Abstract and Applied Analysis

Volume 2013 (2013), Article ID 845146, 9 pages

http://dx.doi.org/10.1155/2013/845146

Research Article

Best Polynomial Approximation in -Norm and -Growth of Entire Functions

Mohamed El Kadiri¹ and Mohammed Harfaoui²

¹Department of Mathematics, University Mohammed V-Agdal, 4 Avenue Ibn Battouta, BP 1014 RP, Rabat, Morocco

²Laboratory of Mathematics, Cryptography and Mechanical, F.S.T., University Hassan II Mohammedia, BP 146, 20650 Mohammedia, Morocco

Received 7 March 2012; Revised 16 December 2012; Accepted 16 December 2012

Academic Editor: Natig M. Atakishiyev

Copyright © 2013 Mohamed El Kadiri and 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 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 Siciak’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 ,

Siciak’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-Ampère equation (see [12]):

If , the Monge-Ampère 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 Siciak’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.

 () Bernstein-Markov inequality: for all , there exists a constant such that

 for every polynomial of complex variables of degree at most . () 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]):

 () ,  () and , where .

For , put, for and ,

Lemma 3 (see [2]). With the above notations one has the following results:

 () and ,   () ,   () ,  ()

  ()

 ()

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 of   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 Hölder inequality, that we have for any

for every .

Step 2. If , let such , then . By the Hölder 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

But on , thus by Lemma 11 we have .

Conversely, suppose now that is a function of such that the relation (105) holds.

(1) Let , then we have because , () and is a basis of as in Section 3. Consider in the series . By (90) of Lemma 12 one may easily check that this series converges normally on every compact subset of to an entire function denoted (this result is a direct consequence of the inequality (BM) and the inequality on coefficients ). We have obviously -a.s on , and by Theorem 8, the -order of is

By Lemma 12 we check that so the proof is completed for .

(2) Now let and , by (BM) inequality and Hölder inequality we have again the inequality (96) and (102), and by the previous arguments we obtain the result.

The proof of the second assertion follows in a similar way of the proof of the first assertion with the help of Theorem 8 and the arguments discussed above, hence we omit the details.

Remark 14. (1) If and , using the results of Theorem 13 we obtain the result of Winiarski (see [6]):

(2) If , , and , using the results of Theorem 13 we obtain the result of Nguyen (see [13]):

Remark 15. The above result holds for (see [14]).

Let ; of course, for , the -norm does not satisfy the triangle inequality. But our relations (92) and relation (102) are also satisfied for , because by using Holder’s inequality we have, for some and all ( fixed),

Using the inequality

we get

We deduce that satisfies the Bernstein-Markov inequality. For there is a constant such that for all (analytic) polynomials we have

Thus if satisfies the Bernstein-Markov inequality for one , then (92) and (95) are satisfied for all .

References

R. P. Boas, Entire Functions, Academic Press, New York, NY, USA, 1954.

S. K. Bajpai, O. P. Juneja, and G. P. Kapoor, “On the -order and lower -order of an entire function,” Journal für die Reine und Angewandte Mathematik, vol. 282, pp. 53–67, 1976.

S. N. Bernstein, Lessons on the Properties and Extremal Best Approximation of Analytic Functions of One Real Variable, Gautier-Villars, Paris, Farnce, 1926.

A. R. Reddy, “Approximation of an entire function,” Journal of Approximation Theory, vol. 3, pp. 128–137, 1970.

A. R. Reddy, “Best polynomial approximation to certain entire functions,” Journal of Approximation Theory, vol. 5, pp. 97–112, 1972.

T. Winiarski, “Approximation and Interpolation of entire functions,” Annales Polonici Mathematici, vol. 23, pp. 259–273, 1970.

M. Harfaoui, “Generalized order and best approximation of entire function in -norm,” International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 360180, 15 pages, 2010.

A. Zeriahi, “Families of almost everywhere bounded polynomials,” Bulletin des Sciences Mathématiques, vol. 107, no. 1, pp. 81–91, 1983.

J. Siciak, “On some extremal functions and their applications in the theory of analytic functions of several complex variables,” Transactions of the American Mathematical Society, vol. 105, pp. 322–357, 1962.

J. Siciak, “Extremal plurisubharmonic functions in ,” in Proceedings of the First Finnish-Polish Summer School in Complex Analysis, pp. 115–152, University of Lodz, Lodz, Poland, 1977.

A. Zeriahi, “Best increasing polynomial approximation of entire functions on affine algebraic varieties,” Annales de l'Institut Fourier, vol. 37, no. 2, pp. 79–104, 1987.

E. Bedford and B. A. Taylor, “A new capacity for plurisubharmonic functions,” Acta Mathematica, vol. 149, no. 1-2, pp. 1–40, 1982.

T. V. Nguyen, “Growth and best polynomial approximation of entire functions of several variables,” Annales Polonici Mathematici, vol. 24, pp. 325–333, 1982.

D. Kumar, “Generalized growth and best approximation of entire functions in -norm in several complex variables,” Annali dell'Universitá di Ferrara VII, vol. 57, no. 2, pp. 353–372, 2011.