IOSR Journal of Mathematics (IOSRJM) eISSN: 22783008, pISSN:23197676. Volume 10, Issue 2 Ver. II (MarApr. 2014), PP 97105 www.iosrjournals.org
The Best Growth and Approximation of Entire Functions of Two Complex Variables in Banach Spaces Dr.Deepti Goyal, Dr.D.P.Singh RKGIT, Ghaziabad ,DIT School of engineering Gr.Noida
Abstract: In this Paper we are studding the polynomial approximation of entire functions of two complex
) order of entire variables in Banach spaces; concept is depend on indexpair. The characterizations of ( functions of two complex variables have been studied in terms of approximation errors. The results can be extended to mvariables but to reduce the mechanical labour we have considered only two variables. Key Words: Approximation error, order, type, entire function, indexpair.
I.
Introduction:
) ∑ Let ( { ( )} be a function of the complex variables z1 and z 2 , regular ) represents an entire function of for   If and can be taken arbitrarily large, then ( ) as complex variables z1 and z 2 . Following Bose and Sharma [1] we define the maximum modulus of ( ( The order
)

 (

)
of the entire function (
) is defined as [1]; (
(
) )
Bose and Sharma [1], obtained the following characterizations for order of entire functions of two complex variables. Theorem 1.1. The entire function (
)
(
∑
{
(
)
(

)
) is equal to . is finite and then the order of ( Let denote the space of functions (     { } Such that ‖ ‖
) analytic in the unit bidisc ( )
Where ( functions ( ‖ ‖
) 2 ∫ ∫  ( ) 3 ) analytic in and satisfying the condition 2
∫

∫

)} is of finite order if and only if
 (
)
3
⁄
and let
denote the space of
⁄
Set ‖ ‖
) { ( } and are Banach spaces for In analogy with spaces of functions of one variable, we call and the Hardy and Bergman spaces respectively. ) analytic in Following the Vakarchuk and Zhir [4] we say that the function ( belongs to the space ( ) where and if ‖ ‖
8∫ ∫ *( ‖ ‖
The space (
)(
)+
.
8*(
) is a Banach space for
/
)(
( )+. and
) /
9 (
)
9
otherwise it is a Frechet space. Further, we have
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The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach .
/
.
(1.1)
) Let Y is a Banach space and let by elements of ( ) be the best approximation of a function ( the space that consists of algebraic polynomials of degree in two complex variables. *‖ + ‖ ( ) Recently, Ganti and Srivastava [2] characterized the order and type in terms of the approximation errors ( ( ) But their results leave to study a big class of entire functions such as slow ( )) and ) order introduced growth and fast growth. To bridge this gap in this chapter we pick up the concept of ( by Juneja et al. [3] and consider it for entire functions of two variables. Roughly speaking, this concept is a modification of the classical definition of order obtained by replacing logarithms by iterated logarithms, where the degrees of iteration are determined by p and q. ) order of entire functions of two To the best of our knowledge, characterizations for the ( complex variables in Banach spaces have not been obtained so far. We define the ( p, q) order of an ) by entire function ( (
(
)
)
(
Where p and q are integers such that
.
Notations: We are using the following notations in this paper. , , , (i) ( , ) ( , , Provided that ⁄
(ii)
)
0(
.
(
/1
⁄
0(
)
⁄
0(
)
.
/1
(
0(
)
.
/1
(
⁄
.
(1.3)
)
)
)
, 
.
)
(
/1
(
) ) )
II. Basic results: In this section we have given some lemmas as basic results, which have been used in the sequel. Lemma 2.1. If ( ( )
) (
∑ { ( ) be defined by (1.3) then ( )
Where
,
(
)
,

)}
be an entire function and for a pair of integers
( (
{(
2
))
)
}

 3
(
)
4.
( (
))
(
( ( (
) ) ))
{ And {
(
(
)
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)
(
)
98  Page
The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach III. Theorem3.1: If ( ( )
Main Results: In this section we prove our main results.
) ∑ { ( be defined by (1.3) then ( ) (
)} be an entire function and for a pair of integers . ( ,
)
,
)/ where  2(
)
{
0
3 (
(
))1
Proof. We prove the above result in two steps, first we consider the space ( ) ( ) be of ( ) order ( and .Let ( ( ) such that there exists a natural number (


,
6

,
2

Now first we show that ( entire function. If ( (
)
{(
)
}3
. (
)
(
(
)/ If (
)
. (
)
(3.1)
}
) ) From (2.1), for any
)
7
(3.2)
)
then (
)/ since (
)
or else (
) is not an
) is nonnegative then we have
)
We denote the partial sum of the Taylor series of a function ( (
)
) by
∑∑
and (
( 8∫ ∫ *(
)(
Where ∑ ∑

∑ Since (
.
)+
/
.∑ ∑
∑  are bounded and ))
( )‖
‖

∑

(
))
 /
9
∑

(
 ,
∑ ∑  and therefore (3.3) becomes )
6∫ {(
.
/
}
(
)
)

7{ ∑

∑ 
 }
Where )
6∫ {(
.
/
}
(
)
7 )
6∫ {(
.
/
(
}
)
7
6∫ {(
) 2∑
∑
.
)
/
}
(
)
7
Therefore (
(
))
(
) (
where is a constant and ( In view of (3.2), we have ∑
∑ 

)(
)

 3
denotes the beta function. (
∑
∑
,
8

{
,

(
)}
,

{
,

(
)}
(
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)
)
9 (
( )6
(3.4)
)
7 99  Page
The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach Using the above inequality in (3.4) we get ( )) ( ( ) (
) (
,
6

,
{

(
)}
(
)
7
)
(
)
The result for has been obtained by Ganti and Srivastava [2]. ) ( ) Now consider for ( ( )) ( ( ) ( ) (
*(
6
)+
(
)
7
)
Or (
(
))
(
) (
*(
6
)+
(
(
)
(
)
)
7
)
7. Or (
(
(
))
)
(
)
*(
8
)+
(
)
9
Or (
(
))
(
)
(
)
(
Or (
)
(
) 0
{
)
}
)
(
))
(
)]}
(
)+ (
*(
{
(
)
(
)}1
Since )
{ [(
â „(
)
Now proceeding to limits, we get (
(
(
) [
) (
(
Or . (
)
))]
(
)/ [
(
(
)
) (
))]
(
)
Or . (
)/
(
)
8. Or (
)
. ( )/ ) ( ) Now for ( ( )) (
( ) ( ) we have from (3.5) that ( ) ( )
(
)
( ,
6

,
{

(
)}
(
)
)
7
Or ( (
( )
)) 6
( ,

{
) ,

(
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(
) )}
(
)
7 100  Page
The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach Taking (3.6) into account, we get [ Or
(
( ,

[
))](
)
,

{
,

(
))](
)
{
,

(
Or
,
(

)
(
)}
(
)
(
)}
(
)
)
, [
(
( Proceeding to limits, we obtain (
(
))](
) ,
) , 0
(
)
(
))1(
(
(
)
)
9. Combining other results for (
)
(
(
)
)
(
) with (3.8) we get
. (
)/
(
To prove reverse inequality consider (eq.2.4 [2]) which gives   ( ) ( ) Or 

(
)
(
(
))](
Now using Lemma (2.1), since (
(
) ,

(
, [
)
(
))
) ](
the inequality for ( )/ .
)
))
(
)
( and for
(
(
Again (3.6) taking into account in above inequality, we obtain , ( ) , [
)
)
gives
it gives
( ) ( ) ) )/ Hence combining above results we get ( . ( This is the proof of first step. ) Now we consider the space ( we have ( )) ‖ (
∫ ∫ *( { Where ∑ ∑
)(
)+
.
/
:∑ ∑

(
)
( )‖
 ;
(
)
} 
∑ Since (
∑

∑  are bounded and (
))
6∫ {(
∑

∑ ∑  and therefore (3.11) becomes )
.
/
}
(
)
7{ ∑
 , 

∑ 
 }
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101  Page
The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach )
8∫ {(
.
/
}
(
)
9 .
)
8∫ {(
/
(
}
)
.
)
9 8∫ {(
/
(
}
)
9
Therefore (
(
))
(
) (
is constant and (
Where
){ ∑
∑ 
 }
(
) is Euler’s integral of the first kind. By using (3.2) we have (
∑
∑ 

)
∑
∑
,
8

,
{

(
)}
(
( ,
( )6

,
{
using this inequality in (3.12), we get ( )) ( (

(
) (
)}
(
)}
(
)
7
)
) (
,
6

,
{
)
9
)

(
)
7
)
(
)
) ( ) the result has been proved by Ganti and Srivastava [2] . For ( ) ( )we have from (3.13) that Now ( (
(
(
))
(
) (
)
(
8
)
(
)
9
)
Or (
(
))
(
)
(
)
(
(
8
)
(
)
9
)
Or (
(
))
(
(
) )
8
(
)
(
)
(
)
9
)
9
Since 0(
)
.
(
/1
((
)( )(
) ( . )
.
/) /)
and {
[(
)
(
(
)]}
)
(
)
Therefore from above inequality, we get (
(
))
(
)
8
(
)
(
Or
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The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach (
(
(
))
(
8
)
(
)
9
)
Or [
(
(
))]
(
Or (
(
)
(
) , [
(
(
))]
(
) , [
)
(
)
) (
(
Or . (
(
)
Proceeding to limits, we get (
)
))]
(
)/ , [
(
)
) (
(
))]
(
)
Or . (
)/
(
)
Or ( ) . ( )/ ( ) ( ) from (3.13) we get ( ) ( )
) Now consider the case ( ( )) (
(
)
(
6
,
(
(

,
{

(
)}
(
)
7
)
Using (3.14), we obtain ))
(
)
,
6
*
,
(
)+
(
(
)
)
7
14. Or [
(
(
))]
(
,
)
Or
,
(

) , [
Proceeding to limits, immediately we get (
)

(
)/
))] (
To prove reverse inequality taking (3.12) into account this gives   ( ) ( ) Or ( )) (   Again using (3.14), we get , [
( (
(
)}
(
)
)
( (
) (

) (
(
. (
,
{
(
))
)
(
( ))]
(
(
)
, 
)
)
) 
(
)
In view of lemma (2.1), we obtain (
)
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The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach Or ( Since ( gives
)
)
(
)
( gives (
the inequality for (
)
)
(
)
)
(
(
.
)/ and for
it
)
Combining (3.15), (3.16), (3.17) and (3.18) the proof of second step is immediate. Now consider the third step. Let and Since . where
(
(
)/
)
(
6 4(
(
(
))
and the condition (3.1) is already proved for the space ( , ) {( }
and , [
(
(
))](
) hence
)
,

)
{(
, [
Now let
)
)57
}
(
(
(
))](
)
)
Since (
)
(
)
.
Therefore .
(
8∫ ∫ *(
)/ [
)(
 (
)+
) (
.
/
9
)]
16. where ,
(
) ,

, [
{( (
 Hence we get ) } (
))](
) ,
 {( ,  [
)
}
](
(3.20)
)
In view of (3.19), (3.20) and Lemma 2.1, we get the required result. This is complete proof of the theorem. ) ∑ Theorem 3.2. If ( { ( )} be an entire function, then for a pair of ) ) ) order ( ) if and only if ( ) integers ( the function ( is of ( )) ( ( where , ) {( } ( ) ( ) , { ( )] } [ ( ) ) ∑ Proof. Let ( { ( )} be an entire transcendental function. Since is entire, we have (
 and and (
therefore )
(
)
( ) ( ⁄
where K is a constant independent of
))
) we have (
and
)
) (
(
(
)
(
)
In the case of space
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The Best Growth And Approximation Of Entire Functions Of Two Complex Variables In Banach (
(
))
(
)
(
)
From (3.23) we have ,
( )
,

)
{(
{
(
(

(
[
) ,
,
}

{ (
)] }
{( [
)
)
)
}
(
( ⁄
))] }
(3.25)
using (3.24) we prove inequality (3.25) for the case For the reverse inequality, we have (
)
( )
∑
,
∑
∑ 

using (3.2) we have
(
(
)
( )
∑
8

,
{

(
)
}
Or
(
)
(
(
)
,
( )6

,
{

(
)
}
(
)
)
9 )
7
18. which gives
,
(

)
)
{(
, [
(
}
)](
)
Proceeding to limits we get (
)
( )
(
)
In the consequence of Theorem 3.1 with (3.25) and (3.26) we obtain the result immediately. Now to prove sufficiency, assume that the condition (3.21) is satisfied. Then it follows that 6
(
)
7
(
)
It gives 6
(
( ) This relation and the estimate is an entire transcendental function. Hence the proof is completed.
(
)
7
(
)
) yield the relation (3.22). It follows that
(
)
REFERENCES [1] [2]
Bose, S.K. and Sharma, D., Integral functions of two complex variables, Compositio Math. 15 (1963), 210226. Ganti, R. and Srivastava, G.S., Approximation of entire functions of two complex variables in Banach spaces, J. Inequalities in Pure & Applied Mathematics, 7, Issue 2, (2006), 111.
[3]
Juneja, O.P., Kapoor, G.P. and Bajpai, S.K., On the
[4]
Angew. Math. 282 (1976), 5367. Vakarchuk, S.B. and Zhir, S.I., On some problems of polynomial approximation of entire transcendental functions, Ukarainian Mathematical Journal, (54) (9) (2002), 13931401.
( p, q) order and lower ( p, q) order of an entire function, J. Reine
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