Download MD2521102111

Nawneet Hooda / International Journal of Engineering Research and Applications (IJERA)
ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.2110-2111
Necessary and sufficient condition for the L1-convergence of a
modified trigonometric sum
Nawneet Hooda
Department of Mathematics, DCR University of science and Technology, Murthal-131039 (India)
Abstract
We obtain a necessary and sufficient
condition for L1-convergence of a modified cosine
sum which generalizes a result of Kumari and Ram
[4]. We also deduce a result of Garrett and
Stanojević [1] as a corollary of our result.
Keywords and phrases : L1-Convergence, Class
S, Class C .
2000 AMS Mathematics Subject Classification :
42A 20, 42A 32.
1. Introduction
Definition 1.1. A null sequence {ak} is said to be
of bounded variation BV if  |ak| <  , where
ak = akak+1.
Definition 1.2. A null sequence {ak} belongs to the
class S[5] if there exists

a monotone sequence {Ak} such that  Ak <  ,
k=1
and |ak|  Ak, for all k .
Definition 1.3. A null sequence {ak} belongs to
class C[1] if for every
 > 0, there exists  > 0 ,
independent of n, and such that


Cn() =   ak Dk(x) dx <  , for all n ,
0
k=n
where Dk(x) denotes Dirichlet’s kernel.
Let us consider the cosine series

(1.1)
f(x) = (a0/2) +  ak cos kx ,
k=1
and let Sn(x) denote the partial sum of (1.1) and
limn Sn(x) = f(x).
Theorem B. Let {ak}  BV  C.
Then
||Sn(x)  f(x)|| = o(1), if and only if an log n = o(1),
n .
Later on, Garrett, Rees and Stanojević [2] proved
that S  C  BV.
Kumari and Ram [4] introduced a new
modified cosine sum
(1.2)
n
n
k=1
j=k
fn(x) = (a0/2) +   (aj/j) k cos kx,
and proved the following result :
Theorem C.
Let {ak}  S and anlog n = o(1),
n, then
||f(x)  fn(x)|| = o(1) , n .
In this paper, we obtain a necessary and
sufficient condition for the L1-convergence of the
limit of (1.2) which generalizes Theorem C and
deduce Theorem B as a corollary from our result.
2. Lemma
We require the following lemma for the
proof of our theorem :
Lemma 1[3]. Let Dn(x), Dn(x) and Fn(x) denote
Dirichlet, conjugate Dirichlet and Fejér kernels
respectively, then
Fn(x) = Dn(x)  [1/(n+1)]Dn(x) .
3. Result
We shall prove
Theorem 1.
Let {ak}  BV  C, then
||f(x)  fn(x)|| = o(1), if and only if an log n = o(1),
n .
Proof. Using Lemma 1 and summation by parts, we
get
n
n
fn(x) = (a0/2) +   (aj/j) k cos kx
Regarding L1-convergence of (1.1), Teljakovskii [6]
proved the following result :
(3.1)
Theorem A. Let {ak}  S . Then, || Sn(x)  f(x)||
= o(1), if and only if anlog n = o(1), n.
= (a0/2) +  ak cos kx  (an+1 / (n+1))Dn (x) .
Garrett and Stanojević [1] extended Theorem A in
the following manner :
= (a0/2) +  ak cos kx  an+1 Dn(x) + an+1 Fn(x)
k=1 j=k
n
k=1
n
k=1
= Sn(x)  an+1 Dn(x) + an+1 Fn(x)
n1
=  ak Dk(x) + anDn(x) + an+1 Fn(x) .
k=0
2110 | P a g e
Nawneet Hooda / International Journal of Engineering Research and Applications (IJERA)
ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.
Now,



 | f(x)  fn(x) | dx =  |  akDk(x) anDn(x)
0
0
k=n


0
0
  |f(x)  fn(x)|dx +  | an+1 Dn(x) | dx
 an+1 Fn(x) | dx


+ |an+1Fn(x)|dx

  |  ak Dk(x) | dx
0
k= n
0


0
0
  |f(x)  fn(x)|dx +  |an+1Dn(x) | dx + (/2)an+1 .


+  |anDn(x)|dx + |an+1 |  Fn(x) dx
Also,
0
0



=  |  ak Dk(x) |dx +  |  akDk(x) |dx
0

k=n


0
0
 |an+1Dn(x)|dx   |fn(x)  Sn(x)|dx

k=n

+ |an+1 Fn(x)| dx

0
+  |anDn(x)|dx + | an+1| (/2)
0


<  |f(x)  Sn(x) | dx + (/2) an+1 .

0
 (/4) +  |ak|  csc (x/2) dx +

k=n

|an|  |Dn(x)|dx + (/4)

Since  |an+1 Dn(x)| dx behaves like an log n for large
0
0
values of n, and by our


= (/4) +  |ak| [2 log |csc (/2)  cot (/2)| ]
theorem limn  |f(x)  fn(x)| = 0, Theorem B of
0
k=n
Garrett and Stanojević follows.

+ |an|  |Dn(x)|dx + (/4) <  ,
References
0

since  |Dn(x)| dx behaves like log n for large n and
[1]
0
{ak}  BV  C. Also, using (3.1)


 |an+1 Dn(x)| dx   |Sn(x)  fn(x)|dx +
0
[2]
0

|an+1|  Fn(x) dx
0
[3]

  |f(x)  fn(x)|dx + (/2) an+1
0
This completes our result.
Corollary 1.
Let {ak}  BV  C,
then
||Sn(x)  f(x) || = o(1), if and only if an log n = o(1),
n .
Proof. We have


 |f(x)  Sn(x)|dx =  |f(x)  fn(x) + fn(x)  Sn(x)|dx
0
0

[4]
[5]
[6]
J.W. Garrett and C.V. Stanojević,
Necessary and sufficient conditions for L1Convergence of trigonometric series, Proc.
Amer. Math. Soc. 60 (1976), 68-71.
J.W. Garrett, C.S. Rees
and C.V.
Stanojević, L1-convergence of Fourier
series with coefficients of bounded
variation, Proc. Amer. Math. Soc., 80
(1980), 423-430.
N. Hooda and B. Ram, Convergence of
certain modified cosine sum, Indian J.
Math. (2001) (to appear).
S. Kumari and B. Ram, L1-convergence of
a modified cosine sum, Indian J. pure
appl. Math. , 19(11) (1988), 110-1104.
S. Sidon, Hinreichende bedingugen fur
den
Fourier-charakter
einer
trigonometrischen Reine, J. London Math.
Soc. , 14 (1939), 158-160.
S.A. Teljakovskii, A certain sufficient
condition of Sidon for the integrability of
trigonometric series, Math. Zametki 14
(1973),
317-328.

  |f(x)  fn(x)|dx +  |fn(x)  Sn(x) | dx
0
0
2111 | P a g e