Applied Mathematics E-Notes, 11(2011), 61-66 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Integrability Of Cosine Trigonometric Series With
Coe¢ cients Of Bounded Variation
Xhevat Z. Krasniqiy
Received 19 March 2010
Abstract
In this paper condition of integrability of cosine trigonometric series with
coe¢ cients of bounded variation of order p is obtained. The results generalize
some previous results of Telyakovski¼¬.
1
Introduction
In the literature, there are many studies related to the trigonometric series, and particularly the cosine series. We refer to the excellent monograph by R. P. Boas, Jr.
[1].
The …rst results pertaining to the series
1
a0 X
+
ak cos kx
2
(1)
k=1
considered the case of monotone coe¢ cients. Later, some authors investigated the
series (1) with quasi-monotone coe¢ cients (an+1 an (1 + =n); n n0 ; > 0).
Several papers have been written on the series (1) when the sequence fak g is a
null-sequence and convex or quasi-convex, i.e. 42 ak 0 or
1
X
k=1
(k + 1)j42 ak j < 1;
(2)
where 42 ak = 4 (4ak ), 4ak = ak ak+1 .
Furthermore, when fak g is a null-sequence of bounded variation, i.e.
1
X
k=1
j4ak j < 1;
is also considered.
We shall consider the series (1) whose coe¢ cients tend to zero and satisfy any
condition that provides the convergence of the series (1) on (0; ]. Let us denote its
Mathematics Subject Classi…cations: 42A20, 42A32.
of Mathematics, University of Prishtina, Prishtinë 10000, Republic of Kosova
y Department
61
62
Integrability of Cosine Series with Coe¢ cients of Bounded Variation
sum by f (x). If the coe¢ cients ak are quasi-convex, it is well-known that f is an
integrable function on [0; ] (see for example [2], page 264), and the following estimate
is valid
Z
1
X
jf (x)jdx
(k + 1)j42 ak j:
0
k=1
In a similar direction, among others, Telyakovski¼¬ [2] obtained some estimates of
the integrals of the following form
Z =`
jf (x)jdx; 1 ` m; (`; m 2 N);
(3)
=(m+1)
expressed in terms of the P
coe¢ cients ak , where he used null-sequences of bounded
1
variation of second order ( k=1 j42 ak j < 1), instead of quasi-convex null-sequences.
It is obvious that the condition
1
X
k=1
j42 ak j < 1
(4)
is a weaker condition than the condition (2).
The following de…nition is introduced in [4]: A sequence fak g is of bounded variation
of integer order p 0 if
1
X
j4p ak j < 1;
(5)
k=1
p
where 4 ak = 4 4 ak = 4 ak 4p 1 ak+1 , and we agree with 40 ak = ak .
In [4] an example is given to show that (5) is an e¤ective generalization of the null
sequences of bounded variation. This fact motivates the author to consider the series
(1) with coe¢ cients that satisfy the condition (5).
The main goal of the present note is to use (5), p 2, instead of (4), to prove some
estimates of the form (3) that shall generalize some results of Telyakovski¼¬in [2].
We write g(u) = Op (h(u)), u ! 0, if there exists a positive constant Ap , that
depends only on p, such that g(u)
Ap h(u) in a neighborhood of the point u = 0.
The constant Ap may be, in general, di¤erent in di¤erent estimates.
2
p 1
p 1
Main Results
We need the following notations
B01 (x)
=
Bk1 (x)
=
Bkp (x)
=
1
;
2
1
+ cos x +
+ cos kx for k 1;
2
k
X
B p 1 (x) for p = 2; 3; : : : and k
=0
and inequalities (see [3], page 20):
0;
Xh. Z. Krasniqi
(i) Bkp (x)
(ii)
Bkp (x)
(iii)
Bkp (x)
63
0,
= O ((k + 1)p ),
=O
1
xp
,
8p
2,
x
;
8p
1,
x
;
8p
1,
0<x
.
THEOREM 1. If ak ! 0 as k ! 1 and (5) is satis…ed, then the series (1) converges
on (0; ], uniformly on ["; ], for every " > 0, and for p 2, 1 ` m, the sum function
f (x) satis…es
!
Z =`
` 1
m + 1 ` X (k + 1)p 1 p 1
jf (x)jdx = Op
j4 ak j
m
`
=(m+1)
k=0
!
1
X
p 2
p
+Op
min(k + 1 `; m + 1 `)(k + 1) j4 ak j :(6)
k=`
PROOF. Applying Abel’s transformation p times we get that
n
p 1
n
Xp
X
a0 X
+
ak cos kx =
4p ak Bkp (x) +
4j an
2
j=0
k=1
j+1
j Bn j (x):
k=0
, it is obvious
In view of the hypotheses of our Theorem and Bkp (x) = O x1p , 0 < x
that the series (1) converges uniformly on ["; ], " > 0, and the following representation
holds
1
X
f (x) =
4p ak Bkp (x):
(7)
k=0
Let i be a positive integer and x 2
i 1
X
k=0
4p ak Bkp (x) =
i+1 ; i
i 1
X
k=0
4p
1
. Using the equality
ak Bkp
1
(x)
4p
1
ai Bip 1 (x);
from (7) we have
f (x) =
i 1
X
k=0
Since jBkp
1
4p
1
ak Bkp
1
(x)j = Op (k + 1)p
=i
=(i+1)
jf (x)jdx = Op
1
X
k=i
1
i 1
X
k=0
4p ak Bkp (x)
Bip 1 (x) :
and
jBkp (x)
we have
Z
(x) +
Bip 1 (x)j = Op
p 1
j4
1
xp
;
(k + 1)p 1
ak j
+ (i + 1)p
i(i + 1)
2
1
X
k=i
p
!
j4 ak j :
64
Integrability of Cosine Series with Coe¢ cients of Bounded Variation
Thus
Z =`
=(m+1)
m X
i 1
X
jf (x)jdx = Op
1
m
p 1
j4
i=` k=0
(k + 1)p 1 X X
ak j
+
(k + 1)p
i(i + 1)
2
i=` k=i
p
!
j4 ak j :
(8)
For the …rst term in the parentheses of the right-hand side of (8) we have
m X
i 1
X
j4p
1
ak j
j4p
1
ak j
(k + 1)p
1
i=` k=0
=
m X
` 1
X
i=` k=0
=
` 1
X
k=0
+
m
X1
m X
i 1
X
(k + 1)p 1
+
j4p
i(i + 1)
(k + 1)p
1
i=`+1 k=`
j4p
1
1
1
`
1
m+1
ak j
1
k+1
ak j
j4p
1
` X (k + 1)p
`
1
k=`
m+1
m
(k + 1)p 1
i(i + 1)
` 1
k=0
j4p
1
ak j
(k + 1)p 1
i(i + 1)
1
m+1
ak j +
m X
1
X
(j + 1)p
2
k=` j=k
j4p aj j:
(9)
Finally, the last term in (9) can be written as
m X
1
X
i=` k=i
(k + 1)p
2
j4p ak j =
=
m X
m
X
(k + 1)p
i=` k=i
m
X
(k + 1
2
j4p ak j +
`)(k + 1)p
k=`
+(m + 1
`)
1
X
2
m
1
X
X
i=` k=m+1
2
j4p ak j
j4p ak j
(k + 1)p
k=m+1
(k + 1)p
2
j4p ak j:
(10)
The proof of theorem follows from (8), (9) and (10).
Now we estimate the integral in (6) only in terms of p-th order di¤erence of the
sequence fak g.
COROLLARY 1. If the coe¢ cients of the series (1) satisfy conditions of the Theorem 1, then
!
Z =`
1
m+1 ` X
(k + 1)2
p 2
p
jf (x)jdx = Op
min
; k + 1; m (k + 1) j4 ak j ;
m
`
=(m+1)
k=0
holds, where 1 ` m, p 2.
PROOF. To deduce from (6) the required relation, we use the identity
4p
1
ak =
1
X
i=k
(4p ai ):
Xh. Z. Krasniqi
65
We have
` 1
X
(k + 1)p
`
k=0
` 1
X
(k + 1)p
`
k=0
=
` 1X
i
X
i=0 k=0
`
` 1
X
(i + 1)2
i=0
j4p
`
1
1
1 X
ak j
j4p ai j
i=k
p 1
(k + 1)
`
` 1
X
(i + 1)p
i=0
1
j4p ai j +
j4p ai j + `p
(i + 1)p
2
1
1 X
` 1
X
(k + 1)p
`
i=` k=0
1
X
i=`
1
j4p ai j
j4p ai j
j4p ai j + `
1
X
(i + 1)p
2
i=`
j4p ai j:
(11)
If k < m, then we can estimate the second term in (6) by means of the fact that
k+1
`
k+1
`
m `
k+1
=
(k + 1):
m
m
From the above and (11) along with (6) we obtain the required estimate.
The following Corollaries 2 and 3 are immediate from Theorem 1 and Corollary 1,
respectively.
COROLLARY 2. ([2]) If ak ! 0 as k ! 1 and (4) holds, then the series (1)
converges on (0; ], uniformly on ["; ], for every " > 0, and for 1 ` m, f satis…es
Z
=`
=(m+1)
jf (x)jdx = O
+O
m+1
m
1
X
!
` 1
` X k+1
j4ak j
`
k=0
min(k + 1
k=`
`; m + 1
2
!
`)j4 ak j :
COROLLARY 3. ([2]) If the coe¢ cient sequence of the series (1) tends to zero and
satis…es condition (4), then the following estimate
Z
holds, 1
=`
=(m+1)
`
jf (x)jdx = O
m+1
m
1
`X
k=0
min
!
(k + 1)2
2
; k + 1; m j4 ak j ;
`
m.
Acknowledgment. The author would like to express heartily many thanks to the
anonymous referee for careful corrections to the original version of this paper.
66
Integrability of Cosine Series with Coe¢ cients of Bounded Variation
References
[1] R. P. Boas Jr, Integrability theorems for trigonometric transforms, Springer-Verlag,
Ergebnisse 38, Berlin, 1967.
[2] S. A. Telyakovski¼¬, Localizing the conditions of integrability of trigonometric series
(Russian), Theory of functions and di¤erential equations, Collection of articles. To
ninetieth birthday of Academician S. M. Nikolskii, Trudy Mat. Inst. Steklov., 210,
Nauka, Moscow, 1995, 264–273. (Russian)
[3] T. M. Vukolova, Certain properties of trigonometric series with monotone coe¢ cients (English. Russian original), Mosc. Univ. Math. Bull., 39(6)(1984), 24–30;
translation from Vestn. Mosk. Univ., Ser. I. 1984, No.6, 18-23.
[4] J. W. Garret, C. S. Rees and C. V. Stanojevic, L1 -convergence of Fourier series with
coe¢ cients of bounded variation, Proc. Amer. Math. Soc., 80(3)(1980), 423–430.