𝑝𝑛
𝜃𝑛𝑘−1
∣𝜆𝑛 ∣𝑘 ∣𝜑𝑛 ∣𝑘 < ∞,
(2.2)
𝑃
𝑛
𝑛=1
∞
∑
𝑘
𝑘
𝜃𝑛𝑘−1 ∣𝜆𝑛 ∣ ∣Δ𝜑𝑛 ∣ < ∞,
(2.3)
𝑛=1
𝑛−1
∑
𝜃𝑣1−1/𝑘 ∣𝜑𝑣 ∣
𝑣=1
and
𝑚+1
∑
𝑛=𝑣+1
(
𝜃𝑛 𝑝 𝑛
𝑃𝑛
)𝑘−1
(
𝑃𝑣
𝑝𝑣
)(1/𝑘)−1
𝑝𝑛
=𝑂
𝑃𝑛 𝑃𝑛−1
∣Δ𝜆𝑣 ∣ < ∞,
((
𝜃𝑣 𝑝 𝑣
𝑃𝑣
)𝑘−1
(2.4)
1
𝑃𝑣
)
,
(2.5)
∞
∑
then the summability 𝑁 , 𝑝𝑛 , 𝜃𝑛 𝑘 of the series
𝐶𝑛 (𝑡)𝜆𝑛 𝜑𝑛 at a point can be
𝑛=1
ensured by local property of f.
Proof. Let (𝑇𝑛 ) denote the (𝑁 , 𝑝𝑛 ) mean of the series
𝐶𝑛 (𝑡)𝜆𝑛 𝜑𝑛 . Then, we
𝑛=1
have
𝑇𝑛 − 𝑇𝑛−1
∞
∑
=
=
=
=
𝑛
∑
𝑝𝑛
𝑃𝑣−1 𝜆𝑣 𝜑𝑣 𝑎𝑣
𝑃𝑛 𝑃𝑛−1 𝑣=1
( 𝑣
)
( 𝑛
)
𝑛−1
∑ ∑
∑
𝑝𝑛
𝑝𝑛
𝑎𝑣 Δ (𝑃𝑣−1 𝜆𝑣 𝜑𝑣 ) +
𝑎𝑣
𝜆𝑛 𝜑𝑛
𝑃𝑛 𝑃𝑛−1 𝑣=1 𝑟=1
𝑃
𝑛
𝑣=1
𝑛−1
∑
𝑝𝑛
𝑝𝑛
(−𝑠𝑣 𝑝𝑣 𝜆𝑣 𝜑𝑣 + 𝑠𝑣 𝑃𝑣 Δ𝜆𝑣 𝜑𝑣 + 𝑠𝑣 𝑃𝑣 𝜆𝑣+1 Δ𝜑𝑣 ) + 𝑠𝑛 𝜆𝑛 𝜑𝑛
𝑃𝑛 𝑃𝑛−1 𝑣=1
𝑃𝑛
𝑇𝑛1 + 𝑇𝑛2 + 𝑇𝑛3 + 𝑇𝑛4 .
In order to complete the proof, by Minkowski′ s inequality, it is sufficient to show
that
∞
∑
𝑘
< ∞,
𝜃𝑛𝑘−1 𝑇𝑛𝑟
𝑟 = 1, 2, 3, 4.
𝑛=1
30
W.T. SULAIMAN
Applying Holder’s inequality,
𝑚+1
∑
𝑛=2
𝜃𝑛𝑘−1 ∣𝑇𝑛1 ∣
𝑘
=
𝑚+1
∑
𝑛=2
𝑚+1
∑
𝜃𝑛𝑘−1 ∣𝑠𝑣 𝑝𝑣 𝜆𝑣 𝜑𝑣 ∣
𝑘
(𝑛−1
)
∑ 𝑝𝑣 𝑘−1
𝑘
𝑘
𝑘
∣𝑠
∣
𝑝
∣𝜆
∣
∣𝜑
∣
𝑣
𝑣
𝑣
𝑣
𝑃𝑛−1
𝑃𝑛−1
𝑛=2
𝑣=1
𝑣=1
( )𝑘
𝑚
𝑚+1
∑
∑
𝑘
𝑘
1
= 𝑂(1)
𝑝𝑣 ∣𝜆𝑣 ∣ ∣𝜑𝑣 ∣
𝜃𝑛𝑘−1 𝑃𝑝𝑛𝑛
𝑃𝑛−1
𝑣=1
𝑛=𝑣+1
(
)𝑘−1
𝑚
∑
1
= 𝑂(1)
𝑝𝑣 ∣𝜆𝑣 ∣𝑘 ∣𝜑𝑣 ∣𝑘 𝜃𝑃𝑣 𝑝𝑣𝑣
𝑃𝑣
𝑣=1
( )𝑘
𝑚
∑
𝑘
𝑘
= 𝑂(1)
𝜃𝑣𝑘−1 𝑃𝑝𝑣𝑣 ∣𝜆𝑣 ∣ ∣𝜑𝑣 ∣ = 𝑂(1).
≤
𝜃𝑛𝑘−1
(
𝑝𝑛
𝑃𝑛
)𝑘
𝑛−1
∑
1
𝑣=1
𝑚+1
∑
𝑘
𝜃𝑛𝑘−1 ∣𝑇𝑛2 ∣ =
𝑛=2
𝑚+1
∑
𝜃𝑛𝑘−1 ∣𝑠𝑣 𝑃𝑣 Δ𝜆𝑣 𝜑𝑣 ∣
𝑘
𝑛=2
≤
𝑚+1
∑
𝜃𝑛𝑘−1
𝑛=2
×
(𝑛−1
∑
(
𝑝𝑛
𝑃𝑛 𝑃𝑛−1
𝜃𝑣1−1/𝑘
∣𝜑𝑣 ∣
𝑣=1
𝑚
∑
(
)𝑘 𝑛−1
∑
∣𝑠𝑣 ∣
𝑘
𝑃𝑣𝑘
∣Δ𝜆𝑣 ∣ ∣𝜑𝑣 ∣ 𝜃𝑣(1−1/𝑘)(1−𝑘)
𝑣=1
𝑃𝑣
𝑝𝑣
)(1/𝑘)−1
∣Δ𝜆𝑣 ∣
(
𝑃𝑣
𝑝𝑣
)(𝑘−1)(1−1/𝑘)
)𝑘−1
)(𝑘−1)(1−1/𝑘) 𝑚+1
∑
)𝑘
𝑝𝑛
= 𝑂(1)
𝑃𝑛 𝑃𝑛−1
𝑣=1
𝑛=𝑣+1
( )(𝑘−1)(1−1/𝑘) 𝑚+1
( )𝑘
𝑚
∑
∑
𝑝𝑛
1
𝑃𝑣
= 𝑂(1)
𝑃𝑣 ∣Δ𝜆𝑣 ∣ ∣𝜑𝑣 ∣ 𝜃𝑣(1−1/𝑘)(1−𝑘)
𝜃𝑛𝑘−1
𝑝
𝑃
𝑃
𝑣
𝑛
𝑛−1
𝑣=1
𝑛=𝑣+1
(
)
(
)𝑘−1
𝑚
(𝑘−1)(1−1/𝑘)
∑
𝑃𝑣
𝜃𝑣 𝑝 𝑣
= 𝑂(1)
∣Δ𝜆𝑣 ∣ ∣𝜑𝑣 ∣ 𝜃𝑣(1−1/𝑘)(1−𝑘)
𝑝𝑣
𝑃𝑣
𝑣=1
(
)
𝑚
(1/𝑘)−1
∑
𝑃𝑣
= 𝑂(1)
𝜃𝑣1−1/𝑘 ∣𝜑𝑣 ∣
∣Δ𝜆𝑣 ∣ = 𝑂(1).
𝑝𝑣
𝑣=1
𝑚+1
∑
𝑘
𝜃𝑛𝑘−1 ∣𝑇𝑛3 ∣ =
𝑛=2
𝑚+1
∑
𝑃𝑣𝑘
∣Δ𝜆𝑣 ∣ ∣𝜑𝑣 ∣ 𝜃𝑣(1−1/𝑘)(1−𝑘)
𝜃𝑛𝑘−1 ∣𝑠𝑣 𝑃𝑣 𝜆𝑣+1 Δ𝜑𝑣 ∣
(
𝑃𝑣
𝑝𝑣
𝜃𝑛𝑘−1
(
𝑘
𝑛=2
≤
𝑚+1
∑
𝜃𝑛𝑘−1
𝑛=2
= 𝑂(1)
(
𝑝𝑛
𝑃𝑛
𝑚
∑
𝑃𝑣𝑘
𝑝𝑘−1
𝑣=1 𝑣
𝑚
∑
𝑃𝑣𝑘
= 𝑂(1)
𝑝𝑘−1
𝑣=1 𝑣
= 𝑂(1)
𝑚
∑
𝑣=1
(𝑛−1
)𝑘−1
∑ 𝑝𝑣
𝑃𝑣𝑘
𝑘
𝑘
∣𝑠𝑣 ∣ 𝑘−1 ∣𝜆𝑣+1 ∣ ∣Δ𝜑𝑣 ∣
𝑃𝑛−1 𝑣=1
𝑃
𝑝𝑣
𝑣=1 𝑛−1
(
)
𝑚+1
𝑘
∑
𝑝𝑛
1
∣𝜆𝑣+1 ∣𝑘 ∣Δ𝜑𝑣 ∣𝑘
𝜃𝑛𝑘−1
𝑃𝑛
𝑃𝑛−1
𝑛=𝑣+1
(
)𝑘−1
𝜃𝑣 𝑝 𝑣
1
𝑘
𝑘
∣𝜆𝑣 ∣ ∣Δ𝜑𝑣 ∣
𝑃𝑣
𝑃𝑣
)𝑘
1
𝑘
𝑛−1
∑
𝑘
𝑘
𝜃𝑣𝑘−1 ∣𝜆𝑣 ∣ ∣Δ𝜑𝑣 ∣ = 𝑂(1).
ON LOCAL PROPERTY OF FACTORED FOURIER SERIES
𝑚+1
∑
𝜃𝑛𝑘−1
𝑘
∣𝑇𝑛4 ∣ =
𝑛=2
=
𝑚+1
∑
𝑛=2
𝑚+1
∑
𝑛=2
𝜃𝑛𝑘−1
𝜃𝑛𝑘−1
𝑘
𝑝𝑛
𝑠𝑛 𝜆𝑛 𝜑𝑛 𝑃𝑛
(
𝑝𝑛
𝑃𝑛
)𝑘
𝑘
31
𝑘
∣𝜆𝑛 ∣ ∣𝜑𝑛 ∣ = 𝑂(1).
Since the behavior of the Fourier series concerns the convergence for a particular
value of 𝑥 depends on the behavior on the function in the immediate neighborhood
of this point only, this justifies (1.2) and valid. This completes the proof.
□
Remark. The result of [7] follows from theorem 2.1 by putting
𝜑𝑛 = 𝑃𝑛 /𝑛𝑝𝑛 ,
Δ𝜑𝑛 = 𝑂(1/𝑛).
Acknowledgments. The authors would like to thank the anonymous referee for
his/her comments that helped us improve this article.
References
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Acad. Sinica. 17 (1980) 165–170.
[3] H. Bor, On the local property of Fourier series. Bull. Math. Anal. Appl. 1 (2009), 15–21.
[4] H. C. Chow, On the summability factors of infinite series, J. London Math. Soc. 16 (1941)
215-220.
[5] K. Matsumoto, Local property of the summability ∣𝑅, 𝜆𝑛 , 1∣ , Thoku Math. J. 2 (8) (1956)
114-124.
[6] K. N. Mishra, Multipliers for 𝑁 , 𝑝𝑛 summability of Fourier series, Bull. Inst. Math. Acad.
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[7] M.A. Sarigöl, On the local property of the factored Fourier series, Bull. Math. Anal. Appl. 1
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waad Sulaiman, Department of Computer Engineering, College of engineering, University of Mosul, Iraq
E-mail address: waadsulaiman@hotmail.com