Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 2 Issue 4(2010), Pages 45-53.
𝐿1 -CONVERGENCE OF THE 𝑟 − 𝑡ℎ DERIVATIVE OF CERTAIN
COSINE SERIES WITH 𝑟-QUASI CONVEX COEFFICIENTS
(DEDICATED IN OCCASION OF THE 70-YEARS OF
PROFESSOR HARI M. SRIVASTAVA)
NAIM L. BRAHA
Abstract. We study 𝐿1 -convergence of 𝑟 − 𝑡ℎ derivative of modified cosine
sums introduced in [3]. Exactly it is proved the 𝐿1 - convergence of 𝑟 − 𝑡ℎ
derivative of modified cosine sums with 𝑟-quasi convex coefficients.
1. Introduction
It is well known that if a trigonometric series converges in 𝐿1 -metric to a function
𝑓 ∈ 𝐿1 , then it is the Fourier series of the function f. Riesz [2] gave a counter
example showing that in a metric space 𝐿1 we cannot expect the converse of the
above said result to hold true. This motivated the various authors to study 𝐿1 convergence of the trigonometric series. During their investigations some authors
introduced modified trigonometric sums as these sums approximate their limits
better than the classical trigonometric series in the sense that they converge in 𝐿1 metric to the sum of the trigonometric series whereas the classical series itself may
not. In this contest we will show 𝐿1 -convergence of the 𝑟 − 𝑡ℎ derivative of certain
cosine series with 𝑟-quasi convex coefficients by this modified cosine series(given in
[3]) by relation
𝑛 ∑
𝑛
∑
1
(Δ5 𝑎𝑗−3 − Δ5 𝑎𝑗−2 ) cos 𝑘𝑥−
𝑁𝑛(3) (𝑥) = − (
)
6
2 sin 𝑥2 𝑘=1 𝑗=𝑘
𝑎1 (15 − 6 cos 𝑥 + cos 2𝑥) 𝑎2 (6 − cos 𝑥)
𝑎3
+ (
−(
(1.1)
(
)
)
)6
𝑥 6
𝑥 6
2 sin 2
2 sin 2
2 sin 𝑥2
In the sequel we will briefly describe the notation and definitions which are used
throughout the paper. Let
∞
𝑎0 ∑
+
𝑎𝑘 cos 𝑘𝑥
(1.2)
2
𝑘=1
2000 Mathematics Subject Classification. 42A20, 42A32.
Key words and phrases. cosine sums, 𝐿1 -convergence, quasi semi-convex null sequences, 𝑟quasi convex null sequences.
c
⃝2010
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted October 10, 2010. Published November 18, 2010.
45
46
N.L.BRAHA
be cosine trigonometric series with its partial sums denoted by
𝑛
𝑆𝑛 (𝑥) =
𝑎0 ∑
+
𝑎𝑘 cos 𝑘𝑥,
2
𝑘=1
and let 𝑔(𝑥) = lim𝑛→∞ 𝑆𝑛 (𝑥).
Definition 1.1. A sequence of scalars (𝑎𝑛 ) is said to be quasi semi-convex if 𝑎𝑛 → 0
as 𝑛 → ∞, and
∞
∑
𝑛∣Δ2 𝑎𝑛−1 − Δ2 𝑎𝑛 ∣ < ∞, (𝑎0 = 0, 𝑎−1 = 0, 𝑎−2 = 0),
(1.3)
𝑛=1
where Δ2 𝑎𝑛 = Δ𝑎𝑛 − Δ𝑎𝑛+1 , Δ𝑎𝑛 = 𝑎𝑛 − 𝑎𝑛+1 .
Definition 1.2. A sequence of scalars (𝑎𝑛 ) is said to be 𝑟-quasi convex if 𝑎𝑛 → 0
as 𝑛 → ∞, and
∞
∑
𝑛𝑟+1 ∣Δ𝑎𝑛−1 − Δ𝑎𝑛 ∣ < ∞, (𝑎0 = 0, 𝑎−1 = 0, 𝑎−2 = 0, 𝑟 ≥ 0).
(1.4)
𝑛=1
˜ 𝑛 (𝑥) we will denote the Dirichlet and its conjugate
As usually with 𝐷𝑛 (𝑥) and 𝐷
kernels defined by
𝑛
𝐷𝑛 (𝑥) =
1 ∑
+
cos 𝑘𝑥,
2
˜ 𝑛 (𝑥) =
𝐷
𝑘=1
𝑛
∑
sin 𝑘𝑥.
𝑘=1
In [3] is studied the 𝐿1 -convergence of modified cosine sums given by relation
(1) with third semi-convex coefficients proving the following theorem.
(3)
Theorem 1.3. Let (𝑎𝑛 ) be a third semi-convex null sequence, then 𝑁𝑛 (𝑥) converges to 𝑔(𝑥) in 𝐿1 -norm.
The main goal of the present work is to study the 𝐿1 -convergence of 𝑟 − 𝑡ℎ
derivative of these new modified cosine sums with 𝑟-quasi convex null coefficients
and to deduce the sufficient condition of Theorem 1.3 as corollaries. We point out
here that a lot of authors investigated the 𝐿1 -convergence of the series (2), see for
example [1],[8],[4],[5],[6],[7].
Everywhere in this paper the constants in the 𝑂-expression denote positive constants and they may be different in different relations. All other notations are like
as in [9], [2].
2. preliminaries
To prove the main results we need the following lemmas:
Lemma 2.1. If 𝑥 ∈ [𝜖, 𝜋], 𝜖 > 0 and 𝑚 ∈ ℕ, then the following estimate holds
¯Ã
)(𝑟) ¯¯
¯
˜ 𝑚 (𝑥)
¯ 𝐷
¯
(
)
¯
¯ = 𝑂𝑟,𝜖 𝑚𝑟+1 , (𝑟 = 0, 1, 2 . . . )
¯ 2 sin6 𝑥
¯
¯
¯
2
where 𝑂𝑟,𝜖 depends only on 𝑟 and 𝜖.
𝐿1 -CONVERGENCE OF THE 𝑟 − 𝑡ℎ DERIVATIVE OF CERTAIN ...
Proof. By Leibniz formula we have
Ã
)(𝑟)
Ã
𝑟 ( )
∑
˜ 𝑚 (𝑥)
𝐷
𝑟
1
=
6 𝑥
𝑖
2 sin 2
2 sin6
𝑖=0
Ã
𝑟 ( )
∑
𝑟
1
=
6
𝑖
2
sin
𝑖=0
)(𝑟−𝑖)
𝑥
2
)(𝑟−𝑖)
(
˜ 𝑚 (𝑥)
𝐷
47
)(𝑖)
(
)
𝑖𝜋
𝑖
𝑗
sin
𝑗𝑥
+
𝑥
2
2
𝑗=1
)(𝑟−𝑖)
Ã
𝑟 ( )
∑
1
𝑟
𝑟+1
= 𝑂(1)𝑚
.
(2.1)
𝑖
2 sin6 𝑥2
𝑖=0
(
)(𝑟)
𝑃𝑟 (cos 𝑥
2)
Using mathematical induction we will prove the equality 2 sin16 𝑥
= sin
𝑟+6 𝑥 ,
2
2
where 𝑃𝑟 is a cosine polynomial of degree 𝑟.
(
)′
𝑥
3⋅sin5 𝑥
𝑃 (cos 𝑥 )
2 cos 2
Namely, we have 2 sin16 𝑥 = − 12 sin12
= 1sin7 𝑥2 , so that for 𝑟 = 1 the
𝑥
2
2
2
above equality is true.
(
)(𝑟)
𝑃𝑟 (cos 𝑥
2)
Assume that the equality 𝐹 (𝑥) := 2 sin16 𝑥
= sin
holds. For the (𝑟 +
𝑟+6 𝑥
1) − 𝑡ℎ derivative of
1
2 sin6
𝑚
∑
2
𝑥
2
2
we get
′
𝐹 (𝑥) :=
=
=
(
)
′
𝑃𝑟 (cos 𝑥2 ) − 12 sin𝑟+7 𝑥2 − 𝑃𝑟 (cos 𝑥2 )(𝑟 + 6) sin𝑟+5
− 12 sin𝑟+5
sin𝑟+5
𝑥
2
⋅ sin𝑟+7
1 −𝑃𝑟 (cos 𝑥2 ) 12 (1 − cos 𝑥) − 𝑃𝑟 (cos
2
sin𝑟+7 𝑥2
′
=
=
=
⋅
1
2
⋅ cos 𝑥2
sin2𝑟+12 𝑥2
[ ′
]
2 𝑥
𝑥
𝑥
𝑥
𝑥
𝑃
(cos
)
sin
+
𝑃
(cos
)(𝑟
+
6)
cos
𝑟
𝑟
2
2
2
2
2
′
=
𝑥
2
𝑥
2
𝑥
2 )(𝑟
+ 6) cos 𝑥2
′
1 − 12 𝑃𝑟 (cos 𝑥2 ) + cos 𝑥 ⋅ 12 𝑃𝑟 (cos 𝑥2 ) − 𝑃𝑟 (cos 𝑥2 )(𝑟 + 6) cos 𝑥2
2
sin𝑟+7 𝑥2
1 (−1/2)𝐻𝑟−1 (cos 𝑥2 ) + (1/2)𝐻𝑟−1 (cos 𝑥2 ) cos 𝑥 − (𝑟 + 6)𝑃𝑟 (cos 𝑥2 ) cos 𝑥2
2
sin𝑟+7 𝑥2
𝑇𝑟+1 (cos 𝑥2 )
1 𝑄𝑟+1 (cos 𝑥2 ) − (𝑟 + 6)𝑅𝑟+1 (cos 𝑥2 )
=
,
(2.2)
2
sin𝑟+7 𝑥2
sin𝑟+7 𝑥2
where 𝐻𝑟−1 , 𝑄𝑟+1 , 𝑅𝑟+1 , 𝑇𝑟+1 are cosine polynomials of degree 𝑟 − 1 and 𝑟 + 1
respectively. Therefore for 𝑥 ∈ [𝜖, 𝜋], 𝜖 > 0, from (5) and (6) we obtain
¯Ã
)(𝑟) ¯¯
¯
¯
𝑟 ( ) ¯¯
𝑥 ¯
∑
˜ 𝑚 (𝑥)
¯ 𝐷
¯
)
(
)
𝑃
(cos
𝑟
𝑟−𝑖
𝑟+1
2
¯
¯ = 𝑂(1)𝑚
= 𝑂𝑟,𝜖 𝑚𝑟+1 .
¯ 2 sin6 𝑥
¯
𝑟−𝑖+6
𝑥
𝑖
sin
¯
¯
2
2
𝑖=0
□
Lemma 2.2. If 𝑥 ∈ [𝜖, 𝜋], 𝜖 > 0 and 𝑚 ∈ ℕ, then the following estimate holds
¯Ã
)(𝑟) ¯¯
¯
¯
¯ 𝐷𝑚 (𝑥)
(
)
¯ = 𝑂𝑟,𝜖 𝑚𝑟+1 , (𝑟 = 0, 1, 2 . . . ),
¯
¯
¯ 2 sin6 𝑥
¯
¯
2
48
N.L.BRAHA
where 𝑂𝑟,𝜖 depends only on 𝑟 and 𝜖.
Proof is similar to the Lemma 2.1.
3. Results
In what follows we prove the main result of the paper:
(
)(𝑟)
(3)
Theorem 3.1. Let (𝑎𝑛 ) be a 𝑟-quasi convex null sequence, then 𝑁𝑛 (𝑥)
converges to 𝑔 (𝑟) (𝑥) in 𝐿1 -norm.
Proof. From definition of 𝑆𝑛 (𝑥) we have:
𝑆𝑛 (𝑥) = − (
1
2 sin
𝑛
∑
)
𝑥 6
2
𝑎−2 cos 𝑥
Δ6 𝑎𝑘−3 cos 𝑘𝑥 + (
)6 +
2 sin 𝑥2
𝑘=1
𝑎0
𝑎𝑛−2 cos (𝑛 + 1)𝑥 𝑎𝑛−1 cos (𝑛 + 2)𝑥 𝑎𝑛 cos (𝑛 + 3)𝑥
𝑎−1 cos 2𝑥
−
− (
(
) +(
) −
(
)6
(
)6
)6 −
𝑥 6
𝑥 6
2 sin 2
2 sin 2
2 sin 𝑥2
2 sin 𝑥2
2 sin 𝑥2
6𝑎−1 cos 𝑥 6𝑎0 cos 2𝑥 6𝑎𝑛−1 cos (𝑛 + 1)𝑥 6𝑎𝑛 cos (𝑛 + 2)𝑥
15𝑎0
+
+(
(
) −(
) +
(
)
(
)
)6 −
𝑥 6
𝑥 6
𝑥 6
𝑥 6
2 sin 2
2 sin 2
2 sin 2
2 sin 2
2 sin 𝑥2
15𝑎𝑛 cos (𝑛 + 1)𝑥
15𝑎1
15𝑎𝑛+1 cos 𝑛𝑥
6𝑎1 cos 𝑥
6𝑎2
−(
+(
(
)
) + (
)
) +(
)6 −
𝑥 6
𝑥 6
𝑥 6
𝑥 6
2 sin 2
2 sin 2
2 sin 2
2 sin 2
2 sin 𝑥2
6𝑎𝑛+1 cos (𝑛 − 1)𝑥 6𝑎𝑛+2 cos 𝑛𝑥
𝑎1 cos 2𝑥
𝑎2 cos 𝑥
𝑎3
− (
(
)
) −(
) −(
) −(
)6 +
𝑥 6
𝑥 6
𝑥 6
𝑥 6
2 sin 2
2 sin 2
2 sin 2
2 sin 2
2 sin 𝑥2
𝑎𝑛+1 cos (𝑛 − 2)𝑥 𝑎𝑛+2 cos (𝑛 − 1)𝑥 𝑎𝑛+3 cos 𝑛𝑥
+
+ (
(
)6
(
)6
)6
2 sin 𝑥2
2 sin 𝑥2
2 sin 𝑥2
Applying Abel’s transformation, we obtain
𝑛−1
5
∑
˜ 𝑛 (𝑥)
1
Δ5 𝑎𝑛−2 ) ⋅ 𝐷
˜ 𝑘 (𝑥)+ (Δ 𝑎𝑛−3 −
𝑆𝑛 (𝑥) = − (
⋅
(Δ5 𝑎𝑘−3 − Δ5 𝑎𝑘−2 )𝐷
−
)
(
)
6
4
2 sin 𝑥2 𝑘=1
2 sin 𝑥2
𝑎𝑛−2 cos (𝑛 + 1)𝑥 𝑎𝑛−1 cos (𝑛 + 2)𝑥 𝑎𝑛 cos (𝑛 + 3)𝑥 6𝑎𝑛−1 cos (𝑛 + 1)𝑥
−
− (
+
+
(
)6
(
)6
)6
(
)6
2 sin 𝑥2
2 sin 𝑥2
2 sin 𝑥2
2 sin 𝑥2
15𝑎1
15𝑎𝑛+1 cos 𝑛𝑥 6𝑎1 cos 𝑥
6𝑎2
6𝑎𝑛 cos (𝑛 + 2)𝑥 15𝑎𝑛 cos (𝑛 + 1)𝑥
−
−(
(
)
(
)
) + (
) +(
) +(
)6 −
𝑥 6
𝑥 6
𝑥 6
𝑥 6
𝑥 6
2 sin 2
2 sin 2
2 sin 2
2 sin 2
2 sin 2
2 sin 𝑥2
6𝑎𝑛+1 cos (𝑛 − 1)𝑥 6𝑎𝑛+2 cos 𝑛𝑥
𝑎1 cos 2𝑥
𝑎2 cos 𝑥
𝑎3
− (
(
)
) −(
) −(
) −(
)6 +
𝑥 6
𝑥 6
𝑥 6
𝑥 6
2 sin 2
2 sin 2
2 sin 2
2 sin 2
2 sin 𝑥2
𝑎𝑛+1 cos (𝑛 − 2)𝑥 𝑎𝑛+2 cos (𝑛 − 1)𝑥 𝑎𝑛+3 cos 𝑛𝑥
+
+ (
(
)6
(
)6
)6
2 sin 𝑥2
2 sin 𝑥2
2 sin 𝑥2
Therefore
𝑆𝑛(𝑟) (𝑥) = − (
1
2 sin 𝑥4
)6 ⋅
𝑛−1
∑
𝑘=1
Ã
(Δ5 𝑎𝑘−3 −
˜ (𝑟) (𝑥)
Δ5 𝑎𝑘−2 )𝐷
𝑘
+ (Δ5 𝑎𝑛−3 − Δ5 𝑎𝑛−2 ) ⋅
˜ 𝑛 (𝑥)
𝐷
2 sin6 𝑥2
)(𝑟)
𝐿1 -CONVERGENCE OF THE 𝑟 − 𝑡ℎ DERIVATIVE OF CERTAIN ...
49
Ã
)(𝑟)
Ã
)(𝑟)
Ã
)(𝑟)
Ã
)(𝑟)
cos (𝑛 + 1)𝑥
cos (𝑛 + 2)𝑥
cos (𝑛 + 3)𝑥
cos (𝑛 + 1)𝑥
−𝑎𝑛−2
−𝑎𝑛−1
−𝑎𝑛
+6𝑎𝑛−1
+
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
Ã
)(𝑟)
Ã
)(𝑟)
Ã
)(𝑟)
cos (𝑛 + 1)𝑥
1
cos (𝑛 + 2)𝑥
− 15𝑎𝑛
− 15𝑎1
+
6𝑎𝑛
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
Ã
)(𝑟)
Ã
)(𝑟)
Ã
)(𝑟)
Ã
)(𝑟)
cos 𝑛𝑥
cos 𝑥
1
cos (𝑛 − 1)𝑥
+15𝑎𝑛+1
+6𝑎1
+6𝑎2
−6𝑎𝑛+1
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
)(𝑟)
Ã
)(𝑟)
Ã
)(𝑟)
Ã
)(𝑟)
Ã
cos 2𝑥
cos 𝑥
1
cos 𝑛𝑥
− 𝑎1
− 𝑎2
− 𝑎3
+
−6𝑎𝑛+2
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
Ã
)(𝑟)
Ã
)(𝑟)
Ã
)(𝑟)
cos (𝑛 − 2)𝑥
cos (𝑛 − 1)𝑥
cos 𝑛𝑥
𝑎𝑛+1
+ 𝑎𝑛+2
+ 𝑎𝑛+3
. (3.1)
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
On the other hand we have:
𝑁𝑛(3) (𝑥) = − (
1
2 sin
𝑛 ∑
𝑛
∑
(Δ5 𝑎𝑗−3
)
𝑥 6
𝑘=1 𝑗=𝑘
2
− Δ5 𝑎𝑗−2 ) cos 𝑘𝑥−
𝑎1 (15 − 6 cos 𝑥 + cos 2𝑥) 𝑎2 (6 − cos 𝑥)
𝑎3
+ (
−(
(
)
)
)6 ,
𝑥 6
𝑥 6
2 sin 2
2 sin 2
2 sin 𝑥2
respectively
𝑛
∑
˜ 𝑛 (𝑥)
1
Δ5 𝑎𝑛−2 𝐷
5
𝑁𝑛(3) (𝑥) = − (
Δ
𝑎
cos
𝑘𝑥
+
)
(
)6 −
𝑘−3
6
2 sin 𝑥2 𝑘=1
2 sin 𝑥2
𝑎1 (15 − 6 cos 𝑥 + cos 2𝑥) 𝑎2 (6 − cos 𝑥)
𝑎3
+ (
−(
(
)
)
)6 .
𝑥 6
𝑥 6
2 sin 2
2 sin 2
2 sin 𝑥2
Now applying Abel’s transformation we get the following relation:
𝑛−1
5
5
∑
˜
˜
1
5
5
˜ 𝑘 (𝑥)+ Δ 𝑎(𝑛−3 ⋅ 𝐷)𝑛 (𝑥) + Δ 𝑎(𝑛−2 ⋅ 𝐷)𝑛 (𝑥) −
𝑁𝑛(3) (𝑥) = − (
(Δ
𝑎
−
Δ
𝑎
)
𝐷
)
𝑘−3
𝑘−2
𝑥 6
𝑥 6
𝑥 6
2 sin 2 𝑘=1
2 sin 2
2 sin 2
𝑎1 (15 − 6 cos 𝑥 + cos 2𝑥) 𝑎2 (6 − cos 𝑥)
𝑎3
+ (
−(
(
)
)
)6 ,
𝑥 6
𝑥 6
2 sin 2
2 sin 2
2 sin 𝑥2
(3.2)
and the 𝑟-derivative of the relation (8) is the following relation:
(
𝑁𝑛(3) (𝑥)
)(𝑟)
1
𝑛−1
∑
Ã
˜ (𝑟) (𝑥)+Δ5 𝑎𝑛−3
Δ5 𝑎𝑘−2 )𝐷
𝑘
˜ 𝑛 (𝑥)
𝐷
2 sin6 𝑥2
)(𝑟)
(Δ5 𝑎𝑘−3 −
+
)6
2 sin 𝑥2 𝑘=1
Ã
)(𝑟)
Ã
)(𝑟)
Ã
)(𝑟)
Ã
)(𝑟)
˜ 𝑛 (𝑥)
𝐷
15
−
6
cos
𝑥
+
cos
2𝑥
6
−
cos
𝑥
1
Δ5 𝑎𝑛−2
−𝑎1
+𝑎2
−𝑎3
.
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
(3.3)
By Lemma 2.1 and since (𝑎𝑛 ) are 𝑟-quasi-convex null sequences, we obtain:
= −(
50
N.L.BRAHA
¯
¯)
ï
Ã
)(𝑟) ¯¯
¯
∞
¯
∑
˜
¯( 5
¯
)¯)
( 5
)¯¯
(¯ 𝑟+1 ( 5
) 𝐷𝑛 (𝑥)
¯
𝑟+1
5
¯ = 𝑂𝑟,𝜖 ¯𝑛
Δ 𝑎𝑛−2 ¯ = 𝑂𝑟,𝜖 ¯𝑛
Δ 𝑎𝑘−2 − Δ 𝑎𝑘−1 ¯
2 ¯¯ Δ 𝑎𝑛−2
¯
¯
¯
2 sin6 𝑥2
¯
¯
𝑘=𝑛
̰
)
̰
)
∑
∑
𝑟+1
5
5
𝑟+1
= 𝑂𝑟,𝜖
𝑘 ∣Δ 𝑎𝑘−2 − Δ 𝑎𝑘−1 ∣ = 𝑂𝑟,𝜖
𝑘 ∣Δ𝑎𝑘−2 − Δ𝑎𝑘−1 ∣ +
4𝑂𝑟,𝜖
Ã
+4𝑂𝑟,𝜖
𝑘=𝑛
̰
∑
)
𝑘
𝑟+1
∣Δ𝑎𝑘−1 − Δ𝑎𝑘 ∣
𝑘=𝑛
∞
∑
𝑘
Ã
+ 6𝑂𝑟,𝜖
)
𝑟+1
Ã
∣Δ𝑎𝑘+1 − Δ𝑎𝑘+2 ∣ +𝑂𝑟,𝜖
𝑘=𝑛
𝑘=𝑛
∞
∑
𝑘
)
𝑟+1
∣Δ𝑎𝑘 − Δ𝑎𝑘+1 ∣ +
𝑘=𝑛
∞
∑
𝑘
)
𝑟+1
∣Δ𝑎𝑘+2 − Δ𝑎𝑘+3 ∣
= 𝑜(1), 𝑛 → ∞.
𝑘=𝑛
(3.4)
After some calculations and by virtue of the Lemma 2.1 we obtain
)(𝑟)
Ã
)(𝑟)
Ã
cos (𝑛 + 3)(𝑥)
cos 𝑛𝑥
− 𝑎𝑛+3 ⋅
=
𝑎𝑛 ⋅
2 sin6 𝑥2
2 sin6 𝑥2
⎡Ã
⎡Ã
)(𝑟) Ã
)(𝑟) ⎤
)(𝑟) Ã
)(𝑟) ⎤
𝐷
(𝑥)
𝐷
(𝑥)
𝐷
(𝑥)
𝐷
(𝑥)
(𝑛−1)
𝑛
𝑛+3
𝑛+2
⎦−𝑎𝑛+3 ⎣
⎦
= 𝑎𝑛 ⎣
−
−
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
2 sin6 𝑥2
(
)
(
)
= 𝑎𝑛 𝑂𝑟,𝜖 (𝑛 + 3)𝑟+1 − (𝑛 + 2)𝑟+1 − 𝑎𝑛+3 𝑂𝑟,𝜖 𝑛𝑟+1 − (𝑛 − 1)𝑟+1 =
Ã
𝑟
(𝑎𝑛 −𝑎𝑛+3 )𝑂𝑟,𝜖 (𝑛 ) = 𝑂𝑟,𝜖
𝑟
𝑛
∞
∑
(
2
2
2
)
2
(Δ 𝑎𝑘 − Δ 𝑎𝑘+1 ) − (Δ 𝑎𝑘+1 − Δ 𝑎𝑘+2 ) + 3
𝑘=𝑛
Ã
= 𝑂𝑟,𝜖
Ã
𝑂𝑟,𝜖
∞
∑
𝑘=𝑛
)
𝑘 𝑟+1 ∣Δ𝑎𝑘 − Δ𝑎𝑘+1 ∣
𝑘=𝑛
∞
∑
𝑘
Ã
+ 2𝑂𝑟,𝜖
∣Δ𝑎𝑘+2 − Δ𝑎𝑘+3 ∣
∞
∑
)
𝑘 𝑟+1 ∣Δ𝑎𝑘+1 − Δ𝑎𝑘+2 ∣ +
𝑘=𝑛
)
𝑟+1
Ã
+ 3𝑂𝑟,𝜖
𝑘=𝑛
∞
∑
)
𝑘
𝑟+1
∣Δ𝑎𝑘+1 − Δ𝑎𝑘+2 ∣
𝑘=𝑛
= 𝑜(1) + 2𝑜(1) + 𝑜(1) + 3𝑜(1) = 𝑜(1), 𝑛 → ∞.
In the same way we can conclude that two other expressions
)(𝑟)
Ã
)(𝑟)
Ã
cos (𝑛 − 1)𝑥
cos (𝑛 + 2)(𝑥)
− 𝑎𝑛+2 ⋅
,
𝑎𝑛−1 ⋅
2 sin6 𝑥2
2 sin6 𝑥2
Ã
𝑎𝑛−2 ⋅
cos (𝑛 + 1)(𝑥)
2 sin6 𝑥2
)(𝑟)
Ã
− 𝑎𝑛+1 ⋅
cos (𝑛 − 2)𝑥
2 sin6 𝑥2
(3.5)
)(𝑟)
are 𝑜(1), where 𝑛 → ∞. In what follows we will estimate this expressions:
Ã
6𝑎𝑛−1 ⋅
∞
∑
cos (𝑛 + 1)(𝑥)
2 sin6 𝑥2
)(𝑟)
Ã
− 6𝑎𝑛+1 ⋅
cos (𝑛 − 1)𝑥
2 sin6 𝑥2
)(𝑟)
,
)
(Δ𝑎𝑘+1 − Δ𝑎𝑘+2 )
𝐿1 -CONVERGENCE OF THE 𝑟 − 𝑡ℎ DERIVATIVE OF CERTAIN ...
Ã
6𝑎𝑛 ⋅
cos (𝑛 + 2)(𝑥)
2 sin6 𝑥2
)(𝑟)
Ã
− 6𝑎𝑛+2 ⋅
cos 𝑛𝑥
2 sin6 𝑥2
51
)(𝑟)
and
Ã
15𝑎𝑛 ⋅
cos (𝑛 + 1)(𝑥)
2 sin6 𝑥2
)(𝑟)
Ã
− 15𝑎𝑛+1 ⋅
cos 𝑛𝑥
2 sin6 𝑥2
)(𝑟)
.
Let us start from the first expression
Ã
)(𝑟)
Ã
)(𝑟)
cos (𝑛 + 1)(𝑥)
cos (𝑛 − 1)𝑥
6𝑎𝑛−1 ⋅
− 6𝑎𝑛+1 ⋅
=
2 sin6 𝑥2
2 sin6 𝑥2
Ã
𝑟
= 6(𝑎𝑛−1 − 𝑎𝑛+1 )𝑂𝑟,𝜖 (𝑛 ) = 6𝑂𝑟,𝜖
Ã
= 6𝑂𝑟,𝜖
𝑟
𝑛
∞
∑
)
((Δ𝑎𝑘−1 + Δ𝑎𝑘 ) − (Δ𝑎𝑘 + Δ𝑎𝑘+1 ))
𝑘=𝑛
∞
∑
)
𝑘 𝑟+1 ∣Δ𝑎𝑘−1 − Δ𝑎𝑘 ∣
Ã
+ 6𝑂𝑟,𝜖
𝑘=𝑛
∞
∑
)
𝑘 𝑟+1 ∣Δ𝑎𝑘 − Δ𝑎𝑘+1 ∣
𝑘=𝑛
= 𝑜(1) + 𝑜(1) = 𝑜(1), 𝑛 → ∞.
(3.6)
In the same way we can conclude that
Ã
)(𝑟)
)(𝑟)
Ã
cos (𝑛 + 2)(𝑥)
cos 𝑛𝑥
6𝑎𝑛 ⋅
− 6𝑎𝑛+2 ⋅
= 𝑜(1), 𝑛 → ∞
2 sin6 𝑥2
2 sin6 𝑥2
And now we are able to estimate the last expression
Ã
)(𝑟)
Ã
)(𝑟)
cos (𝑛 + 1)(𝑥)
cos 𝑛𝑥
15𝑎𝑛 ⋅
− 15𝑎𝑛+1 ⋅
=
2 sin6 𝑥2
2 sin6 𝑥2
⎡Ã
)(𝑟) Ã
)(𝑟) ⎤
𝐷
(𝑥)
𝐷
(𝑥)
𝑛
𝑛+1
⎦ − 15
−
= 15𝑎𝑛 ⎣
2 sin6 𝑥2
2 sin6 𝑥2
⎡Ã
)(𝑟) Ã
)(𝑟) ⎤
𝐷
(𝑥)
𝐷
(𝑥)
𝑛
𝑛−1
⎦
𝑎𝑛+1 ⎣
−
2 sin6 𝑥2
2 sin6 𝑥2
(
)
(
)
= 15𝑎𝑛 ⋅ 𝑂𝑟,𝜖 (𝑛 + 1)𝑟+1 − 𝑛𝑟+1 − 15𝑎𝑛+1 ⋅ 𝑂𝑟,𝜖 𝑛𝑟+1 − (𝑛 − 1)𝑟+1
Ã
)
∞
∑
𝑟
𝑟
= 15𝑂𝑟,𝜖 (𝑛 (𝑎𝑛 − 𝑎𝑛+1 )) = 15𝑂𝑟,𝜖 𝑛
(Δ𝑎𝑘 − Δ𝑎𝑘+1 )
Ã
= 15𝑂𝑟,𝜖
∞
∑
)
𝑘 𝑟 ∣Δ𝑎𝑘 − Δ𝑎𝑘+1 ∣
𝑘=𝑛
= 𝑜(1), 𝑛 → ∞,
𝑘=𝑛
Therefore
𝑔 (𝑟) (𝑥) = lim
𝑛→∞
(
)(𝑟)
𝑁𝑛(3) (𝑥)
= lim 𝑆𝑛(𝑟) (𝑥) =
𝑛→∞
(3.7)
52
N.L.BRAHA
∞
∑
1
˜ (𝑟) (𝑥)
= −(
(Δ5 𝑎𝑘−3 − Δ5 𝑎𝑘−2 )𝐷
)
𝑘
6
𝑥
2 sin 2 𝑘=1
Ã
)(𝑟)
Ã
)(𝑟)
Ã
15 − 6 cos 𝑥 + cos 2𝑥
6 − cos 𝑥
1
− 𝑎1
+ 𝑎2
− 𝑎3
6 𝑥
6 𝑥
2 sin 2
2 sin 2
2 sin6
)(𝑟)
𝑥
2
. (3.8)
Using Lemma 2.1, relations (10), (11), (12), (13) and (14) we get the following
estimation:
¯
(
)(𝑟) ¯¯
¯ (𝑟)
¯ 𝑑𝑥
¯𝑔 (𝑥) − 𝑁𝑛(3) (𝑥)
¯
¯
−𝜋
¯Ã
)(𝑟) ¯
∫ 𝜋∑
∞
¯ 𝐷
¯
˜ 𝑘 (𝑥)
¯ 5
¯
¯
¯
¯Δ 𝑎𝑘−3 − Δ5 𝑎𝑘−2 ¯ ¯
¯𝑑𝑥
6 𝑥
¯
¯
2 sin 2
0 𝑘=𝑛
̰
)
∑
¯
¯
𝑂𝑟,𝜖
𝑘 𝑟+1 ¯Δ5 𝑎𝑘−3 − Δ5 𝑎𝑘−2 ¯ = 𝑜(1), 𝑛 → ∞,
∫
=
=
𝜋
𝑘=𝑛
which proves the theorem.
□
Corollary 3.2. Let (𝑎𝑛 ) be a 𝑟-quasi convex null sequence, then the necessary and
sufficient condition for 𝐿1 -convergence of the 𝑟 − 𝑡ℎ derivative of the series (2) is
𝑛𝑟+1 ∣𝑎𝑛 ∣ = 𝑜(1), as 𝑛 → ∞.
Proof. We have
°
(
)(𝑟) °
° (𝑟)
°
°𝑔 (𝑥) − 𝑁𝑛(3) (𝑥)
° ≤
°
°
°
° °
(
)(𝑟) °
°
° (𝑟)
° ° (𝑟)
(3)
°
𝑆
(𝑥)
−
𝑁
(𝑥)
°𝑔 (𝑥) − 𝑆𝑛(𝑟) (𝑥)° + °
𝑛
° 𝑛
°
°
°
)(𝑟) °
)(𝑟)
Ã
)(𝑟) °
Ã
Ã
°
° °
°
˜ 𝑛 (𝑥)
°( 5
° °
°
) 𝐷
cos
𝑛𝑥
cos
(𝑛
+
3)(𝑥)
°
°
°
°
= 𝑜(1)+2 ° Δ 𝑎𝑛−2
− 𝑎𝑛+3 ⋅
+°𝑎𝑛 ⋅
6 𝑥
6 𝑥
6 𝑥
°
°
2 sin 2
2 sin 2
2 sin 2
°
° °
°
°
)(𝑟)
)(𝑟) °
Ã
Ã
°
°
°
°
cos
(𝑛
−
1)𝑥
cos
(𝑛
+
2)(𝑥)
°
°+
+ °𝑎𝑛−1 ⋅
− 𝑎𝑛+2 ⋅
6 𝑥
6 𝑥
°
2 sin 2
2 sin 2
°
°
°
°
)(𝑟)
Ã
)(𝑟) °
Ã
°
°
°
cos
(𝑛
−
2)𝑥
cos
(𝑛
+
1)(𝑥)
°+
°𝑎𝑛−2 ⋅
− 𝑎𝑛+1 ⋅
6 𝑥
6 𝑥
°
°
2 sin 2
2 sin 2
°
°
°
Ã
)(𝑟)
Ã
)(𝑟) °
°
°
°
°
cos
(𝑛
+
1)(𝑥)
cos
(𝑛
−
1)𝑥
°6𝑎𝑛−1 ⋅
°+
− 6𝑎𝑛+1 ⋅
6 𝑥
6 𝑥
°
°
2 sin 2
2 sin 2
°
°
°
°
Ã
)(𝑟)
)(𝑟) °
Ã
°
°
°
cos 𝑛𝑥
°6𝑎𝑛 ⋅ cos (𝑛 + 2)(𝑥)
°
− 6𝑎𝑛+2 ⋅
6 𝑥
6 𝑥
°
°
2 sin 2
2 sin 2
°
°
°
)(𝑟)
Ã
)(𝑟) °
Ã
°
°
°
°
cos
𝑛𝑥
cos
(𝑛
+
1)(𝑥)
°
°
− 15𝑎𝑛+1 ⋅
+ °15𝑎𝑛 ⋅
6 𝑥
6 𝑥
°
2 sin 2
2 sin 2
°
°
𝐿1 -CONVERGENCE OF THE 𝑟 − 𝑡ℎ DERIVATIVE OF CERTAIN ...
53
In similar way like as in proof of the Theorem 3.1 (relations (11), (12) and (13))we
get that all the above summands are 𝑜(1), respectively
°
(
)(𝑟) °
° (𝑟)
°
°𝑔 (𝑥) − 𝑁𝑛(3) (𝑥)
° = 𝑜(1),
°
°
where 𝑛 → ∞.
□
From Theorem 3.1 and Corollary 3.2 we deduce the following corollaries (𝑟 = 0):
Corollary 3.3. If (𝑎𝑛 ) is 0-quasi convex null sequence of scalars, then the necessary
and sufficient condition for 𝐿1 -convergence of the cosine series (2) is lim𝑛→∞ 𝑛 ⋅ ∣𝑎𝑛 ∣ =
0.
References
[1] Bala R. and Ram B., Trigonometric series with semi-convex coefficients, Tamkang J. Math.
18 (1) (1987) 75-84.
[2] Bary K. N., Trigonometric series, Moscow, (1961)(in Russian.)
[3] Braha, N., On 𝐿1 convergence of certain cosine sums with third semi-convex coefficients
(submitted for publications).
(2)
[4] Braha, N., 𝐿1 -convergence of 𝑁𝑛 (𝑥) cosine sums with quasi hyper convex coefficients,Int.
J. Math. Anal. 3 (2009), no. 17-20, 863-870.
[5] Braha, N. L. On 𝐿1 convergence of certain cosine sums with twice quasi semi-convex coefficients. Appl. Sci. 12 (2010), 30-36.
[6] Braha, Naim L. On 𝐿1 -convergence of certain cosine sums with third semi-convex coefficients.
Int. J. Open Probl. Comput. Sci. Math. 2 (2009), no. 4, 562-571.
[7] Braha, N.L.; Krasniqi, Xh.Z. On 𝐿1 convergence of certain cosine sums. Bull. Math. Anal.
Appl. 1(2009), No. 1, 55-61.
[8] Kaur K., On 𝐿1 -convergence of a modified sine sums, An electronic J. of Geogr. and Math.
Vol. 14, Issue 1, (2003).
[9] A. Zygmund, Trigonometric series, 2nd ed., Cambridge University Press, 1959.
N.L.Braha
Department of Mathematics and Computer Sciences, Avenue ”Mother Theresa ” 5, Prishtinë, 10000, Kosova.
E-mail address: nbraha@yahoo.com