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

 
Approximations of Entire Functions in Locally Global Norms 

 
 

S.K.Jas sim, N.J.Mohamed  
Departme nt of Mathematics, College of Science, Unive rsity of Al-Mustansirya  
Departme nt of Mathematics, College of Education I bn Al-Haitham , Unive rsity 
of Baghdad 
Received  in :  3    February  2011  
Accepte d  in :  10  May          2011 
 
Abstract 
        The p urp ose of this p aper is to evaluate the error of the app roximation of an entire 
function by  some discrete op erators in locally global quasi-norms (L,p-sp ace), we intend to 
establish new theorems concerning that Jackson p olynomial and Valee-Poussin op erator 
remain within the same bounds as bounded and p eriodic entire function in locally global 
norms (L,p), (0 < p   1). 
Key words : Entire functions, bounded masurable functions, quasi-normed sp ace. 

  
Introduction and Preliminaries 

        Al-Abdulla, A. [1], Al-Saidy , S.K. [2] and E.S.Bhayah [3] gave estimation for 
app roximation of bounded measurable functions with some discrete op erators in Lp-norm           
(0 < p   1). 
        Here, we give an est imation for app roximation of entire functions in L,p-sp ace. 
        Let X = [–,] we denote the set of all 2-p eriodic bounded measurable function with 
usual sup -norm by L, such that 
L(X) = {f : f is 2-p eriodic 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 p olynomial 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 op erator. 

be the valee-p oussin discrete op erator of 2-p eriodic bounded measurable function. 
 
        The unique linear trigonometric p olynomial which is interp olating a given function          
f Lp(X) at the point Xj  is denote by  In(t) which has t he rep resentation: 

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

 
        Now the k

th
 average modulus of smoothness for f  L,p are defined by  the following 

resp ectively, [6], [7] 

where the k
th
 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:- 
 

The orem 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 


 . 

 
 

2 n n n 1 2 n

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

n 1
  


 …(1.6) 

3 n

2 n ,3n j 2 n j
j 0

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

3n 1 
 


  …(1.7) 

2 n

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 
 

The orem 1.2: [3] 
        If f  2-p eriodic bounded measurable functions, t hen for (0 < p   1) 

n 1 pp

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

n


   . 

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

2 n ,3 n k pp

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

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

k and ℓ. 
 
The orem 1.4: [3] 
        Let f be 2-p eriodic 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 dep ends on p , k and ℓ. 

 

Main Re sults 

        We shall p rove direct inequality  to find the degree of app roximation of 2-p eriodic entire 
function by  some discrete op erators in (L,p) sp aces, (0 < p   1). 
 
Lemma 2.1:  
        Let f be 2-p eriodic 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 pp
k

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





 
 
 
 

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

 

  
        

  















 
The orem 2.2:  
        Let f be 2-p eriodic 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 const ant dep ends 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

 











 

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

2 n ,3 n k ,p,p

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

2n
   

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

Proof:  
By  using theorem (1.1), we get 

1

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

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

  . 

Since 
1

n
  , then 

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


 

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

2 n ,3 n k p,p

k ,p

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

2n
1

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

 















 

 
The orem 2.4:  
        Let f be 2-p eriodic 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 dep ends 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 app roximation of entire functions by  using Jackson, Vallee 
Pouson and interpolation polynomials in locally quasi-norms L,p ( 0 < p  < 1). 
 

Re ferences 
1. Al-Abdullah, A. (2005), On Equi-Ap p roximation of Bounded -M easurable Functions in 

Lp()-Sp ace, Thesis, University  of Baghdad. 
2. Al-Saidy , S.K. (2002), Best One-Sided Ap p roximation with Algebraic Poly nomials in Lp-

Sp aces, Ibn Al-Haitham J. for p ure and app lied Science., 15 :(3). 
3. Bhayah, E.S. (1999), A Study  on Ap p roximation of Bounded M easurable Functions with 

some Discrete Series in Lp-Sp aces (0 < p   1), Thesis. 
4. Zy gmund, A. (1958), Trigonometric Series, I:II, Cambridge. 
5. Verhey , C.B., Complex Variables and Ap p lication, Third Edition, Toky o, Jap an. 
6. Dry anov, D. (1991), Equi Convergence and Equi Ap p roximation for Entire Functions. 

Const ructive Theory  of Functions' 91, International Conference, Varna, M ay 28-June 3. 
7. Sendov, B. and Pop ov, V.A., (1983), Average M odulus of Smoothness, Sofia. 

  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  



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

  طة المتعددات المتقطعة في الفضاءات المحلیةاتقریب الدوال الداخلیة بواس

  
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k  الوسیط لاباستعم المحلیة شبھ الفضاءات p
1

τ , )
n

.  

  
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