Bulletin of Mathematical analysis and Applications
ISSN:1821-1291, URL: http://www.bmathaa.org
Volume 1, Issue 1, (2009) Pages 55-61
On L1-convergence of certain cosine sums
∗
N. L. Braha and Xh. Z. Krasniqi
Abstract
In this paper a criterion for L1 - convergence of a certain cosine sums with
quasi semi-convex coefficients is obtained. Also a necessary and sufficient
condition for L1 -convergence of the cosine series is deduced as a corollary.
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 introduce new modified cosine series
given by relation
Nn (x) = −
n X
n
X
1
2 sin
x 2
2
(∆2 aj−1 − ∆2 aj ) cos kx +
k=1 j=k
a1
2 sin x2
2 ,
and for this modified cosine series we will prove L1 -convergence, under conditions that coefficients (an ) are quasi semi-convex.
2
Preliminaries
In what follows we will denote by
∞
g(x) =
a0 X
+
ak cos kx,
2
k=1
∗ Mathematics Subject Classifications: 42A20, 42A32.
Key words: cosine sums, L1 -convergence, quasi semi-convex null sequences.
c
2009
Universiteti i Prishtines, Prishtine, Kosovë.
Submitted November, 2009. Published March, 2009.
55
(1)
56
On L1 -convergence of certain cosine sums
with partial sums defined by
n
Sn (x) =
a0 X
+
ak cos kx,
2
(2)
k=1
and
g(x) = lim Sn (x).
n→∞
(3)
In the sequel we will mention some notations which are useful for the further
work. First let us denote
n
sin n + 12 t
1 X
cos kt =
Dn (t) = +
2
2 sin 2t
k=1
and
e n (t) =
D
n
X
cos kt.
k=1
For all other notations see [11].
Definition 2.1 A sequence of scalars (an ) is said to be semi-convex if an → 0
as n → ∞, and
∞
X
n|∆2 an−1 + ∆2 an | < ∞, (a0 = 0),
(4)
n=1
where ∆2 an = ∆an − ∆an+1 .
Definition 2.2 A sequence of scalars (an ) is said to be quasi-convex if an → 0
as n → ∞, and
∞
X
n|∆2 an−1 | < ∞, (a0 = 0),
(5)
n=1
Definition 2.3 A sequence of scalars (an ) is said to be quasi semi-convex if
an → 0 as n → ∞, and
∞
X
n|∆2 an−1 − ∆2 an | < ∞, (a0 = 0),
(6)
n=1
where ∆2 an = ∆an − ∆an+1 .
Kolmogorov in [5], proved the following theorem:
Theorem 2.4 If (an ) is a quasi-convex null sequence, then for the L1 -convergence
of the cosine series (1), it is necessary and sufficient that limn→∞ an · log n = 0.
N. L. Braha and Xh. Z. Krasniqi
57
The case in which sequence (an ) is convex, of this theorem was established
by Young (see [10]). That is why, sometimes, this theorem is known as YoungKolmogorov Theorem.
Remark 2.5 If (an ) is a quasi-convex null scalar sequence, then it is quasi
semi-convex scalars sequence too.
Bala and Ram in [1] have proved that Theorem 2.4 holds true for cosine
series with semi-convex null sequences in the following form:
Theorem 2.6 If (an ) is a semi-convex null sequence, then for the convergence
of the cosine series (1) in the metric space L, it is necessary and sufficient that
ak−1 log k = 0(1), k → ∞.
Garret and Stanojevic in [3], have introduced modified cosine sums
n
gn (x) =
n
n
XX
1X
∆ak +
(∆aj ) cos kx.
2
k=0
(7)
k=1 j=k
The same authors (see [4]), 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 [7], introduced new modified cosine and
sine sums as
n
n
aj
a0 X X
∆
+
cos kx
(8)
fn (x) =
2
j
k=1 j=k
and
Gn (x) =
n X
n
X
k=1 j=k
∆
aj
j
sin kx,
(9)
and have studied their L1 -convergence under the condition that the coefficients
(an ) belong to different classes of sequences. Later one, Kulwinder in [6], introduced new modified sine sums as
n
Kn (x) =
n
1 XX
(∆aj−1 − ∆aj+1 ) sin kx,
2 sin x
(10)
k=1 j=k
and have studied their L1 -convergence under the condition that the coefficients
(an ) are semi-convex null.
3
Results
In this paper we introduce the following modified cosine sums
Nn (x) = −
1
2 sin
n X
n
X
(∆
x 2
k=1
j=k
2
2
aj−1 − ∆2 aj ) cos kx +
a1
2 sin x2
2 .
(11)
58
On L1 -convergence of certain cosine sums
The aim of this paper is to study the L1 -convergence of this modified cosine
sums with quasi semi-convex coefficients and to give necessary and sufficient
condition for L1 -convergence of the cosine series defined by relation (1).
Theorem 3.1 Let (an ) a the quasi semi-convex null sequence, then Nn (x) converges to g(x) in L1 norm.
Proof We have
n
n
X
a0 X
1
x 2
+
·
Sn (x) =
ak · cos kx =
ak · cos kx · 2 sin
2
2
2
2 sin x2
k=1
k=1
1
=−
·
x 2
2 sin
=−
1
2 sin
·
x 2
2
n
X
2
n
X
ak [· cos (k + 1)x − 2 cos kx + cos(k − 1)x]
k=1
(ak−1 − 2ak + ak+1 ) · cos kx −
k=1
a1
2 sin
Sn (x) = −
1
2 sin
x 2
2
·
n
X
x 2
2
−
an cos (n + 1)x
a0 cos x
+
2 +
x 2
2 sin 2
2 sin x2
an+1 cos nx
2 ⇒
2 sin x2
a0 cos x
an cos (n + 1)x
+
2 +
x 2
2 sin 2
2 sin x2
k=1
a1
an+1 cos nx
−
2 .
x 2
2 sin 2
2 sin x2
∆2 ak−1 cos kx −
Applying Abel’s transformation, we have
Sn (x) = −
−
1
2 sin
·
x 2
2
n−1
X
e k (x) +
(∆2 ak−1 − ∆2 ak )D
k=1
e n (x)
∆2 an−1 · D
2
2 sin x2
an cos (n + 1)x
a1
an+1 cos nx
a0 cos x
+
+
−
2 .
x 2
x 2
x 2
2 sin 2
2 sin 2
2 sin 2
2 sin x2
e n (x) is uniformly bounded on every segment [, π − ], for every > 0,
Since D
g(x) = lim Sn (x) = −
n→∞
1
2 sin
x 2
2
·
∞
X
e k (x) +
(∆2 ak−1 − ∆2 ak )D
k=1
a1
2 sin x2
Also
Nn (x) = −
1
2 sin
n X
n
X
(∆
x 2
k=1 j=k
2
2
aj−1 − ∆2 aj ) cos kx +
a1
2 sin x2
2
respectively
Nn (x) = −
n
X
1
2 sin
x 2
2
k=1
∆2 ak−1 cos kx +
e n (x)
∆2 an · D
a1
+
2 .
x 2
2 sin 2
2 sin x2
2
N. L. Braha and Xh. Z. Krasniqi
59
Now applying Abel’s transformation we get the following relation:
Nn (x) = −
n−1
X
1
2 sin
x 2
2
2
2
e
e
e k (x)+ ∆ an−1 · Dn (x) + ∆ an · Dn(x)
(∆2 ak−1 − ∆2 ak )D
2
2
2 sin x2
2 sin x2
k=1
+
a1
2 sin x2
2
From above relation we will have:
g(x)−Nn (x) = −
∞
X
1
2 sin
x 2
2
2
2
e
e
e k (x)− ∆ an−1 · Dn (x) − ∆ an · Dn(x) .
(∆2 ak−1 − ∆2 ak )D
2
2
2 sin x2
2 sin x2
k=n+1
Thus, we have
Z
π
|g(x) − Nn (x)|dx → 0,
0
for n → ∞, and definition 1.3.
Corollary 3.2 Let (an ) be a quasi-convex null sequence, then Nn (x) converges
to g(x) in L1 norm.
Proof Proof of the corollary follows directly from Theorem 3.1 and Remark
2.5.
Corollary 3.3 If (an ) is a quasi semi-convex null sequence of scalars, then the
necessary and sufficient condition for L1 -convergence of the cosine series (1) is
limn→∞ an log n = 0.
Proof Let us start from this estimation:
||Sn (x)−g(x)||L1 ≤ ||Sn (x)−Nn (x)||L1 +||Nn (x)−g(x)||L1 = ||Nn (x)−g(x)||L1 +
a cos (n + 1)x a
e n (x) ∆2 an · D
n
n+1 cos nx
−
−
x 2
x 2 x 2
2 sin 2
2 sin 2
2 sin 2
On the other hand
a cos (n + 1)x a
e n (x) ∆2 an · D
n
n+1 cos nx
−
−
=
x 2
x 2
x 2 2 sin 2
2 sin 2
2 sin 2
||Nn (x) − Sn (x)|| ≤ ||Nn (x) − g(x)|| + ||g(x) − Sn (x)||,
and
∆2 an =
∞
X
(∆2 ak − ∆2 ak+1 ) =
k=n
∞
X
k 2
(∆ ak − ∆2 ak+1 ) ≤
k
k=n
∞
1X 2
1
2
(∆ ak − ∆ ak+1 ) = o
.
n
n
k=n
(12)
60
On L1 -convergence of certain cosine sums
Since
π
Z
e n (x)
D
2 = O(n),
2 sin x2
0
therefore
∆2 an ·
Z
π
0
e n (x)
D
2 = o(1).
2 sin x2
For the rest of the expression (11) we have this estimation:
Z
Z π cos (n + 1)x
π
cos nx an cos (n + 1)x an+1 cos nx an −
2
2 ≤
−
=
x 2
2 sin x 2
0
0 2 sin x2
2 sin x2
2
sin
2
2
Z π e n (x) − 1 dx ∼ (an log n).
=
an D
2
0
From Theorem 3.1 it follows that
||Nn (x) − g(x)|| = o(1), n → ∞.
Finally we get this estimation
Z
lim
n→∞
π
|g(x) − Sn (x)| = o(1),
0
if and only if
lim an log n = 0,
n→∞
with which was proved corollary.
Corollary 3.4 If (an ) is a quasi-convex null sequence of scalars, then the necessary and sufficient condition for L1 -convergence of the cosine series (1) 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] Bary K. N., Trigonometric series, Moscow, (1961)(in Russian.)
[3] J. W. Garrett and C. V. Stanojevic, On integrability and L1 - convergence
of certain cosine sums, Notices, Amer. Math. Soc. 22(1975), A-166.
[4] J. W. Garrett and C. V. Stanojevic, On L1 - convergence of certain cosine
sums, Proc. Amer. Math. Soc. 54(1976), 101-105.
[5] 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) 8386.
N. L. Braha and Xh. Z. Krasniqi
61
[6] Kulwinder Kaur, On L1 - convergence of modified sine sums, An electronic
journal of Geography and Mathematics, Vol. 14, Issue 1, (2003), 1-6.
[7] Kumari Suresh and Ram Babu, L1 -convergence modified cosine sum, Indian J. Pure appl. Math. 19(11) (1988), 1101-1104.
[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] W.H.Young, On the Fourier series of bounded functions, Proc.London
Math. Soc. 12(2)(1913), 41-70.
[11] A. Zygmund, Trigonometric series, Vol. 1, Cambridge University Press,
Cambridge, 1959.
N. L. Braha
Department of Mathematics and Computer Sciences,
Avenue ”Mother Theresa ” 5, Prishtinë, 10000, Kosova
E-mail address: nbraha@yahoo.com
Xh. Z. Krasniqi
Department of Mathematics and Computer Sciences,
Avenue ”Mother Theresa ” 5, Prishtinë, 10000, Kosova
E-mail address:xheki00@hotmail.com