#### Abstract

The classical growth has been characterized in terms of approximation errors for a continuous function on by Reddy (1970), and a compact of positive capacity by Nguyen (1982) and Winiarski (1970) with respect to the maximum norm. The aim of this paper is to give the general growth (-growth) of entire functions in by means of the best polynomial approximation in terms of -norm, with respect to the set , where is the Siciak's extremal function on an -regular nonpluripolar compact is not pluripolar.

#### 1. Introduction

Let be a nonconstant entire function and . It is well known that the function is indefinitely increasing convex function of . To estimate the growth of precisely, Boas (see [1]) has introduced the concept of order, defined by the number ():

The concept of type has been introduced to determine the relative growth of two functions of the same nonzero finite order. An entire function, of order , , is said to be of type , , if

If is an entire function of infinite or zero order, the definition of type is not valid and the growth of such function cannot be precisely measured by the above concept. Bajpai et al. (see [2]) have introduced the concept of index-pair of an entire function. Thus, for , they have defined the number , where if and if , where , and , for .

The function is said to be of index-pair if is nonzero finite number. The number is called the -order of .

Bajpai et al. have also defined the concept of the -type , for , by

In their works, the authors established the relationship of -growth of with respect to the coefficients in the Maclaurin series of .

We have also many results in terms of polynomial approximation in classical case. Let be a compact subset of the complex plane of positive logarithmic capacity and a complex function defined and bounded on . For , put where the norm is the maximum on and is the th Chebytchev polynomial of the best approximation to on .

Bernstein showed (see [3, page 14]), for , that there exists a constant such that is finite, if and only if is the restriction to of an entire function of order and some finite type.

This result has been generalized by Reddy (see [4, 5]) as follows: if and only if is the restriction to of an entire function of order and type for .

In the same way Winiarski (see [6]) generalized this result to a compact of the complex plane of positive logarithmic capacity, denoted as follows.

If is a compact subset of the complex plane , of positive logarithmic capacity, then if and only if is the restriction to of an entire function of order () and type .

Recall that the capacity of is and the capacity of a unit disc is .

The authors considered, respectively, the Taylor development of with respect to the sequence and the development of with respect to the sequence defined by where is the th extremal points system of (see [6, page 260]).

We remark that the above results suggest that the rate at which the sequence tends to zero depends on the growth of the entire function (order and type).

Harfaoui (see [7]) obtained a result of generalized order in terms of approximation in -norm for a compact of .

The aim of this paper is to generalize the growth (-order and -type), studied by Reddy (see [4, 5]) and Winiarski (see [6]), in terms of approximation in -norm for a compact of satisfying some properties which will be defined later.

We also obtain a general result of Harfaoui (see [7]) in term of -order and -type for the functions

So we establish relationship between the rate at which , for , tends to zero in terms of best approximation in -norm, and the generalized growth of entire functions of several complex variables for a compact subset of , where is a compact well selected and where is the family of all polynomials of degree and is the well selected measure (the equilibrium measure associated to a -regular compact ) (see [8]) and , , is the class of all functions such that

In this work we give the generalization of these results in , replacing the circle by the set , where is the Siciakâ€™s extremal function of , a compact of satisfying some properties (see [9, 10]), and using the development of with respect to the sequence constructed by Zeriahi (see [11]).

Recall that in the paper of Winiarski (see [6]) the author used the Cauchy inequality. In our work we replace this inequality by an inequality given by Zeriahi (see [11]).

#### 2. Definitions and Notations

Before we give some definitions and results which will be frequently used in this paper, let be a compact of and let denote the maximum norm on .

Multivariate polynomial inequalities are closely related to the Siciak extremal function associated with a compact subset of ,

Siciakâ€™s function establishes an important link between polynomial approximation in several variables and pluripotential theory.

It is known (see [10]) that where is the Lelong class of plurisubharmonic functions with logarithmic growth at infinity. If is nonpluripolar (i.e., there is no plurisubharmonic function such that ), then the plurisubharmonic function is the unique function in the class which vanishes on except perhaps for a pluripolar subset and satisfies the complex Monge-AmpÃ¨re equation (see [12]):

If , the Monge-AmpÃ¨re equation reduces to the classical Laplace equation.

For this reason, the function is considered as a natural counterpart of the classical Green function with logarithmic pole at infinity and it is called the pluricomplex Green function associated with .

*Definition 1 (Siciak [10]). *The function
is called the Siciakâ€™s extremal function of the compact .

*Definition 2. *A compact in is said to be -regular if the extremal function, , associated to is continuous on .

Regularity is equivalent to the following Bernstein-Markov inequality (see [9]).

For any , there exists an open such that for any polynomial

In this case we take .

Regularity also arises in polynomial approximation. For , we let where is the set of polynomials of degree at most . Siciak showed that (see [10]).

If is -regular, then if and only if has an analytic continuation to

It is known that if is a compact -regular of , there exists a measure , called extremal measure, having interesting properties (see [9, 10]), in particular, we have the following properties.â€‰() Bernstein-Markov inequality: for all , there exists a constant such that â€‰for every polynomial of complex variables of degree at most .â€‰() Bernstein-Walsh (BW) inequality: for every set -regular and every real we have

Note that the regularity is equivalent to the Bernstein-Markov inequality.

Let be a bijection such that

Zeriahi (see [11]) has constructed according to the Hilbert-Schmidt method a sequence of monic orthogonal polynomials according to an extremal measure (see [9]), , called extremal polynomial, defined by such that We need the following notations and lemma which will be used in the sequel (see [2]): â€‰() ,â€‰ () and , where .

For , put, for and ,

Lemma 3 (see [2]). *With the above notations one has the following results:*â€‰*() and , *â€‰* () ,*â€‰â€‰* () ,*â€‰â€‰*()
*â€‰â€‰*()
*â€‰*()
*

For more details of these results, see [2].

*Definition 4. *Let be a compact -regular and put

An entire function is said to be of -order if it is of index-pair such that

If , the -type is defined by
with if and and .

#### 3. -Growth in terms of the Coefficients of the Development with respect to Extremal Polynomials

The object of this section is to establish the relationship of -growth of an entire function with respect to the set and the coefficients of entire function on of the development with respect to the sequence of extremal polynomials.

The -growth of an entire function is defined by -order and -type of .

Let be the basis of extremal polynomials associated to the set defined by (25). Recall that is a basis of the vector space of entire functions, hence if is an entire function, then

To prove the aim result of this section we need Brernstein-Walsh inequality and the following lemmas which have been proved by Zeriahi (see [11]).

Lemma 5. *Let be a compact -regular subset ofâ€‰â€‰ and let be an entire function such that . Then for every , there exists an integer and a constant such that
**
where and are constant not depending on . *

Lemma 6. *If is an -regular, then the sequence of extremal polynomials satisfies
**
for every , and
*

Recall that the second assertion (37) of Lemma 5 replaces the Cauchy inequality for complex function defined on the complex plane .

Theorem 7. *Let be an entire function. Then is said of a finite -order if and only if
**
and , where
**
for with . *

*Proof. *Put . Let us prove that . If is of finite -order , then we have
Thus for every there exists such that for every
Using the inequalities (37) of Lemma 5 and (39) of Lemma 6, one has, for every , there exist and such that for every and
for and . But for and we have

Then, by proceeding to limits as , we get for sufficiently large

(i) For with , let
Then if we replace in the equality (46) by , we get easily that for sufficiently large
After passing to the upper limit, we get for

(ii) For , the inequality (46) gives (because for ).

(iii) For , choose and in the same way we show that
for sufficiently large, thus

By combining (i), (ii), and (iii) we have . This result holds obviously if .

We prove now reverse inequality . By the definition of , for every there exists such that for every
where , for simplification.

Let be a positive integer such that, for ,
and , by (39), (37), (BM) and (BW) inequalities, there exists , such that

Indeed,
(because satisfies ).

By (38) and (39), for sufficiently large we have

Therefore, for sufficiently large we have
where is a polynomial of degree not exceeding . By using (46) we get

By (52) the series (1) is convergent, and (2) is obviously convergent, hence we have for sufficiently large
where is a constant. Thus, for sufficiently large we obtain
Therefore, for sufficiently large

For sufficiently large let
where is the integer part of . Replacing in the inequality (61) we get

To prove the result we proceed in three steps.*Step *1. For , we have
Then
Proceeding to the upper limit we get .*Step *2. For , since , we get .*Step *3. For , the relation is equivalent to
Then, for sufficiently large
Passing to the upper limit after division by we obtain .

By combining (i), (ii), and (iii) we obtain for , . The inequality is obviously true for .

Theorem 8. *Let be an entire function of -order . Then is of finite -type if and only if
**
and , where if , if , if , if , , and
*

*Proof. *Let us first prove that . By the definition of , for every there exists such that for every ,

Let be a positive integer such that
and . By the estimate (39) from Lemma 6 and the (BM) and (BW) inequalities, there exists such that
Put
where

By repeating the argument used in the proof of Theorem 7 one may easily check that

For example we will show that . By the relation (69) we have

The maximum of the function is reached for , where is the solution of the equation

For the relation (75) becomes
Thus
Therefore and and by (69) we get
which gives for sufficiently large
Passing to the upper limit when we get

For we have
Therefore
Hence
Then, for sufficiently large,
We obtain the result after passing to the upper limit.

*Remark 9. *If and , we know that
Then by using Theorems 7 and 8 we get
which gives the result of Winiarski.

*Remark 10. *The notion of the type associated to a compact in was considered by Nguyen (see [13]). In this work the concept of the general type seems to be a new result for a compact in , (), which is not Cartesian product. Also the generalized order is independent of the norm but not the generalized type.

#### 4. Best Polynomial Approximation in terms of -Norm

Let be a bounded function defined on a -regular compact of .

The object of this section is to study the relationship between the rate of the best polynomial approximation of in -norm and the -growth of an entire function such that .

To our knowledge, no similar result is known according to polynomial approximation in -norm () with respect to a measure on in . To prove the aim results we use the results obtained in the second section to give relationship between the general growth of and the sequence which extend the classical results of Reddy and Winiarsk in . We need the following lemmas.

Lemma 11. *If is compact -regular in , then every function can be written in the form
**
where is the closed subspace of generated by the restrictions to of polynomials and is the sequence defined by (25). *

Lemma 12. *Let be the sequence defined by (25) and an element of , for , then
*

*Proof of Lemma 12. *The proof is done in two steps ( and ). Let be an element of .*Step *1. If with , then with convergence in , where , and therefore (because ). Since , we obtain easily, using Bernstein-Walsh inequality and de HÃ¶lder inequality, that we have for any
for every .*Step *2. If , let such , then . By the HÃ¶lder inequality we have
But , therefore, by the (BM) inequality, we have
Hence
Thus in both cases we have
where is a constant which depends only on . After passing to the upper limit (96) gives
To prove the other inequality we consider the polynomial of degree
then
By the Bernstein-Walsh inequality we have
for and . If we take as a common factor , the other factor is convergent, thus we have
and by (39) of Lemma 6 we have then
We then deduce that
This inequality is a direct consequence of (102) and the inequality on coefficients given by

Applying the above lemma we get the following main result.

Theorem 13. *Let be an element of , then*(1)* is the restriction to of an entire function in of finite -order if and only if
and .*(2)* is the restriction to of an entire function of -order () and of -type if and only if
and , where if and if .*

*Proof. *Suppose that is -a.s the restriction to of an entire function of -order () and show that . We have , and = in , where , . From (40) of Theorem 7 we get , where