ON THE BEHAVIOR OF r−DERIVATIVE NEAR THE
ORIGIN OF SINE SERIES WITH CONVEX
COEFFICIENTS
Sine Series With Convex
Coefficients
Xh. Z. Krasniqi and N. L. Braha
vol. 8, iss. 1, art. 22, 2007
Xh. Z. KRASNIQI AND N. L. BRAHA
Department of Mathematics and Computer Sciences,
Avenue "Mother Theresa " 5, Prishtinë,
10000, Kosova-UNMIK
EMail: xheki00@hotmail.com and nbraha@yahoo.co
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21 December, 2006
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Communicated by:
H. Bor
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2000 AMS Sub. Class.:
42A15, 42A32.
Key words:
Sine series, Convex coefficients.
Abstract:
In this paper we will give the behavior of the r−derivative near origin of sine
series with convex coefficients.
Received:
04 August, 2006
Accepted:
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Contents
1
Introduction and Preliminaries
3
2
Results
5
Sine Series With Convex
Coefficients
Xh. Z. Krasniqi and N. L. Braha
vol. 8, iss. 1, art. 22, 2007
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1.
Introduction and Preliminaries
Let us denote by
(1.1)
∞
X
an sin nx,
n=1
the sine series of the function f (x) with coefficients an such that an ↓ 0 or an → 0
and ∆2 an = ∆an − ∆an+1 ≥ 0, ∆an = an − an+1 . It is a known fact that under
these conditions, series (1.1) converges uniformly in the interval δ ≤ x ≤ 2π − δ,
∀δ > 0 (see [2, p. 95]). In the following we will denote by g(x) the sum of the series
(1.1), i.e
(1.2)
g(x) =
∞
X
Sine Series With Convex
Coefficients
Xh. Z. Krasniqi and N. L. Braha
vol. 8, iss. 1, art. 22, 2007
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an sin nx.
n=1
Many authors have investigated the behaviors of the series (1.1), near the origin
with convex coefficients. Young in [9] gave the estimation for |g(x)| near the origin
from the upper side. Later Salem (see [4], [5]) proved the following estimation for
the behavior of the function g(x) near the origin
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g(x) ∼ mam ,
for
π
π
<x≤ ,
m+1
m
Hartman and Winter (see [3]), proved that
m = 1, 2, . . . .
∞
g(x) X
=
nan ,
lim
x→0 x
n=1
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holds for an ↓ 0. In this context Telyakovskii (see [7]) has proved the behavior near
the origin of the sine series with convex coefficients. He has compared his own
results with those of Shogunbenkov (see [6]) and Aljancic et al. (see [1]).
In the sequel we will mention some results which are useful for further work.
Dirichlet’s kernels are denoted by
n
sin n + 12 t
1 X
Dn (t) = +
cos kt =
,
2 k=1
2 sin 2t
e n (t) =
D
n
X
k=1
and
sin kt =
cos 2t − cos n +
2 sin 2t
1
2
t
Sine Series With Convex
Coefficients
Xh. Z. Krasniqi and N. L. Braha
vol. 8, iss. 1, art. 22, 2007
,
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1
t
1
e n (t) = − cos n + 2 t .
Dn (t) = − cot + D
2
2
2 sin 2t
P
P
Let En (t) = 12 + nk=1 eikt and E−n (t) = 21 + nk=1 e−ikt , then the following
holds:
Lemma 1.1 ([8]). Let r be a non-negative integer. Then for all 0 < x ≤ π and all
n ≥ 1 the following estimates hold
r
(r)
1. E−n (x) ≤ 4πn
;
|x|
e (r) 4πnr
2. D
n (x) ≤ |x| ;
(r) 4πnr
1
3. Dn (x) ≤ |x| + O |x|r+1 .
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2.
Results
Theorem 2.1. Let an be a sequence of scalars such that:
1. an ↓ 0;
P
r
2. ∞
n=1 n ∆an < ∞, for r = 0, 1, 2, . . . ,
π
m+1
<x≤
π
,
m
m = 1, 2, . . . the following estimate is valid
( m
)
m
3
X
X n
rπ
r
g (r) (x) =
nr an nx +
+O
an n r
+
+ n3 mr−3
+ o(m).
2
m
2
n=1
n=1
then for
Proof. Applying Abel’s transform we obtain
(2.1)
g(x) =
∞
X
e n (x) = Pn sin kx is Dirichlet’s conjugate kernel. Let us denote by
where D
k=1
g (r) (x) the r−th derivatives for the function g. Let
(2.2)
e n (r) (x),
∆an D
n=1
be the r-th derivatives of the series in the relation (2.1).
From the given conditions in the theorem and Lemma 1.1(2), series (2.2) converges uniformly in (0, π], so the following relation holds
(2.3)
(r)
g (x) =
∞
X
n=1
Xh. Z. Krasniqi and N. L. Braha
vol. 8, iss. 1, art. 22, 2007
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e n (x),
∆an D
n=1
∞
X
Sine Series With Convex
Coefficients
e n (r) (x).
∆an D
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From the last relation we have
m
∞
X
X
e n (r) (x) +
e n (r) (x) = I1 (x) + I2 (x).
(2.4)
g (r) (x) =
∆an D
∆an D
n=1
n=m+1
In the following we will estimate sums I1 (x) and I2 (x). Let us start with estimation
of the second sum. From the second condition in Lemma 1.1, the second condition
π
π
of the theorem and fact that m+1
<x≤ m
, we have
I2 (x) ≤ 4π ·
(2.5)
∞
X
∆an
n=m+1
∞
X
nr
≤ 8m
nr ∆an = o(m).
x
n=m+1
For the first sum we have the following estimation
m
m
h
i
X
X
(r)
(r)
(r)
e
e
e (r) (x),
e
I1 (x) =
∆an Dn (x) =
an Dn (x) − Dn−1 (x) − am+1 D
m
n=1
n=1
(r)
e 0 (x) = 0. Knowing that
where D
rπ −
= n sin nx +
,
2
taking into consideration Lemma 1.1 and the conditions in Theorem 2.1, we have
m
X
rπ r
I1 (x) =
n sin nx +
+ O(mr+1 am ).
2
n=1
e (r) (x)
D
n
(r)
e n−1
D
(x)
r
In the last relation we can use the known fact that sin x = x + O(x3 ) for x → 0. The
following relation then holds
" m
#
m
3
X
X
rπ
rπ
I1 (x) =
nr an nx +
+O
+ 8mr+1 am .
nr an nx +
2
2
n=1
n=1
Sine Series With Convex
Coefficients
Xh. Z. Krasniqi and N. L. Braha
vol. 8, iss. 1, art. 22, 2007
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Taking into consideration the fact that an is a monotone sequence we obtain
m
4 X 3
mam ≤ 3
n an ,
m n=1
from which it follows that
mr+1 am ≤ 4mr−3
m
X
Sine Series With Convex
Coefficients
n 3 an .
Xh. Z. Krasniqi and N. L. Braha
n=1
From the above relations we have the following estimation for I1 (x),
( m
)
m
3
X X
rπ
rπ
+O
an nr nx +
+ n3 mr−3
.
(2.6) I1 (x) =
nr an nx +
2
2
n=1
n=1
Now proof of Theorem 2.1 follows from (2.4), (2.5) and (2.6).
Remark 1. The above result is a generalization of that given in [7].
Corollary 2.2. Let an be sequence of scalars such that an ↓ 0. Then for
π
, m = 1, 2, . . . , the following relation holds
m
!
m
m
X
1 X 3
nan x + O
n an .
g(x) =
3
m
n=1
n=1
π
m+1
<x≤
Theorem 2.3. Let (an ) be a sequence of scalars such that the following conditions
hold:
1. an → 0 and ∆an ≥ 0
P
r+1 2
2. ∞
∆ an < ∞, for r = 0, 1, 2, . . . .
n=1 n
vol. 8, iss. 1, art. 22, 2007
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π
<x≤ m
, m = 1, 2, . . . the following estimate is valid
(
)
m−1
n
X
r
g (r) (x) ≤ M (r) mr+2 [am + ∆am ] +
nr+1
+
∆an + o(m) ,
m 2
n=1
Then for
π
m+1
where M (r) is a constant which depends only on r.
Proof. Applying Abel’s transform we obtain
∞
∞
n
∞
X
X
X
X
r
2
r
n ∆an =
∆ an
i ≤
nr+1 ∆2 an < ∞.
n=1
n=1
n=1
i=1
P∞
From the convergence of the series
obtain that
∞
X
n=1
Sine Series With Convex
Coefficients
Xh. Z. Krasniqi and N. L. Braha
vol. 8, iss. 1, art. 22, 2007
r
n ∆an and Condition 2 in Lemma 1.1 we
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e n(r) (x)
∆an D
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n=1
converges uniformly in (0, π], so the following relation is valid
∞
X
e (r) (x).
g (r) (x) =
∆an D
n
n=1
From the other side we have that
e n(r) (x)
D
1
x (r)
(r)
=
cot
+ Dn (x),
2
2
respectively,
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m−1
∞
X
am x (r) X
e n (r) (x) +
cot
+
∆an D
∆an Dn (r) (x)
2
2
n=1
n=m
x (r)
am cot
+ J1 (x) + J2 (x).
=
2
2
g (r) (x) =
(2.7)
JJ
For
π
m+1
<x≤
π
,
m
we will have the following estimation
(2.8)
x (r)
M
cot
≤ r+1 ≤ M (r)mr+2 .
2
x
On the other hand it is known that
n
n
X
rπ r
rπ e n(r) (x) =
≤ nr+1 nx +
≤ πnr+1
+
.
D
ir sin ix +
2
2
m
2
i=1
From last two relations we have the following estimation for J1 (x),
J1 (x) ≤ π
(2.9)
m−1
X
n
n
r
+
∆an .
m 2
r+1
n=1
J2 (x) =
=
n=m
∞
X
n=m
∆2 an
n
X
∆ an
vol. 8, iss. 1, art. 22, 2007
Contents
(r)
Di (x) − ∆am
m−1
X
( n
X
i=0
(r)
Di (x)
i=0
i=0
2
Xh. Z. Krasniqi and N. L. Braha
Title Page
In the following we will estimate the second sum J2 (x). Applying the Abel transform
we have
∞
X
Sine Series With Convex
Coefficients
(r)
Di (x) −
m−1
X
)
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(r)
Di (x) ,
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i=0
P
2
because ∞
n=m ∆ an = ∆am .
Taking into consideration Lemma 1.1, we have the following estimation
n n
n
X
X
X
1
ir
(r) +M
≤ M (r)mnr+1 .
Di (x) ≤ 4π
r+1
x
x
i=0
i=0
i=0
Close
In a similar way we can prove that
m−1
X
(r) Di (x) ≤ M (r)mr+2 .
i=0
Now the estimation of J2 (x) can be expressed in the following way
( ∞
)
X
(2.10)
|J2 (x)| ≤ M (r) m
nr+1 ∆2 an + mr+2 ∆am
Sine Series With Convex
Coefficients
Xh. Z. Krasniqi and N. L. Braha
n=m
r+2
vol. 8, iss. 1, art. 22, 2007
The proof of the theorem follows from relations (2.7), (2.8), (2.9) and (2.10).
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Remark 2. The above theorem is a generalization of the result obtained in [7], from
the upper side for the case m ≥ 11.
Contents
= M (r){m
∆am + o(m)}.
Corollary 2.4. Let an → 0 be a convex sequence of scalars. If
π
π
< x ≤ , m ≥ 11
m+1
m
then the following estimation holds
m−1
m−1
am
x
1 X 2
am
x
6 X 2
cot +
n ∆an ≤ g(x) ≤
cot +
n ∆an .
2
2 2m n=1
2
2 m n=1
Remark 3. Telyakovskii compared his own results with those given by Hartman,
Winter (see [3]), then with results given by Salem (see [4], [5]). Taking into consideration Corollary 2.2 and Corollary 2.4 for the case r = 0, we can compare our
results with the results mentioned above.
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References
[1] S. ALJANCIC, R. BOJANIC AND M. TOMIC, Sur le comportement asymtotique au voisinage de zero des series trigonometrique de sinus a coefficients
monotones, Publ. Inst. Math. Acad. Serie Sci., 10 (1956), 101–120.
[2] N.K. BARY, Trigonometric Series, Moscow, 1961 (in Russian).
[3] Ph. HARTMAN AND A. WINTER, On sine series with monotone coefficients,
J. London Math. Soc., 28 (1953), 102–104.
Sine Series With Convex
Coefficients
Xh. Z. Krasniqi and N. L. Braha
vol. 8, iss. 1, art. 22, 2007
[4] R. SALEM, Determination de l’order de grandeur a l’origine de certaines series
trigonometrique, C.R. Acad. Paris, 186 (1928), 1804–1806.
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[5] R. SALEM, Essais sur les series Trigonometriques, Paris, 1940.
[6] Sh.Sh. SHOGUNBENKOV, Certain estimates for sine series with convex coefficients (in Russian), Primenenie Funktzional’nogo analiza v teorii priblizhenii,
Tver’ 1993, 67–72.
[7] S.A. TELYAKOVSKI, On the behaivor near the origin of sine series with convex
coefficients, Pub. De L’inst. Math. Nouvelle serie, 58(72) (1995), 43–50.
[8] Z. TOMOVSKI, Some results on L1 -approximation of the r−th derivateve of
Fourier series, J. Inequal. Pure and Appl. Math., 3(1) (2002), Art. 10. [ONLINE:
http://jipam.vu.edu.au/article.php?sid=162].
[9] W.H. YOUNG, On the mode of oscillation of Fourier series and of its allied
series, Proc. London Math. Soc., 12 (1913), 433–452.
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