Theoretical Mathematics & Applications, vol.2, no.1, 2012, 179-188
ISSN: 1792- 9687 (print), 1792-9709 (online)
International Scientific Press, 2012
Assessment Fourier coefficients
of a function of class L( p, )
Ismet Temaj1
Abstract
In this paper, we’ll give a necessary and sufficient condition that a function

f ( x)   an cos nx , where coefficient an (n  1, 2,...) are quasi-monotone, to be
n 1
of class L( p,  ) .
Mathematics Subject Classification :
42A16
Keywords: Fourier series, Fourier coefficients, quasi-monotone coefficients
1
Introduction
The studying of order of decrease of Fourier coefficients of a function
1
Prishtina University Education Faculty, Prishtina, Kosovo,
e-mail: itemaj63@yahoo.com
Article Info: Received : December 21, 2011. Revised : January 27, 2012
Published online : March 15, 2012
Assessment Fourier coefficients of a function of class L( p, )
180
belonging to different subclasses of class L p ( p  1 ) represents one of the
fundamental issues of Fourier theory. This paper deals with the Fourier
coefficients ( a n  0 and quasi-monotone) of a function of class L( p,  ) , where
1  p   , 1   p  p  1 .
2
Preliminary Notes
That’s why, first of all, we’ll represent the main statements needed for
representation of the results of this paper.
bn  0 , and n   bn  0
Definition 2.1 A sequence {bn } is quasi-monotone if
for some   0 .
Definition 2.2 Let
1  p   , we say that function f
with period 2 is in
class L p if
f
p
2

   | f ( x) | p dx
0

1
p

So
1


p
2






L p   f ( x) / f ( x) p    | f ( x)| p dx   
 0





Definition 2.3
A function f ( x) is said to belong to the class
f
p ,


p
  f ( x) (sin x) p dx 
0

1
p

where 1  p   , 1   p  p  1 .
So
1
L( p,  )   f ( x ) / f
p ,


p
   f ( x) (sin x) p dx   
0

p

L( p,  ) , if:
Ismet Temaj
181
The following affirmation gives necessary condition receptively adequate
that is necessary to complete Fourier coefficients in order that function belongs to
class L p ( L( p,  ) ).
Theorem 2.4 (Hausdorff-Young) [2, p. 211] Let 1  p  2
and
q
p
p 1
( 2  q   ). The following estimate holds true
cn n 
1) If f  L p and
are Fourier coefficients of function, then
 

q
  | cn | 
|n| 0

2) If

cn n

1
q
 A( p) f p
is sequence of numbers such that

 cn
|n| 0
p

then there exists function f  Lq with Fourier coefficients {cn } the inequality

f q  A'(q )   cn
n  0
p


1
p
holds true.
Theorem 2.5 ( Hardy- Littlewood ) [2, p. 657] The necessary and sufficient

condition that
a
n 1
n
cos nx
f  L p , p  1 is that the series
a n  0 be the Fourier series of a function

a
n 1
p
n
n p  2   .
Theorem 2.6 [3] The necessary and sufficient condition that the

a
n
cos nx
n 1
where {an } is positive and quasi-monoton be Fourier series of a function
Assessment Fourier coefficients of a function of class L( p, )
182
f  L( p,  ) , where 1  p   , 1   p  p  1 is that the series

 a 
n
n 1
p
n pp2  
In [1] given the following theorem concerning the Fourier coefficients of a
function belonging to L p class.
f  L p ( p  1 ), function given with Fourier series
Theorem 2.7 [1] Let

f ( x)   an cos nx a n  0
n 1
Then
S1 S 2 S 3
,
,
,...
1 2 3
n
are also Fourier coefficients of a function of class L p , where S n   a k .
k 1
As you can see from Theorem 2.7 the connection is becoming between
coefficients
a n 
An    1  a n  , where
n
and
n
k 1

an  0
An  0 .
A question is settled down if the coefficients a n  are quasi-monoton will the
coefficients
An    1  a n 
n
n
k 1

be quasi-monoton. A following lemma gives the
positive answer to this question.
An    1  ak 
n
Lemma 2.8 [4] If {an} is positive and quasi-monoton, then
is also positive and quasi-monoton.
n
k 1

Ismet Temaj
3
183
Main Results
The purpose of this paper is to reformulate the Theorem 4. in case when
function f ( x)  L( p,  ) and appropriate coefficients are quasi-monoton.
Theorem 3.1 Let f ( x)  L( p,  ) ( 1  p   ,
 1  p  p  1 ), function given
with Fourier series

f ( x)   an cos nx ,
n 1
where a n  is positive and quasi-monoton. Then series

A
n
n 1
where

n

 An    n1  a k  ,
 k 1

cos nx
will be Fourier series of a function F ( x) of class
L ( p,  ) .
Proof: Let
f ( x)  L( p,  ) (1  p   ,
 1  p  p  1 , function given with
Fourier series

f ( x)   an cos nx ,
n 1
where a n  is positive and quasi-monoton.
Since {an} is positive and quasi-monoton and due to Lemma 2.8
1 n

 k 1

 An    n  a k 
is positive and quasi-monoton. To proof theorem we have to show that

  An  p n p p2  
, then by Theorem 2.6 follows that series
n 1
Fourier series of a function F ( x) of class L( p,  ) .

A
n 1
n
cos nx is
Assessment Fourier coefficients of a function of class L( p, )
184
Let
x
f1 ( x)   f ( x) dx
x
and
0
f 2 ( x )   f1 ( x) dx
0
Then

n
k 1
k 1
f 2 ( x)   ak [1  cos kx]  k 2   ak [1  cos kx]k 2
for


x
4(n  1)
4n
we have
n
f 2 ( x)  B1  n 2   ak  B1  n 1 An
k 1
for same constant B1 . So An  B  n  f 2 ( x ) for same constant B .
Thus:


n p  p  2 [ An ] p  B 
n 1

n
p  p  2
 n p [ f 2 ( x)] p 
n 1
 B

n
2 p  p  2

n 1
4n
n 1


[ f 2 ( x)] p 
p

(sin x)
p  p
4 ( n 1 )
 /4
 B
 x  4n


 B
min
4 ( n 1)
(sin x)
p  p

 f ( x) 
  2  dx 
 x 

 x f 2 ( x) dx  B( , p ) 
1
p
 (sin x)
0
0
 B( , p) 
 /4
 /4

 (sin x)   f ( x)
p
p
dx  
0
A similar method may be used to estimate
p  p


p
 x 1 f 1 ( x) dx
Ismet Temaj
185


 (sin x)   f ( x)
p
p
dx  
/4
So

n
p p  2
[ An] p   .
n 1
This finishes the proof of Theorem 3.1.

The question appears: Is the converse valuable of Theorem 2.7 and 3.1 if the
series

A
n 1
n

a
cos nx is Fourier series, will Fourier series be
n 1
n
cos nx . From
the following example it is proved that the converse of Theorem 2.7 and 3.1
doesn’t worth.
Example 3.2 Let


1
A
cos
nx

( 1) n cos nx


n
n 1
n 1 n
We have


 n 1

1
p
1
1
p p
 1

  1 p
p
n
An    (1)    p   ,

 n 1 n 

 n 1 n
1  p  2.
Hence by the theorem1.(Hausdorf-Young), An is the Fourier coefficients of a
function F ( x )  Lq , where
Now, if

a
n 1
n
1 1
 1, 1 p  2 , q  2 .
p q
cos nx Fourier series of a function e
f ( x )  L p , then we have
1
 
q
by Theorem 2.4 (Hausdorf – Young) necessarily   | a n | q    , where
 n 1

1 1
  1 , 1 p  2 , q  2.
p q
Assessment Fourier coefficients of a function of class L( p, )
186
But An 
1 n
1
a k = ( 1) n so follow

n k 1
n
n
S n   a k  (1) n ,
k 1
a n  S n  S n 1  ( 1) n  ( 1) n 1  ( 1) n 1 ( 1  1)  2  ( 1) n
1
1
1


q
 
q  
  | a n | q     | 2(1) n | q    (2 q ) q   2   .
n 1
n 1
 n 1

 n 1

Therefore


n 1
n 1
 an cos nx   2  (1)n cos nx
is not the Fourier series of a function
f ( x)  Lp .
So the question is settled down. What conditions of coefficients an will be
fulfilled in order that converse is valuable. A following theorem gives answer to
the question.
Theorem 3.3 Let

f ( x)   an cos nx ,
n 1
where {an } is positive and quasi-monoton. Then a necessary and sufficient

condition that
a
n 1
n
cos nx be the Fourier series of function f ( x)  L( p,  ) is
that:

 A cos nx
n 1
n
to be the Fourier series of a function F ( x) be belonging to L ( p,  ) where
1  p   ,  1  p  p  1
and An 
1
n
n
a
k
.
k 1
Proof: The necessary part follows from Theorem 3.1 as a particular case.
Ismet Temaj
187
Sufficiency. Suppose that series

 A cos nx
n
is Fourier series of a function
n 1
F (x )  L ( p,  ) . Since {an } is positive and quasi-monoton, then by Lemma 2.8
follows
{ An } is positive and quasi-monoton. Hence by Theorem 2.6 we have

n
n 1
p p  2
Anp   .
Since sequence {an } is positive and quasi-monoton then for some constant   0 ,
sequence n  an  0 , and fore some constant B1  0 we have n  an  B1k  ak
for
k  n , then it follows that
An 

n
1 n
1 n 
1 1 





a
k
a
k
n
a


k
k
nk 
n k 1
n k 1
B1 n
k 1
1 1 
1
n a n  nn 
an
B1 n
B1
a n  B1  An
So that


n 1
k 1
 n p p 2 a np  ( B1 ) p  n p p 2 Anp   .
Hence by Theorem 2.6 f ( x )  L ( p,  ) and consequently

a
n
cos nx is the
n 1
Fourier series of function f ( x) .

ACKNOWLEDGEMENTS. The author wish to express their thanks to the
worthy referees for their valuable suggestions and encouragement.
Assessment Fourier coefficients of a function of class L( p, )
188
References
[1] G.H. Hardy, Not on some points in integral calculus, Messenger of
Mathematics, 58, (1929), 50-52.
[2] N.K. Bari, Trigonometriçeskie rjade, Moskva, 1961.
[3] R. Askey and R. Wainger, Integrability theorems for Fourier series, Duke
Mathematical Journal, 33(1), (1966), 223-228.
[4] A.K. Gaur, A theorem for Fourier coefficients of a function of class LP,
International Journal of Mathematics and Mathematical Sciences, 13(4),
(1990), 721-726.