LE MATEMATICHE
Vol. LXV (2010) – Fasc. II, pp. 25–32
doi: 10.4418/2010.65.2.3
ON L1 -CONVERGENCE OF REES-STANOJEVIĆ’S SUMS
WITH COEFFICIENTS FROM THE CLASS K
XHEVAT Z. KRASNIQI
In this paper are considered the modified cosine sums introduced by
Rees and Stanojević [2] with coefficients from the class K. In addition,
is proved that the condition limn→∞ |an+1 | log n = 0 is a necessary and
sufficient condition for the L1 -convergence of the cosine series. Also, an
open problem about L1 -convergence for the r−th derivative of the cosine
series is presented.
1.
Introduction and Preliminaries
Let us consider the cosine trigonometric series
g(x) =
∞
a0
+ ∑ am cos mx
2 m=1
with its partial sums denoted by
Sn (x) =
n
a0
+ ∑ am cos mx
2 m=1
and let limn→∞ Sn (x) = g(x).
Entrato in redazione: 27 febbraio 2010
AMS 2010 Subject Classification: 42A20, 42A32.
Keywords: cosine sums, Conjugate Fejer’s kernel, L1 -convergence.
(1)
26
XHEVAT Z. KRASNIQI
A lot of authors investigated the L1 -convergence of the series (1) under different classes of coefficients. Was W. H. Young [9] who showed that condition
limm→∞ am log m = 0 is necessary and sufficient condition for cosine series with
so-called convex (42 am ≥ 0, where 4am = am − am+1 , 42 am = 4am − 4am+1 )
coefficients and A. N. Kolmogorov
[1]
this result to the cosine series
2
extended
4 am−1 < ∞ coefficients.
with quasi-convex ∑∞
m
m=1
Likewise, in various papers are introduced different modified cosine and
sine sums which we shall make for them a list as follows:
C. S. Rees and Č. V. Stanojević [2] have introduced the following modified
cosine sums
n
n
1 n
gn (x) = ∑ 4ak + ∑ ∑ (4a j ) cos mx,
(2)
2 m=0
m=1 j=m
until Kumari and Ram [6], introduced new modified cosine and sine sums as
n
n
aj
a0
m cos mx,
fn (x) = + ∑ ∑ 4
2 m=1 j=m
j
and
n
n
aj
hn (x) = ∑ ∑ 4
j
m=1 j=m
m sin mx.
Later on, N. Hooda, B. Ram and S. S. Bhatia [5] introduced new modified
cosine sums as
!
!
n
n
n
1
2
2
Rn (x) =
a1 + ∑ 4 am + ∑ am+1 + ∑ 4 a j cos mx.
2
j=m
m=0
m=1
Recently K. Kaur, S. S. Bhatia, and B. Ram [4] introduced new modified
sine sums as
Kn (x) =
n
n
1
∑ ∑ (4a j−1 − 4a j+1 ) sin mx,
2 sin x m=1
j=m
and a new class of sequences defined as follows:
A sequence {am } is said to belongs to the class K, if am → 0, m → ∞, and
∞
∑ m 42 am−1 − 42 am+1 < ∞,
(a0 = 0).
m=1
The class K can be generalized in the following manner: A sequence {am }
is said to belongs to the class Kr , (r = 0, 1, . . . ), if am → 0, as m → ∞, and
∞
∑ mr+1 42 am−1 − 42 am+1 < ∞,
m=1
(a0 = 0).
ON L1 -CONVERGENCE OF REES-STANOJEVIĆ’S SUMS WITH ...
27
For r = 0, clearly K ≡ K0 and Kr ⊂ K, r ≥ 1.
Further, wishing to generalize the class Kr we define a certain subclass of
r
K as follows:
Let α := {αm } be a positive monotone sequence tending to infinity. A sequence {am } belongs to the class K(α) if am → 0, as m → ∞, and
∞
∑ αm 42 am−1 − 42 am+1 < ∞,
(a0 = 0).
m=1
If we denote the class K(α), where αm := mr+1 , by Kr , that is, if we introduce the definition Kr := K(mr+1 ), we immediately get the generalization of
the classes Kr , for any positive integer r.
All of the authors, (see also [7], [8]), cited above studied the L1 -convergence
of these cosine and sine sums proving different results. Among others let us
formulate the following statements proved in [4]:
Theorem 1.1. Let the sequence {an } belong to the class K, then Kn (x) converges to g(x) in the L1 -norm.
Corollary 1.2. If {an } belongs to the class K, then the necessary and sufficient
condition for the L1 -convergence of the cosine series (1) is limn→∞ an log n = 0.
As usually with Dn (x) and D̃n (x) we shall denote the Dirichlet and its conjugate kernels defined by Dn (x) = a20 + ∑nm=1 cos mx, D̃n (x) = ∑nm=1 sin mx, respectively.
The object of this paper is to prove analogous statements as the Theorem 1.1
and the Corollary 1.2 when we use the modified cosine sums gn (x) instead of
new modified sine sums Kn (x). More precisely, we shall show that Theorem 1.1
and the Corollary 1.2 hold good for modified cosine sums (2) with coefficients
from the class K.
Throughout this paper the constants in the O-expression denote positive constants and they may be different in different relations.
2.
Main Results
We shall prove the following main result.
Theorem 2.1. Let the sequence {an } belong to the class K, then gn (x) converges
to g(x) in the L1 -norm.
28
XHEVAT Z. KRASNIQI
Proof. During the proof we shall use the same technique as in [4]. We have
gn (x) =
n
a0
+ ∑ am cos mx − an+1 Dn (x)
2 m=1
n
1
∑ 2am sin x cos mx − an+1 Dn (x) (a0 = 0)
2 sin x m=1
n
1
=
am sin(m + 1)x − sin(m − 1)x − an+1 Dn (x)
∑
2 sin x m=1
=
=
n
1
∑ (am−1 − am+1 ) sin mx
2 sin x m=1
(3)
sin nx
sin(n + 1)x
+ an+1
− an+1 Dn (x)
2 sin x
2 sin x
n
1
sin(n + 1)x
=
(am−1 − am+1 ) sin mx + (an − an+1 )
.
∑
2 sin x m=1
2 sin x
+ an
Applying the Abel’s transformation in (3) we get
n
1
∑ (4am−1 − 4am+1 ) D̃m (x)
2 sin x m=1
gn (x) =
+ (an − a2n )
sin(n + 1)x
D̃n (x)
+ (an − an+1 )
,
2 sin x
2 sin x
(4)
and passing on limit when n → ∞ we obtain
lim gn (x) =
n→∞
∞
1
(4am−1 − 4am+1 ) D̃m (x).
∑
2 sin x m=1
(5)
In a similiar fashion we can show that
n
1
(4am−1 − 4am+1 ) D̃m (x)
∑
2 sin x m=1
Sn (x) =
+ (an − a2n )
g(x) = lim Sn (x) =
n→∞
and the series
D̃n (x)
sin nx
sin(n + 1)x
+ an+1
+ an
,
2 sin x
2 sin x
2 sin x
∞
1
∑ (4am−1 − 4am+1 ) D̃m (x),
2 sin x m=1
∞
1
∑ (4am−1 − 4am+1 ) D̃m (x)
2 sin x m=1
(6)
ON L1 -CONVERGENCE OF REES-STANOJEVIĆ’S SUMS WITH ...
29
converges (see [4]).
Therefore limn→∞ gn (x) = g(x) exists, and from (5) and (6) the following
equality
lim gn (x) = lim Sn (x) = g(x)
n→∞
n→∞
holds. Hence
g(x) − gn (x) =
∞
1
(4am−1 − 4am+1 ) D̃m (x)
∑
2 sin x m=n+1
− (an − a2n )
D̃n (x)
sin(n + 1)x
− (an − an+1 )
.
2 sin x
2 sin x
1
Denoting with F̃m (x) = m+1
∑m
i=0 D̃i (x) the conjugate Fejer kernel, then the use
of Abel’s transformation gives
"
`−1
1
g(x) − gn (x) =
lim
(m + 1) 42 am−1 − 42 am+1 F̃m (x)
∑
2 sin x `→∞ m=n+1
#
+ (` + 1) (4a`−1 − 4a`+1 ) F̃` (x) − (n + 1) (4an − 4an+2 ) F̃n (x)
D̃n (x)
sin(n + 1)x
− (an − an+1 )
− (an − a2n )
2 sin x
" 2 sin x
∞
1
=
(m + 1) 42 am−1 − 42 am+1 F̃m (x)
∑
2 sin x m=n+1
#
D̃n (x)
− (n + 1) (4an − 4an+2 ) F̃n (x) − (an − a2n )
2 sin x
− (an − an+1 )
sin(n + 1)x
.
2 sin x
Thus
∞
kg − gn k = O
∑
2
Z
2
(m + 1) 4 am−1 − 4 am+1 π
−π
m=n+1
Z π F̃n (x) dx
+ (n + 1) |4an − 4an+2 |
−π
Z π D̃
(x)
n dx
+ |an − a2n |
−π 2 sin x
Z π sin(n + 1)x + |an − an+1 |
dx.
2 sin x −π !
F̃m (x) dx
30
XHEVAT Z. KRASNIQI
The first and fourth terms tend to zero as n → ∞ based on facts that
Z π −π
F̃m (x)dx = π
and {am } belongs the class K.
Further, for the second term, denoted by Λ(n), for large enough n we obtain
Λ(n) = O (n + 1) |4an − 4an+2 |
∞
2
2
= O (n + 1) ∑ 4 am − 4 am+2 m=n
!
∞ = O (n + 1) ∑ 42 am−1 − 42 am+1 m=n+1
∞
=O
∑
!
2
2
m4 am−1 − 4 am+1 = o(1).
m=n+1
R π D̃n (x) Since −π
2 sin x dx = O (n) then the third term tends to zero, as well. Indeed, we have
∞
Z π D̃n (x) |an − a2n |
dx = O n ∑ (4am − 4am+2 ) −π 2 sin x
m=n
!
∞ = O (n + 1) ∑ 42 am−1 − 42 am+1 m=n+1
!
2
m4 am−1 − 42 am+1 = o(1).
∞
=O
∑
m=n+1
This completes the proof of our theorem.
Corollary 2.2. If {an } belongs to the class K, then the necessary and sufficient
condition for the L1 -convergence of the cosine series (1) is limn→∞ |an+1 | log n =
0.
Proof. Sufficiency. We can write
kg − Sn k ≤ kg − gn k + kSn − gn k
sin
nx
sin(n
+
1)x
= kg − gn k + an+1 2 sin x + 2 sin x
= kg − gn k + |an+1 |
Z π
−π
|Dn (x)| dx.
ON L1 -CONVERGENCE OF REES-STANOJEVIĆ’S SUMS WITH ...
31
π
|D (x)| dx ∼ log n and our assumption
From the well-known relation −π
Rπ n
that an log n = o(1), we obtain |an+1 | −π |Dn (x)| dx = o(1) as n → ∞. Also,
according to the Theorem 2.1 kg − gn k = o(1), as n → ∞. This completes the
sufficient condition.
Necessity. The following holds
R
|an+1 | log n ∼ |an+1 |
Z π
−π
|Dn (x)| dx = kan+1 Dn (x)k
= kSn − gn k ≤ kSn − gk + kg − gn k = o(1)
by our assumption and the Theorem 2.1.
The Corollary 2.2 is proved completely.
We saw in Section 2 how we extended the class K to the class Kr , (r =
0, 1, . . . ). Since the condition limn→∞ |an+1 | log n = 0 is a special case of the condition limn→∞ nr |an+1 | log n = 0, (r = 0), then the following propositions may be
true:
(r)
Theorem 2.3. Let the sequence {an } belong to the class Kr , then gn (x) converges to g(r) (x) in the L1 -norm.
Corollary 2.4. If {an } belongs to the class Kr , then the necessary and sufficient
condition for the L1 -convergence of the r − th derivative of the cosine series (1)
is limn→∞ nr |an+1 | log n = 0.
The question is: Do these propositions are true or not? This is still an open
problem.
REFERENCES
[1] A. N. Kolmogorov, Sur l’ordre de grandeur des coefficients de la series de
Fourier-Lebesgue, Bull. Polon. Sci. Ser. Sci. Math. Astron. Phys. (1923), 73–76.
[2] C. S. Rees - Č. V. Stanojević, Necessary and sufficient condition for intergrability
of certain cosine sums, J. Math. Anal. Appl. 43 (1973), 579–586.
[3] K. Kaur - S. S. Bhatia, Integrability and L1 -convergence of Ress-Stanojević sums
with generalized semiconvex coefficients, IJMMS 30 (11) (2002), 645–650.
[4] K. Kaur - S. S. Bhatia - B. Ram, Integrability and L1 -convergence of modified sine
sums, Georgian Math. J. 11 (1) (2004), 99–104.
[5] N. Hooda - B. Ram - S. S. Bhatia, On L1 -convergence of a modified cosine sum,
Soochow J. of Math. 28 (3) (2002), 305–310.
32
XHEVAT Z. KRASNIQI
[6] S. Kumari - B. Ram, L1 -convergence of modified cosine sums, Indian J. Pure Appl.
Math. 19 (11) (1977), 1101–1104.
[7] X. Z. Krasniqi, A note on L1 -convergence of the sine and cosine trigonometric
series with semi-convex coefficients, Int. J. Open Probl. Comput. Sci. Math. 2 (2)
(2009), 231–239.
[8] N. L. Braha - X. Z. Krasniqi, On L1 convergence of certain cosine sums, Bull.
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[9] W. H. Young, On Fourier series of bounded function, Proc. London Math. Soc. 12
(2) (1913), 41–70.
XHEVAT Z. KRASNIQI
Department of Mathematics and Computer Sciences
University of Prishtina
Avenue ”Mother Theresa ” 5, Prishtinë 10000
Republic of Kosova
e-mail: [email protected]