IBN AL- HAITHAM J. FO R PURE & APPL. SC I.        VOL.24 (3) 2011 

 
Approximations of Entire Functions in Locally Global Norms 

 
 

S.K.Jassim, N.J.Mohamed  
Department of Mathematics, College of Science, University of Al-Mustansirya  
Department of Mathematics, College of Education Ibn Al-Haitham , University 
of Baghdad 
Received  in :  3    February  2011  
Accepted  in :  10  May          2011 
 
Abstract 
        The purpose of this paper is to evaluate the error of the approximation of an entire 
function by some discrete operators in locally global quasi-norms (L,p-space), we intend to 
establish new theorems concerning that Jackson polynomial and Valee-Poussin operator 
remain within the same bounds as bounded and periodic entire function in locally global 
norms (L,p), (0 < p   1). 
Key words : Entire functions, bounded masurable functions, quasi-normed space. 

  
Introduction and Preliminaries 

        Al-Abdulla, A. [1], Al-Saidy, S.K. [2] and E.S.Bhayah [3] gave estimation for 
approximation of bounded measurable functions with some discrete operators in Lp-norm           
(0 < p   1). 
        Here, we give an estimation for approximation of entire functions in L,p-space. 
        Let X = [–,] we denote the set of all 2-periodic bounded measurable function with 
usual sup-norm by L, such that 
L(X) = {f : f is 2-periodic bounded measurable function} with norm 

f sup{ f (x) x X}

      …(1.1) 

and the Lp-norm (1  p < ) of  f  Lp by 
Lp

f , such that 

p p

1
p

p
p p L (X)

X

L (X) f : f ( f(x) dx) ; f f
  

     
  

  …(1.2) 

 
        Now let us consider the Dirich let kernel of degree n, [4] 

n

n
v 1

1
D (u) cos(vu)

2 

   uR, n=0,1,… …(1.3) 

 
Let  

be the Fejer kernel of degree not grater than n. 
 

where k,n

2K
X ,(K 0,1, 2,..., n),

n 1
 




be the so called Jackson polynomial of function f  L. 

n 0 1 n

1
K (u) [D (u) D (u) ... D (u)]

n 1
   


  …(1.4) 

n

n k,n n k,n
k 0

2
J (f, x) f (x )K (X X )

n 1 

 


  …(1.5) 



IBN AL- HAITHAM J. FO R PURE & APPL. SC I.        VOL.24 (3) 2011 
 

 

Let j

2 j
X

3n 1





, j=0,1,…,3n. Then we define the following operator. 

be the valee-poussin discrete operator of 2-periodic bounded measurable function. 
 
        The unique linear trigonometric polynomial which is interpolating a given function          
f Lp(X) at the point Xj is denote by In(t) which has the representation: 

 
        Now let Bn be the set of all entire functions, since the derivative of polynomial exists 
every where, then we get that every polynomial is an entire function [5], so we consider that            
f Bn and Jn(f ) Bn, V2n,3n(f ) Bn. 
        Let n, k be positive integers, (0 < p  1) and ( > 0) are fixed numbers which will be 
used for the degree of approximating polynomial, for the rate order of modulus and for the 
space L,p respectively. 
 
        We consider the locally global norm for ( > 0), (0 < p   ) 

 
        Now the kth average modulus of smoothness for f  L,p are defined by the following 
respectively, [6], [7] 

where the kth modulus of smoothness for f  L,p, k   is defined by 

 

Now, we set 

k
k m

k
m 0h

k
( 1) f (t mh) if t or t kh X

f (t) m

0 otherwise





  
         

 
 

 . 

        In the following we recall some theorems which are needed:- 
 

Theorem 1.1: [6] 
        If f  Bn, then for (0 < p  1) and ( > 0), we have, 

1

1 p p

,p p
f c(p)[(1 n ) (ns)] f 


 . 

 
 

2n n n 1 2n

1
V (t) [D (t) D (t) D (t)]

n 1
  


 …(1.6) 

3n

2n,3n j 2n j
j 0

2
V (f ,X) f(X )V (X X )

3n 1 

 
   …(1.7) 

2n

n j n j
j 0

2
I (f ,X) f (X )D (X X )

2n 1 

 


  …(1.8) 

1
p p

,p
X

f sup f (y) , y x , x dx
2 2

   
          



 
, X[–,].  …(1.9) 

k p k

p

k ,p k

,p

1 1
(f , ) W (f ,., ) ,

n n

1 1
(f , ) W (f ,., )

n n

 
 

 
 
 
  








 …(1.10) 

k
k h

1 k k
W (f , x, ) sup f(t) : t, t kh x , x X

n 2 2

  
          

 
 …(1.11) 



IBN AL- HAITHAM J. FO R PURE & APPL. SC I.        VOL.24 (3) 2011 
 

Theorem 1.2: [3] 
        If f  2-periodic bounded measurable functions, then for (0 < p  1) 

n 1 pp

1
f J (f ) C(p) (f, )

n


   . 

Theorem 1.3: [3] 
        If f  2-periodic bounded measurable function, then for (0 < p  1) 

2n,3n k pp

1
f V (f ) C(p, k, ) (f , )

2n
    , where n=1,2,… and (p,k,ℓ) is a constant depends on p, 

k and ℓ. 
 
Theorem 1.4: [3] 
        Let f be 2-periodic bounded measurable function, then for (0 < p  1), we have 

n k pp

1
f I (f ) C(p, k, ) (f , )

n
    , where p,k,ℓ is a constant depends on p, k and ℓ. 

 

Main Results 

        We shall prove direct inequality  to find the degree of approximation of 2-periodic entire 
function by some discrete operators in (L,p) spaces, (0 < p   1). 
 
Lemma 2.1:  
        Let f be 2-periodic entire function, then for (0 < p  1), we have 

k p k ,p

1 1
(f, ) (f , ) .

n n
    

Proof:  

k p k

p

k
h

p

k
i k

i 0
p

p
k

i k

i 0

1 1
(f , ) W (f ,., )

n n

k k
sup f (t ) ; t, t kh x ,x X

2n 2n

k k k
sup ( 1) f (t ih) ;t , t kh x ,x X

i 2n 2n

k k k
sup ( 1) f (t ih) ;t , t kh x ,x X

i 2n 2n











  
          

    
           

    

    
              







1

p

X

1

p ppk
i k

i 0X

k

h

,p

k

,p

k ,p

dx

k k k k k
sup sup ( 1) f (t ih) ;t , t kh y , y X y x ,x dx

i 2n 2n 2n 2n

k k
sup f (t ) ; t, t kh x ,x X

2n 2n

1
W (f , ., )

n

1
(f , )

n





 
 
 
 

                             
 

  
        

  















 
Theorem 2.2:  
        Let f be 2-periodic bounded measurable entire function, (f  L,p), (0 < p   1), we have 



IBN AL- HAITHAM J. FO R PURE & APPL. SC I.        VOL.24 (3) 2011 
 

n 1 ,p, p

1
f J (f ) C(p) (f, ) ,

n
  

 where C(p) is a constant depends only on p. 

 
Proof:  
By theorem (1.1), we get 

1

1 p p p
n n, p p

f J (f ) C(p)[1 (1_ n ) (n ) ] f J (f ) .   


   

Now since ( > 0), then 

n 1 n, p p
f J (f ) C (p) f J (f ) .  


 

Then by using theorem (1.2) and lemma (2.1), we get that  

n 2 1 p, p

1 ,p

1
f J (f ) C (p) (f , )

n
1

C(p) (f, )
n

 











 

 
Theorem 2.3:  
        Let f be 2-periodic bounded measurable entire function, (f  L,p), (0 < p   1), we have 

2n,3n k ,p,p

1
f V (f ) C(p,k, ) (f, ) ,

2n
   

 where p,k,ℓ is a constant depends on p, k and ℓ. 

Proof:  
By using theorem (1.1), we get 

1

1 p p p
2n,3n 1 2n,3n,p p

f V (f ) C (p)[1 (1 n ) (n ) ] f V (f )    


  . 

Since 
1

n
  , then 

2n,3n 2 2n,3n,p p
f V (f ) C (p) f V (f) .  


 

Now by using theorem (1.3) and lemma (2.1), we have  

2n,3n k p,p

k ,p

1
f V (f ) C(p,k, ) (f, )

2n
1

C(p,k, ) (f, ) .
2n

 















 

 
Theorem 2.4:  
        Let f be 2-periodic bounded measurable entire function, (f  L,p), (0 < p   1), we have 

n k ,p,p

1
f I (f ) C(p, k, ) (f , )

2n


   , where p,k,ℓ is a constant depends on p, k and ℓ. 

Proof:  
By using theorem (1.1), we get 

1 p p
n 1 p n,p p

f I (f ) C (p)[1 (1 n ) (n ) y ] f I (f)    


  . 

Since 
1

n
  , then 

n 2 n,p p
f I (f ) C (p) f I (f ) .  


 

Then by using theorem (1.4) and lemma (2.1), we get 
 
  
 

 



IBN AL- HAITHAM J. FO R PURE & APPL. SC I.        VOL.24 (3) 2011 
 

n k p,p

k ,p

1
f I (f ) C(p,k, ) (f, )

n
1

C(p,k, ) (f, ) .
n

 















 

 
 

Conclusion 

        We found the degree of approximation of entire functions by using Jackson, Vallee 
Pouson and interpolation polynomials in locally quasi-norms L,p ( 0 < p  < 1). 
 

References 
1. Al-Abdullah, A. (2005), On Equi-Approximation of Bounded -Measurable Functions in 

Lp()-Space, Thesis, University of Baghdad. 
2. Al-Saidy, S.K. (2002), Best One-Sided Approximation with Algebraic Polynomials in Lp-

Spaces, Ibn Al-Haitham J. for pure and applied Science., 15 :(3). 
3. Bhayah, E.S. (1999), A Study on Approximation of Bounded Measurable Functions with 

some Discrete Series in Lp-Spaces (0 < p   1), Thesis. 
4. Zygmund, A. (1958), Trigonometric Series, I:II, Cambridge. 
5. Verhey, C.B., Complex Variables and Application, Third Edition, Tokyo, Japan. 
6. Dryanov, D. (1991), Equi Convergence and Equi Approximation for Entire Functions. 

Constructive Theory of Functions' 91, International Conference, Varna, May 28-June 3. 
7. Sendov, B. and Popov, V.A., (1983), Average Modulus of Smoothness, Sofia. 

  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  



  2011) 3( 24للعلوم الصرفة والتطبیقیة               المجلد مجلة ابن الهیثم

  طة المتعددات المتقطعة في الفضاءات المحلیةاتقریب الدوال الداخلیة بواس

  
  صاحب كحیط جاسم ، نادیة جاسم محمد

الجامعة المستنصریة ،كلیة العلوم  ،قسم الریاضیات   

د ،ابن الهیثم  -كلیة التربیة ،قسم الریاضیات جامعة بغدا  

     2011 شباط  3 :في استلم البحث

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k  الوسیط لاباستعم المحلیة شبھ الفضاءات p

1
τ , )

n
.  

  
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