On L1 convergence of certain cosine sums
with twice quasi semi-convex coefficients
N. L. Braha
Abstract. In this paper a criterion for L1 - convergence of a certain cosine sums with twice quasi semi-convex coefficients is obtained. Also a
necessary and sufficient condition for L1 -convergence of the cosine series
is deduced as a corollary.
M.S.C. 2000: 42A20, 42A32.
Key words: cosine sums; L1 -convergence; twice quasi semi-convex null sequences;
twice quasi-convex null sequences.
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 the converse is not true. This motivated
several 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 a sense that
they converge in L1 -metric to the sum of the trigonometric series whereas the classical
series do not. In this contest we introduce the following cosine sum defined by relation
(1.1)
n X
n
X
1
a1 (cos x − 4)
a2
Nn(2) (x) = ¡
(∆4 aj−2 − ∆4 aj−1 ) cos kx + ¡
+¡
¢
¢
¢4 ,
x 4
x 4
2 sin 2 k=1 j=k
2 sin 2
2 sin x2
and will show the L1 -convergence of this modified cosine sums with twice quasi semiconvex coefficients. In the sequel we will briefly describe the notations and definitions
which are used throughout the paper.
In what follows we will denote by
∞
(1.2)
g(x) =
a0 X
ak cos kx,
+
2
k=1
with partial sums defined by
n
(1.3)
Sn (x) =
a0 X
ak cos kx,
+
2
k=1
Applied Sciences, Vol.12, 2010, pp.
30-36.
c Balkan Society of Geometers, Geometry Balkan Press 2010.
°
On L1 convergence of certain cosine sums
31
and
(1.4)
g(x) = lim Sn (x).
n→∞
In the sequel we will mention some results which are useful for the further work.
Dirichlet’s kernels are denoted by
¢
¡
n
sin n + 12 t
1 X
Dn (t) = +
cos kt =
2
2 sin 2t
k=1
e n (t) =
D
n
X
cos kt
k=1
¡
¢
cos 2t − cos n + 12 t
Dn (t) =
sin kt =
2 sin 2t
k=1
¡
¢
cos n + 12 t
1
t
Dn (t) = − cot + Dn (t) = −
2
2
2 sin 2t
n
X
Definition 1.1. A sequence of scalars (an ) is said to be semi-convex if an → 0 as
n → ∞, and
∞
X
(1.5)
n|∆2 an−1 + ∆2 an | < ∞, (a0 = 0),
n=1
where ∆2 an = ∆an − ∆an+1 .
Definition 1.2. A sequence of scalars (an ) is said to be quasi-convex if an → 0 as
n → ∞, and
∞
X
(1.6)
n|∆2 an−1 | < ∞, (a0 = 0),
n=1
Definition 1.3. A sequence of scalars (an ) is said to be twice quasi semi-convex if
an → 0 as n → ∞, and
(1.7)
∞
X
n|∆4 an−1 − ∆4 an | < ∞, (a0 = a−1 = 0),
n=1
where ∆4 an = ∆3 an − ∆3 an+1 .
Definition 1.4. A sequence of scalars (an ) is said to be twice quasi-convex if an → 0
as n → ∞, and
(1.8)
∞
X
n=1
n|∆4 an−1 | < ∞, (a0 = a−1 = 0),
32
N. L. Braha
Remark 1.1. If (an ) is a twice quasi-convex null scalar sequence , then it is twice
quasi semi-convex scalars sequence too.
The L1 -convergence of cosine and sine sums was studied by several authors. Kolmogorov in [6], proved the following theorem:
Theorem 1.2. If (an ) is a quasi-convex null sequence, then for the L1 -convergence
of the cosine series (1.2), it is necessary and sufficient that limn→∞ an · log n = 0.
The case in which sequence (an ) is convex, of this theorem was established by
Young (see [11]).
Bala and Ram in [1] have proved that Theorem 1.2 holds true for cosine series
with semi-convex null sequences in the following form:
Theorem 1.3. If (an ) is a semi-convex null sequence, then for the convergence of the
cosine series (1.2) in the metric space L, it is necessary and sufficient that ak−1 log k =
0(1), k → ∞.
Garret and Stanojevic in [4], have introduced modified cosine sums
n
(1.9)
gn (x) =
n
n
XX
1X
∆ak +
(∆aj ) cos kx.
2
k=0
k=1 j=k
The same authors (see [5]), Ram in [8] and Singh and Sharma in [9] studied the L1 convergence of this cosine sum under different sets of conditions on the coefficients
(an ). Kumari and Ram in [10], introduced new modified cosine and sine sums as
µ ¶
n
n
a0 X X
aj
(1.10)
fn (x) =
+
∆
cos kx
2
j
k=1 j=k
and
(1.11)
Gn (x) =
n X
n
X
k=1 j=k
µ
∆
aj
j
¶
sin kx,
and have studied their L1 -convergence under the condition that the coefficients (an )
belong to different classes of sequences. Later one, Kulwinder in [7], introduced new
modified sine sums as
n
(1.12)
Kn (x) =
n
1 XX
(∆aj−1 − ∆aj+1 ) sin kx,
2 sin x
k=1 j=k
and have studied their L1 -convergence under the condition that the coefficients (an )
are semi-convex null. In [3], was introduced the following modified cosine sums:
Nn (x) = − ¡
1
2 sin
n X
n
X
¢
x 2
2
a1
(∆2 aj−1 − ∆2 aj ) cos kx + ¡
¢2 .
2 sin x2
k=1 j=k
For this cosine sums was studied L1 -convergence under the condition that the coefficients (an ) are quasi semi-convex null.
On L1 convergence of certain cosine sums
2
33
Results
The aim of this paper is to study the L1 -convergence of modified cosine sums given
by relation (1.1), with twice quasi semi-convex coefficients and to give necessary and
sufficient condition for L1 -convergence of the cosine series defined by relation (1.2).
Theorem 2.1. Let (an ) be a twice quasi semi-convex null sequence, then Nn (x)
converges to g(x) in L1 norm.
Proof. We have
Sn (x) =
n
n
³
X
a0 X
1
x ´4
+
ak · cos kx = ¡
·
a
·
cos
kx
·
2
sin
¢
k
4
2
2
2 sin x2
k=1
k=1
n
X
1
=¡
·
ak [cos (k + 2)x − 4 cos (k + 1)x + 6 cos kx − 4 cos (k − 1)x + cos (k − 2)x]
¢
4
2 sin x2 k=1
n
X
1
a−1 cos x a0 cos 2x
=¡
·
(ak−2 − 4ak−1 + 6ak − 4ak+1 + ak+2 ) · cos kx− ¡
¢
¢4 − ¡
¢4 +
x 4
2 sin 2 k=1
2 sin x2
2 sin x2
4a0 cos x
4an cos (n + 1)x
4a1
an−1 cos (n + 1)x an cos (n + 2)x
+ ¡
+¡
−¡
¡
¢
¢
¢ −
¡
¢
¢4 +
x 4
x 4
x 4
x 4
2 sin 2
2 sin 2
2 sin 2
2 sin 2
2 sin x2
4an+1 cos nx
a1 cos x
a2
an+1 cos (n − 1)x an+2 cos nx
+ ¡
− ¡
¡
¢ +¡
¢ +¡
¢ −
¢4
¢4 ⇒
x 4
x 4
x 4
2 sin 2
2 sin 2
2 sin 2
2 sin x2
2 sin x2
Sn (x) = ¡
1
2 sin
¢ ·
x 4
2
n
X
a−1 cos x
a0 cos 2x
∆4 ak−2 cos kx − ¡
¢ −¡
¢4 +
x 4
2 sin 2
2 sin x2
k=1
4a0 cos x
4an cos (n + 1)x
4a1
an−1 cos (n + 1)x an cos (n + 2)x
+ ¡
+¡
−¡
¡
¢
¢
¢ −
¡
¢
¢4 +
x 4
x 4
x 4
x 4
2 sin 2
2 sin 2
2 sin 2
2 sin 2
2 sin x2
4an+1 cos nx
a1 cos x
a2
an+1 cos (n − 1)x an+2 cos nx
+ ¡
− ¡
¢ +¡
¢ +¡
¢ −
¡
¢4
¢4 .
x 4
x 4
x 4
2 sin 2
2 sin 2
2 sin 2
2 sin x2
2 sin x2
Applying Abel’s transformation, we have
n−1
4
X
e n (x)
1
∆4 an−1 ) · D
e k (x) − (∆ an−2 −
Sn (x) = ¡
(∆4 ak−2 − ∆4 ak−1 )D
·
+
¢
¡
¢
4
4
2 sin x2
2 sin x2
k=1
an−1 cos (n + 1)x an cos (n + 2)x 4an cos (n + 1)x
4a1
+ ¡
−
−¡
¡
¢
¢
¡
¢
¢4 +
x 4
x 4
x 4
2 sin 2
2 sin 2
2 sin 2
2 sin x2
a1 cos x
a2
an−1 cos (n + 1)x an+2 cos nx
4an+1 cos nx
+¡
+¡
−
− ¡
+ ¡
¢
¢
¢
¡
¢4
¢4 .
4
4
4
2 sin x2
2 sin x2
2 sin x2
2 sin x2
2 sin x2
e n (x) is uniformly bounded on every segment [², π − ²], for every ² > 0,
Since D
34
N. L. Braha
g(x) = lim Sn (x) = ¡
n→∞
1
2 sin
¢ ·
x 4
2
∞
X
4)
a2
e k (x)+ a1¡(cos x −
(∆4 ak−2 − ∆4 ak−1 )D
¢ +¡
¢4
x 4
2 sin 2
2 sin x2
k=1
Also
Nn(2) (x)
1
=¡
¢4
2 sin x2
n X
n
X
(∆4 aj−2 − ∆4 aj−1 ) cos kx +
k=1 j=k
a2
a1 (cos x − 4)
+¡
¡
¢
¢4
x 4
2 sin 2
2 sin x2
respectively
Nn(2) (x) = ¡
1
2 sin
n
X
¢
x 4
2
∆4 ak−2 cos kx−
k=1
e n (x) a1 (cos x − 4)
a2
∆4 an−1 · D
+ ¡
¡
¢
¢ +¡
¢4 .
x 4
x 4
2 sin 2
2 sin 2
2 sin x2
Now applying Abel’s transformation we get the following relation:
Nn(2) (x) = ¡
1
2 sin
n−1
X
¢
x 4
2
e k (x)−
(∆4 ak−2 − ∆4 ak−1 )D
k=1
e n (x) ∆4 an−1 · D
e n (x)
∆4 an−2 · D
− ¡
+
¡
¢
¢4
4
2 sin x2
2 sin x2
a1 (cos x − 4)
a2
+¡
¡
¢
¢4 .
x 4
2 sin 2
2 sin x2
From the above relations we will have:
(2)
g(x) − Nn (x) =
¡
1
2 sin
+
∞
X
¢
x 4
2
e k (x)+
(∆4 ak−2 − ∆4 ak−1 )D
k=n+1
e n (x) ∆4 an−1 · D
e n (x)
∆4 an−2 · D
+
,
¡
¡
¢
¢4
4
2 sin x2
2 sin x2
whence
Ã
g(x) −
(2)
Nn (x)
=
lim
1
¡
¢4
2 sin x2
m→∞
m
X
!
e
(∆ ak−2 − ∆ ak−1 )Dk (x) +
4
4
k=n+1
e n (x) ∆4 an−1 · D
e n (x)
∆ an−2 · D
+
.
¡
¢
¡
¢
x 4
x 4
2 sin 2
2 sin 2
4
+
Thus, we have
Z
π
0
|g(x) − Nn(2) (x)|dx → 0,
for n → ∞, and definition 1.3.
¤
(2)
Corollary 2.2. Let (an ) be a twice quasi-convex null sequence, then Nn (x) converges
to g(x) in L1 norm.
On L1 convergence of certain cosine sums
35
Proof. Proof of the corollary follows directly from Theorem 2.1 and Remark 1.1. ¤
Corollary 2.3. If (an ) is a twice quasi semi-convex null sequence of scalars, then
the necessary and sufficient condition for L1 -convergence of the cosine series (1.2) is
limn→∞ an log n = 0.
Proof. Let us start from this estimation:
||Sn (x)−g(x)||L1 ≤ ||Sn (x)−Nn(2) (x)||L1 +||Nn(2) (x)−g(x)||L1 ≤ ||Nn(2) (x)−g(x)||L1 +
(2.1)
°
° °
° °
°
° 2∆4 a
° °
°
°
e
an+2 cos nx °
°
° ° an cos (n + 1)x an+1 cos nx °
n−1 Dn (x) ° ° an cos (n + 2)x
+
−
+4
−
° ¡
°
°
°
°
¢
¡
¢
¡
¢
¡
¢
¡
¢
4
4
4 °
4
4 °
°
° °
°
°
2 sin x2
2 sin x2
2 sin x2
2 sin x2
2 sin x2
On the other hand
4
∆ an−1 =
∞
X
∞
X
k 4
(∆ ak − ∆ ak+1 ) =
(∆ ak − ∆4 ak+1 ) ≤
k
4
4
k=n−1
k=n−1
1
n−1
∞
X
k=n−1
Since
µ ¶
1
k(∆ ak − ∆ ak+1 ) = o
.
n
4
Z
π
¡
0
therefore
4
e n (x)
D
¢4 = O(n),
2 sin x2
Z
∆4 an−1 ·
0
π
e n (x)
D
¡
¢4 = o(1).
2 sin x2
For the rest of the expression (2.1) we have this estimation:
¯
¯
Z π ¯¯
Z π ¯¯
¯
cos nx ¯¯
¯ an cos (n + 2)x an+2 cos nx ¯
¯ cos (n + 2)x
− ¡
an ¯ ¡
¯ ¡
¢4
¢4 ¯dx ≤ C1
¢ −¡
¢2 ¯dx ≤
¯
¯ 2 sin x 2
0 ¯
0
2 sin x2
2 sin x2
2 sin x2 ¯
2
¯
Z π ¯
¯
¯
e n (x) − 1 ¯dx ∼ C1 · C2 (an log n).
C1 · C2
an ¯¯D
2¯
0
In similar way we can estimate this expressions:
¯
Z π ¯¯
¯
¯ an cos (n + 1)x an+1 cos nx ¯
−
¯ ¡
¢
¡
¢
4
4 ¯¯dx ∼ C3 (an log n),
0 ¯
2 sin x
2 sin x
2
2
where C1 , C2 and C3 are constants. From Theorem 2.1 it follows that
||Nn(2) (x) − g(x)|| = o(1), n → ∞.
Finally we get this estimation
Z
lim
n→∞
π
|g(x) − Sn (x)| = o(1),
0
36
N. L. Braha
if and only if
lim an log n = 0,
n→∞
which proves the corollary.
¤
Corollary 2.4. If (an ) is a twice quasi-convex null sequence of scalars, then the
necessary and sufficient condition for L1 -convergence of the cosine series (1.2) is
limn→∞ an log n = 0.
References
[1] R. Bala and B. Ram, Trigonometric series with semi-convex coefficients, Tamang
J. Math. 18, 1 (1987), 75-84.
[2] N.K. Bary, Trigonometric Series (in Russian), Fizmatgiz, Moscow 1961.
[3] N.L. Braha and Xh.Z. Krasniqi, On L1 -convergence of certain cosine sums, Bull.
Math. Anal. Appl. 1, 1 (2009), 55-61.
[4] J.W. Garrett and C.V. Stanojevic, On integrability and L1 -convergence of certain
cosine sums, Notices Amer. Math. Soc. 22 (1975), (A-166). Bull. Amer. Math.
Soc. Volume 82, Number 1 (1976), 129-130
[5] J.W. Garrett and C.V. Stanojevic, On L1 -convergence of certain cosine sums,
Proc. Amer. Math. Soc. 54 (1976), 101-105.
[6] A.N. Kolmogorov, Sur l’ordere de grandeur des coefficients de la series de
Fourier-Lebesque, Bull.Polon. Sci. Ser. Sci. Math. Astronom. Phys. (1923), 83-86.
[7] K. Kaur, On L1 -convergence of modified sine sums, Solstice: An Electronic Journal of Geography and Mathematics, Ann Arbor: Institute of Mathematical Geography 14, 1 (2003), 1-6.
[8] B. Ram, Convergence of certain cosine sums in the metric space L, Proc. Amer.
Math. Soc. 66 (1977), 258-260.
[9] N. Singh and K.M. Sharma, Convergence of certain cosine sums in the metric
space L, Proc. Amer. Math. Soc. 75 (1978), 117-120.
[10] K. Suresh and R. Babu, L1 -convergence modified cosine sum, Indian J. Pure
Appl. Math. 19, 11 (1988), 1101-1104.
[11] W.H. Young, On the Fourier series of bounded functions, Proc. London Math.
Soc. 12, 22 (1913), 41-70.
Author’s address:
L. Braha
Department of Mathematics and Computer Sciences,
Av. Mother Theresa 5, Prishtinë, 10000, Kosovo.
E-mail: nbraha@yahoo.com