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International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 360180, 15 pages
http://dx.doi.org/10.1155/2010/360180
Research Article

Generalized Order and Best Approximation of Entire Function in -Norm

University Hassan II-Mohammedia, U.F.R Mathematics and Applications, F.S.T, BP 146, Mohammedia, Morocco

Received 7 March 2010; Accepted 26 December 2010

Academic Editor: Rodica Costin

Copyright © 2010 Mohammed Harfaoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this paper is the characterization of the generalized growth of entire functions of several complex variables by means of the best polynomial approximation and interpolation on a compact with respect to the set , where is the Siciak extremal function of a -regular compact .

1. Introduction

Let be a no constant entire function in complex plane and let It is well known that the function is convex and decreasing of . To estimate the growth of , the concept of order, defined by the number (), such that has been given (see [1]).

The concept of type has been introduced to establish the relative growth of two functions having the same nonzero finite order. So an entire function, in complex plane , 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 can not be precisely measured by the above concept. Bajpai and Juneja (see [2]) have introduced the concept of index-pair of an entire function. Thus, for , they have defined the number

It is easy to show that where if and if .

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

Bajpai and Juneja have also defined the concept of the -type , for , by

In their works, the authors have established the relationship of -growth of with respect to the coefficients in the Maclaurin series of in complex plane (for we obtain the classical case).

We have also many results in terms in polynomial approximation in classical case. Let be a compact subset of the complex plane , of positive logarithmic capacity and be 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 .

It is known (see [3]) that if and only if is the restriction to of an entire function in .

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]) has generalized this result for a compact of the complex plane , of positive logarithmic capacity noted as follows:

If be 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 and capacity of a disk unit is .

The authors considered the Taylor development of with respect to the sequence and the development of with respect to the sequence defined by where is the thth extremal points system of .

The aim of this paper is to establish relationship between the rate at which 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. In this work we give the generalization of these results in , replacing the circle by the set , where is the Siciak's extremal function of a compact of which will be defined later satisfying some properties.

2. Definitions and Notations

Before we give some definitions and results which will be frequently used in this paper.

Definition 2.1 (see Siciak [7]). Let be a compact set in and let denote the maximum norm on . The function is called the Siciak's extremal function of the compact .

Definition 2.2. 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 [8]).

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 (see [7]) showed the following

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

Let be a function defined and bounded on . For put

where is the family of all polynomial of degree and μ the well-selected measure (The equilibrium measure associated to a -regular compact ) (see [9]) and , , is the class of all function such that:

For an entire function we establish a precise relationship between the general growth with respect to the set: and the coefficients of the development of with respect to the sequence , called extremal polynomial (see [10]). Therefore, we use these results to give the relationship between the generalized growth of and the sequence .

Recall that the subset of , , replaces the unit disc in the classical case.

It is known that if is an compact -regular of , there exists a measure μ, called extremal measure, having interesting properties (see [7, 8]), in particular, we have:()Bernstein-Markov Inequality. For all , there exists is a constant such that for every polynomial of complex variables of degree at most .()Bernstein-Waish (B.W) 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

Zériahi (see [10]) has constructed according to the Hilbert-Shmidt method a sequence of monic orthogonal polynomial according to a extremal measure (see [8]), , called extremal polynomial, defined by such that

We need the following notations which will be used in the sequel: (). () and , where .

With that notations and (B.W) inequality, we have where . For more details (see [9]).

Let α and β be two positives, strictly increasing to infinity differentiable functions to such that for every : where a function such that .

Assume that, for every , there exists two constants and such that for every : where means the differential of .

Definition 2.3. Let be a compact -regular, we put
If is an entire function we define the -order and the -type of (or generalized order and generalized type), respectively, by where .

Note that in the classical case and .

In this paper we will consider a more generalized growth to extend the classical results to a large class of entire functions of several variables.

We need the following lemma, see [10].

Lemma 2.4. Let be a compact -regular subset of . Then for every , there exists an integer and a constant such that: Let be an entire function. Then for every , there exists and such that for every and .

Note that the second assertion of the lemma is a consequence of the first assertion and it replaces Cauchy inequality for complex function defined on the complex plane .

3. Generalized Growth and Coefficients of the Development with Respect to Extremal Polynomial

The purpose of this section is to establish this relationship of the generalized growth of an entire function with respect to the set and the coefficient of entire function of the development with respect to the sequence of extremal polynomials.

Let be a basis of extremal polynomial associated to the set defined the relation (2.11). We recall that is a basis of (the set of entire functions on ). So if is an entire function then and we have the following results.

Theorem 3.1. If then the -order of is given by formula

To prove theorem we need the following lemmas.

Lemma 3.2. Let be a compact -regular subset of . Then for every the connected component of ,

Lemma 3.3. For every the maximum of the function is reached for solution of the equation

Consequence 3. This relation is equivalent to
Indeed, the relations give
Then
We verify easily with the relation that for every and .

Proof of Theorem 3.1. Put and show that .
(1) Show that .
By definition of , we have, for all , such that for all
From the second assertion of the Lemma 3.2, we have for every , there exists and such that or
But so for sufficiently large.
Then
Let be a real satisfying
Then
This is equivalent to or hence
 (2) Show that . According to the definition of , we have for every there exists such that for all for sufficiently large.
According to the first assertion of Lemma 3.2 and (BM) and (BW) inequalities, we have and for every entire function
The term (1) is a constant denoted , and
The series (4) is convergent.
Let, for sufficiently, where means the integer part of .
Then
Applying the the relation (3.13) with and if we replace by , we obtain: And so or
Thus and for sufficiently large, we have
But near to the infinity, thus
If we put , then and,
This is true for every hence . Thus the assertion is proved.

4. Best Approximation Polynomial in -Norm

To our knowledge, no similar result is known according to polynomial approximation in -norm () with respect to a measure μ on in .

The purpose of this paragraph is to give the relationship between the generalized order and speed of convergence to 0 in the best polynomial. We need the following lemma.

Lemma 4.1. Let an element of , for , then

Proof of Lemma 4.1. The proof is done in two steps and .Step 1. If where , then with convergence in , hence for and therefore (because ).
Since the relation, satisfied, is easily verified by using inequalities Bernstein-walsh and Holder that, we have for all for all .
Step 2. If , let such that , we have . According to the inequality of Hölder, we have:
But,
This shows, according to inequality (BM), that:
Hence the result
In both cases, we have therefore where is a constant which depends only on .
After passing to the upper limit in the relation (4.10) and Applying the relation (3.5) of the Lemma 3.2 we get
To prove the other inequality we consider the polynomial of degree , then
By Bernstein-Walsh inequality, we have for and . If we take as a common factor the other factor is convergent thus, we have and by (3.5) of Lemma 3.2, we have, then
We deduce
This inequality is a direct consequence of the relation (4.10) and the inequality on coefficients given by

Applying this lemma we get the following main result:

Theorem 4.2. Let , then is μ-almost-surely the restriction to of an entire function in of finite generalized order finite if and only if

Proof. Suppose that is μ-almost-surely the restriction to of an entire function of general order () and show that .
We have , and in Since is an element of then and according to the Theorem 3.1. and with the Lemma 4.1, we have
But on hence
Suppose now that is a function of such that the relation (4.19) is verified.
(1) Let , then , because is an element of ( is decreasing sequence).
Consider in the series , , we verify easily that this series converges normally on all compact of to an entire function denoted . We have , obviously, μ-almost surly on , and by Theorem 3.1, we have Applying the Lemma 4.1, we find
Consider the function , we have μ-almost surely for every in . Therefore the -order of is: (see Theorem 3.1).
By Lemma 4.1, we check so the proof is completed.

(2) Now let and .

By (BM) inequality and Hölder inequality, we have again the inequality the relation (4.10) and by the previous arguments we obtain the result.

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