ON THE BEHAVIOR OF WITH CONVEX COEFFICIENTS Communicated by H. Bor

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Volume 8 (2007), Issue 1, Article 22, 6 pp.
ON THE BEHAVIOR OF r−DERIVATIVE NEAR THE ORIGIN OF SINE SERIES
WITH CONVEX COEFFICIENTS
XH. Z. KRASNIQI AND N. L. BRAHA
D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCES ,
AVENUE "M OTHER T HERESA " 5, P RISHTINË ,
10000, KOSOVA -UNMIK
[email protected]
[email protected]
Received 04 August, 2006; accepted 21 December, 2006
Communicated by H. Bor
A BSTRACT. In this paper we will give the behavior of the r−derivative near origin of sine series
with convex coefficients.
Key words and phrases: Sine series, Convex coefficients.
2000 Mathematics Subject Classification. 42A15, 42A32.
1. I NTRODUCTION AND P RELIMINARIES
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
∞
X
(1.2)
g(x) =
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
g(x) ∼ mam ,
for
π
π
< x ≤ , m = 1, 2, . . . .
m+1
m
212-06
2
X H . Z. K RASNIQI AND N. L. B RAHA
Hartman and Winter (see [3]), proved that
∞
g(x) X
lim
=
nan ,
x→0 x
n=1
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
cos kt =
Dn (t) = +
,
2 k=1
2 sin 2t
n
t
1
X
cos
−
cos
n
+
t
2
2
e n (t) =
D
sin kt =
,
t
2
sin
2
k=1
and
Let En (t) =
1
2
1
cos
n
+
t
1
t
2
e n (t) = −
.
Dn (t) = − cot + D
t
2
2
2 sin 2
P
P
+ nk=1 eikt and E−n (t) = 12 + 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
(r) 1
+
O
.
(3) Dn (x) ≤ 4πn
|x|
|x|r+1
2. R ESULTS
Theorem 2.1. Let an be a sequence of scalars such that:
(1) a
n ↓ 0;
P
r
(2) ∞
n=1 n ∆an < ∞, for r = 0, 1, 2, . . . ,
π
π
then for m+1
<x≤ 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
Proof. Applying Abel’s transform we obtain
g(x) =
(2.1)
∞
X
e n (x),
∆an D
n=1
Pn
(r)
e n (x) =
where D
k=1 sin kx is Dirichlet’s conjugate kernel. Let us denote by g (x) the r−th
derivatives for the function g. Let
∞
X
e n (r) (x),
(2.2)
∆an D
n=1
be the r-th derivatives of the series in the relation (2.1).
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 22, 6 pp.
http://jipam.vu.edu.au/
S INE S ERIES W ITH C ONVEX C OEFFICIENTS
3
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
(r)
g (x) =
(2.3)
∞
X
(r)
e n (x).
∆an D
n=1
From the last relation we have
m
∞
X
X
(r)
(r)
e
e n (r) (x) = I1 (x) + I2 (x).
(2.4)
g (x) =
∆an Dn (x) +
∆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 of the theorem and
π
π
fact that m+1
<x≤ m
, we have
(2.5)
∞
X
∞
X
nr
I2 (x) ≤ 4π ·
∆an ≤ 8m
nr ∆an = o(m).
x
n=m+1
n=m+1
For the first sum we have the following estimation
m
m
h
i
X
X
(r)
(r)
(r)
e
e n(r) (x) − D
e n−1
em
I1 (x) =
∆an Dn (x) =
an D
(x) − am+1 D
(x),
n=1
n=1
e 0(r) (x) = 0. Knowing that
where D
rπ (r)
(r)
r
e
e
Dn (x) − Dn−1 (x) = n sin nx +
,
2
taking into consideration Lemma 1.1 and the conditions in Theorem 2.1, we have
m
X
rπ I1 (x) =
nr sin nx +
+ O(mr+1 am ).
2
n=1
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
X
X
3
rπ
rπ
I1 (x) =
nr an nx +
+O
nr an nx +
+ 8mr+1 am .
2
2
n=1
n=1
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
m
r+1
am ≤ 4m
r−3
m
X
n 3 an .
n=1
From the above relations we have the following estimation for I1 (x),
( m
)
m
3
X
X rπ
rπ
(2.6)
I1 (x) =
nr an nx +
+O
an nr nx +
+ n3 mr−3
.
2
2
n=1
n=1
Now proof of Theorem 2.1 follows from (2.4), (2.5) and (2.6).
Remark 2.2. The above result is a generalization of that given in [7].
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 22, 6 pp.
http://jipam.vu.edu.au/
4
X H . Z. K RASNIQI AND N. L. B RAHA
Corollary 2.3. Let an be sequence of scalars such that an ↓ 0. Then for
m = 1, 2, . . . , the following relation holds
!
m
m
X
1 X 3
g(x) =
nan x + O
n an .
3
m
n=1
n=1
π
m+1
< x ≤
π
,
m
Theorem 2.4. Let (an ) be a sequence of scalars such that the following conditions hold:
(1) a
n → 0 and ∆an ≥ 0
P
r+1 2
(2) ∞
∆ an < ∞, for r = 0, 1, 2, . . . .
n=1 n
π
π
Then for m+1 < x ≤ m
, m = 1, 2, . . . the following estimate is valid
)
(
m−1
n
X
r
+
∆an + o(m) ,
g (r) (x) ≤ M (r) mr+2 [am + ∆am ] +
nr+1
m
2
n=1
where M (r) is a constant which depends only on r.
Proof. Applying Abel’s transform we obtain
∞
∞
n
∞
X
X
X
X
nr ∆an =
∆2 an
ir ≤
nr+1 ∆2 an < ∞.
n=1
From the convergence of the
n=1
i=1
P∞ r
series n=1 n ∆an
∞
X
n=1
and Condition 2 in Lemma 1.1 we obtain that
e n(r) (x)
∆an D
n=1
converges uniformly in (0, π], so the following relation is valid
∞
X
(r)
e n(r) (x).
g (x) =
∆an D
n=1
From the other side we have that
(r)
(r)
e n(r) (x) = 1 cot x
D
+ Dn (x),
2
2
respectively,
m−1
∞
X
am x (r) X
(r)
e
g (x) =
cot
+
∆an Dn (x) +
∆an Dn (r) (x)
2
2
n=1
n=m
(r)
am
x
(2.7)
=
cot
+ J1 (x) + J2 (x).
2
2
π
π
For m+1
<x≤ m
, we will have the following estimation
x (r)
M
(2.8)
cot
≤ r+1 ≤ M (r)mr+2 .
2
x
On the other hand it is known that
n
n
X
rπ rπ r
r
(r)
r+1
r+1
e
Dn (x) =
i sin ix +
≤n
nx +
≤ πn
+
.
2
2
m 2
i=1
(r)
From last two relations we have the following estimation for J1 (x),
m−1
n
X
r
(2.9)
J1 (x) ≤ π
nr+1
+
∆an .
m 2
n=1
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 22, 6 pp.
http://jipam.vu.edu.au/
S INE S ERIES W ITH C ONVEX C OEFFICIENTS
5
In the following we will estimate the second sum J2 (x). Applying the Abel transform we have
J2 (x) =
=
∞
X
2
∆ an
n
X
(r)
Di (x) − ∆am
n=m
i=0
∞
X
( n
X
∆2 an
n=m
m−1
X
(r)
Di (x)
i=0
(r)
Di (x) −
i=0
m−1
X
)
(r)
Di (x) ,
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
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
= M (r){m
n=m
r+2
∆am + o(m)}.
The proof of the theorem follows from relations (2.7), (2.8), (2.9) and (2.10).
Remark 2.5. The above theorem is a generalization of the result obtained in [7], from the upper
side for the case m ≥ 11.
Corollary 2.6. 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 2.7. 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 2.3 and
Corollary 2.6 for the case r = 0, we can compare our results with the results mentioned above.
R EFERENCES
[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.
[4] R. SALEM, Determination de l’order de grandeur a l’origine de certaines series trigonometrique,
C.R. Acad. Paris, 186 (1928), 1804–1806.
[5] R. SALEM, Essais sur les series Trigonometriques, Paris, 1940.
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 22, 6 pp.
http://jipam.vu.edu.au/
6
X H . Z. K RASNIQI AND N. L. B RAHA
[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.
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 22, 6 pp.
http://jipam.vu.edu.au/
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