Best Polynomial Approximation in Lp -Norm and (p,q) -Growth of Entire Functions

Abstract

The classical growth has been characterized in terms of approximation errors for a continuous function on [ -1,1 ] by Reddy (1970), and a compact K 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 ( (p ,q ) -growth) of entire functions in [superscript] ... n[/superscript] by means of the best polynomial approximation in terms of [superscript] L p[/superscript] -norm, with respect to the set [subscript] Ω r[/subscript] = {z ∈[superscript] C n[/superscript] ;exp[subscript] V K[/subscript] (z ) ...4;r } , where [subscript] V K[/subscript] =sup { (1/d)log |[subscript] P d[/subscript] | ,[subscript] P d[/subscript] polynomial of degree ...4;d , ||[subscript] P d[/subscript] [subscript] || K[/subscript] ...4;1 } is the Siciak's extremal function on an L -regular nonpluripolar compact K is not pluripolar.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Mohamed El Kadiri 1 and Mohammed Harfaoui 2

Recommended by Natig M. Atakishiyev

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

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.

1. Introduction

Let f (z ) = ^{∑ k =0 + ∞} ... _{a k} ^{z _{λ k}} be a nonconstant entire function and M (f ,r ) = _{max |z | =r} |f (z ) | . It is well known that the function r ...log (M (f ,r ) ) is indefinitely increasing convex function of log (r ) . To estimate the growth of f precisely, Boas (see [ 1]) has introduced the concept of order, defined by the number ρ ( 0 ...4; ρ ...4; + ∞ ): [figure omitted; refer to PDF]

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 ρ , 0 < ρ < + ∞ , is said to be of type σ , 0 ...4; σ ...4; + ∞ , if [figure omitted; refer to PDF]

If f 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 p ...5;q ...5;1 , they have defined the number [figure omitted; refer to PDF] b ...4; ρ (p ,q ) ...4; + ∞ , where b =0 if p >q and b =1 if p =q , where ^{log [0 ]} (x ) =x , and ^{log [p ]} (x ) =log ( ^{log [p -1 ]} (x ) ) , for p ...5;1 .

The function f is said to be of index-pair (p ,q ) if ρ (p -1 ,q -1 ) is nonzero finite number. The number ρ (p ,q ) is called the (p ,q ) -order of f .

Bajpai et al. have also defined the concept of the (p ,q ) -type σ (p ,q ) , for b < ρ (p ,q ) < + ∞ , by [figure omitted; refer to PDF]

In their works, the authors established the relationship of (p ,q ) -growth of f with respect to the coefficients _{a k} in the Maclaurin series of f .

We have also many results in terms of polynomial approximation in classical case. Let K be a compact subset of the complex plane ... of positive logarithmic capacity and f a complex function defined and bounded on K . For k ∈ ... , put [figure omitted; refer to PDF] where the norm || · _{|| K} is the maximum on K and _{T k} is the k th Chebytchev polynomial of the best approximation to f on K .

Bernstein showed (see [ 3, page 14]), for K = [ -1,1 ] , that there exists a constant ρ >0 such that [figure omitted; refer to PDF] is finite, if and only if f is the restriction to K of an entire function of order ρ and some finite type.

This result has been generalized by Reddy (see [ 4, 5]) as follows: [figure omitted; refer to PDF] if and only if f is the restriction to K of an entire function g of order ρ and type σ for K = [ -1,1 ] .

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

If K is a compact subset of the complex plane ... , of positive logarithmic capacity, then [figure omitted; refer to PDF] if and only if f is the restriction to K of an entire function of order ρ ( 0 < ρ < + ∞ ) and type σ .

Recall that the capacity of [ -1,1 ] is cap ( [ -1,1 ] ) =1 /2 and the capacity of a unit disc is cap (D (0,1 ) ) =1 .

The authors considered, respectively, the Taylor development of f with respect to the sequence ( _{z n ) n} and the development of f with respect to the sequence ( _{W n ) n} defined by [figure omitted; refer to PDF] where ^{η (n )} = ( _{η n0} , _{η n1} , ... , _{η nn} ) is the n th extremal points system of K (see [ 6, page 260]).

We remark that the above results suggest that the rate at which the sequence ( _{E k} (K ,f )k _{) k} 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 ^{L p} -norm for a compact of ^{... n} .

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

We also obtain a general result of Harfaoui (see [ 7]) in term of (p ,q ) -order and (p ,q ) -type for the functions [figure omitted; refer to PDF]

So we establish relationship between the rate at which ( ^{π k p} (K ,f ) ^{) 1 /k} , for k ∈ ... , tends to zero in terms of best approximation in ^{L p} -norm, and the generalized growth of entire functions of several complex variables for a compact subset K of ^{... n} , where K is a compact well selected and [figure omitted; refer to PDF] where _{...AB; k} ( ^{... n} ) is the family of all polynomials of degree ...4;k and μ is the well selected measure (the equilibrium measure μ = (d ^{d c} _{V K} ^{) n} associated to a L -regular compact K ) (see [ 8]) and ^{L p} (K , μ ) , p ...5;1 , is the class of all functions such that [figure omitted; refer to PDF]

In this work we give the generalization of these results in ^{C n} , replacing the circle {z ∈ ... ; |z | =r } by the set {z ∈ ^{... n} ; exp ( _{V K} (z ) ) <r } , where _{V K} is the Siciak's extremal function of K , a compact of ^{... n} satisfying some properties (see [ 9, 10]), and using the development of f with respect to the sequence ( _{A k ) k ∈ ...} 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 K be a compact of ^{... n} and let || · _{|| K} denote the maximum norm on K .

Multivariate polynomial inequalities are closely related to the Siciak extremal function associated with a compact subset K of ^{... n} , [figure omitted; refer to PDF]

Siciak's function establishes an important link between polynomial approximation in several variables and pluripotential theory.

It is known (see [ 10]) that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the Lelong class of plurisubharmonic functions with logarithmic growth at infinity. If K is nonpluripolar (i.e., there is no plurisubharmonic function u such that K ⊂ {u (z ) = - ∞ } ), then the plurisubharmonic function ^{V K *} (z ) =lim _{sup w [arrow right]z V K} (w ) is the unique function in the class [Lagrangian (script capital L)] ( ^{... n} ) which vanishes on K except perhaps for a pluripolar subset and satisfies the complex Monge-Ampère equation (see [ 12]): [figure omitted; refer to PDF]

If n =1 , the Monge-Ampère equation reduces to the classical Laplace equation.

For this reason, the function ^{V K *} 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 K .

Definition 1 (Siciak [ 10]).

The function [figure omitted; refer to PDF] is called the Siciak's extremal function of the compact K .

Definition 2.

A compact K in ^{... n} is said to be L -regular if the extremal function, _{V K} , associated to K is continuous on ^{... n} .

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

For any ... >0 , there exists an open U ⊃K such that for any polynomial P [figure omitted; refer to PDF]

In this case we take U = {z ∈ ^{... n} ; _{V K} (z ) < ... } .

Regularity also arises in polynomial approximation. For f ∈ ...9E; (K ) , we let [figure omitted; refer to PDF] where _{...AB; k} ( ^{... n} ) is the set of polynomials of degree at most d . Siciak showed that (see [ 10]).

If K is L -regular, then [figure omitted; refer to PDF] if and only if f has an analytic continuation to [figure omitted; refer to PDF]

It is known that if K is a compact L -regular of ^{... n} , there exists a measure μ , called extremal measure, having interesting properties (see [ 9, 10]), in particular, we have the following properties.

: ( _{P 1} ) Bernstein-Markov inequality: for all ... >0 , there exists a constant C = _{C [straight epsilon]} such that [figure omitted; refer to PDF]

: for every polynomial of n complex variables of degree at most d .

: ( _{P 2} ) Bernstein-Walsh (BW) inequality: for every set L -regular K and every real r >1 we have [figure omitted; refer to PDF]

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

Let α : ... [arrow right] ^{... n} ,k ... α (k ) = ( _{α 1} (k ) , ... , _{α n} (k ) ) be a bijection such that [figure omitted; refer to PDF]

Zeriahi (see [ 11]) has constructed according to the Hilbert-Schmidt method a sequence of monic orthogonal polynomials according to an extremal measure (see [ 9]), ( _{A k ) k} , called extremal polynomial, defined by [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] We need the following notations and lemma which will be used in the sequel (see [ 2]):

: ( _{N 1} ) _{ν k} = _{ν k} (K ) = || _{A k || ^{L 2} (K , μ )} ,

: ( _{N 2} ) _{a k} = _{a k} (K ) = || _{A k || K} = _{max z ∈K} | _{A k} (z ) | and _{τ k} = ( _{a k} ^{) 1 / _{s k}} , where _{s k} =deg ( _{A k} ) .

For p ∈ ... , put, for p ...5;1 and x >0 , [figure omitted; refer to PDF]

Lemma 3 (see [ 2]).

With the above notations one has the following results:

: ( RR1 ) _{E [ -p ]} (x ) =x / _{⋀ [ p -1 ]} (x ) and _{⋀ [ -p ]} (x ) =x / _{E [ p -1 ]} (x ) ,

: ( RR2 ) (d /dx ) ^{exp [ p ]} ( x ) = _{E [ p ]} (x ) /x =1 / _{⋀ [ -p -1 ]} (x ) ,

: ( RR3 ) (d /dx ) ^{log [ p ]} ( x ) = _{E [ -p ]} (x ) /x =1 / _{⋀ [ p -1 ]} (x ) ,

: ( RR4 ) [figure omitted; refer to PDF]

: ( RR5 ) [figure omitted; refer to PDF]

: ( RR6 ) [figure omitted; refer to PDF]

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

Definition 4.

Let K be a compact L -regular and put [figure omitted; refer to PDF]

An entire function f is said to be of (K ,p ,q ) -order _{ρ K} (p ,q ) if it is of index-pair (p ,q ) such that [figure omitted; refer to PDF]

If ρ ∈ [ β , + ∞ [ , the (K ,p ,q ) -type is defined by [figure omitted; refer to PDF] with β =1 if p =q and β =0 and p >q .

3. (p ,q ) -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 (p ,q ) -growth of an entire function with respect to the set [figure omitted; refer to PDF] and the coefficients of entire function f on ^{... n} of the development with respect to the sequence of extremal polynomials.

The (p ,q ) -growth of an entire function is defined by (K ,p ,q ) -order and (K ,p ,q ) -type of f .

Let ( _{A k ) k} be the basis of extremal polynomials associated to the set K defined by ( 25). Recall that ( _{A k ) k} is a basis of the vector space of entire functions, hence if f is an entire function, then [figure omitted; refer to PDF]

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 K be a compact L -regular subset of ^{... n} and let f be an entire function such that f = ^{∑ k =0 + ∞} ... _{f k A k} . Then for every θ >1 , there exists an integer _{N θ} ...5;1 and a constant _{C θ} such that [figure omitted; refer to PDF] where _{N θ} ∈ ... and _{C θ} >0 are constant not depending on (r ,k ,f ) .

Lemma 6.

If K is an L -regular, then the sequence of extremal polynomials ( _{A k ) k} satisfies [figure omitted; refer to PDF] for every z ∈ ^{... n} , and [figure omitted; refer to PDF]

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

Theorem 7.

Let f = _{∑ k ...5;1} ... _{f k A k} be an entire function. Then f is said of a finite (K ,p ,q ) -order _{ρ K} (p ,q ) if and only if [figure omitted; refer to PDF] and _{ρ K} (p ,q ) = _{P 1} (L (p ,q ) ) , where [figure omitted; refer to PDF] for (p ,q ) ∈ ^{... 2} with p ...5;q .

Proof.

Put ρ = _{ρ K} (p ,q ) . Let us prove that [varrho] ...5; _{P 1} (L (p ,q ) ) . If f is of finite (p ,q ) -order ρ , then we have [figure omitted; refer to PDF] Thus for every [straight epsilon] >0 there exists r ( [straight epsilon] ) such that for every r >r ( [straight epsilon] ) [figure omitted; refer to PDF] Using the inequalities ( 37) of Lemma 5and ( 39) of Lemma 6, one has, for every [straight epsilon] >0 , there exist r ( [straight epsilon] ) and k ( [straight epsilon] ) such that for every r >r ( [straight epsilon] ) and k >k ( [straight epsilon] ) [figure omitted; refer to PDF] for r >r ( [straight epsilon] ) and k >k ( [straight epsilon] ) . But for r >r ( [straight epsilon] ) and k >k ( [straight epsilon] ) we have [figure omitted; refer to PDF]

Then, by proceeding to limits as k [arrow right] ∞ , we get for r sufficiently large [figure omitted; refer to PDF]

(i) For (p ,q ) ...0; (2,2 ) with p >q , let [figure omitted; refer to PDF] Then if we replace in the equality ( 46) r by _{r k} , we get easily that for k sufficiently large [figure omitted; refer to PDF] After passing to the upper limit, we get for p >q [figure omitted; refer to PDF]

(ii) For 3 ...4;p =q < + ∞ , the inequality ( 46) gives ρ ...5;max (1 ,L (p ,q ) ) (because ρ ...5;1 for p =q ).

(iii) For (p ,q ) = (2,2 ) , choose _{r k} = (1 / θ )exp ^{( _{s k} / ( ρ + [straight epsilon] ) ) 1 / ( ρ -1 + [straight epsilon] )} >r ( [straight epsilon] ) and in the same way we show that [figure omitted; refer to PDF] for k sufficiently large, thus [figure omitted; refer to PDF]

By combining (i), (ii), and (iii) we have ρ ...5; _{P 1} (L (p ,q ) ) . This result holds obviously if ρ = + ∞ .

We prove now reverse inequality ρ ...4; _{P 1} (L (p ,q ) ) . By the definition of L (p ,q ) , for every [straight epsilon] >0 there exists k ( [straight epsilon] ) such that for every k ...5;k ( [straight epsilon] ) [figure omitted; refer to PDF] where L =L (p ,q ) , for simplification.

Let k (r ) be a positive integer such that, for k ...5;k (r ) , [figure omitted; refer to PDF] and k (r ) >k ( [straight epsilon] ) , by ( 39), ( 37), (BM) and (BW) inequalities, there exists _{k 0} ∈ ... , such that [figure omitted; refer to PDF]

Indeed, [figure omitted; refer to PDF] (because z satisfies exp ( _{V K} (z ) ) ...4;r ).

By ( 38) and ( 39), for k sufficiently large we have [figure omitted; refer to PDF]

Therefore, for r sufficiently large we have [figure omitted; refer to PDF] where _{A k ( [straight epsilon] )} is a polynomial of degree not exceeding _{k [straight epsilon]} . By using ( 46) we get [figure omitted; refer to PDF]

By ( 52) the series (1) is convergent, and (2) is obviously convergent, hence we have for r sufficiently large [figure omitted; refer to PDF] where _{A 1} is a constant. Thus, for r sufficiently large we obtain [figure omitted; refer to PDF] Therefore, for r sufficiently large [figure omitted; refer to PDF]

For r sufficiently large let [figure omitted; refer to PDF] where E (x ) is the integer part of x . Replacing _{s k} in the inequality ( 61) we get [figure omitted; refer to PDF]

To prove the result we proceed in three steps.

Step 1. For (p ,q ) = (2,2 ) , we have [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] Proceeding to the upper limit we get ρ ...4;1 +L (2,2 ) = _{P 1} (L (2,2 ) ) .

Step 2. For 3 ...4;p =q , since ρ ...5;1 , we get ρ ...4;max (1 (L (p ,p ) ) = _{P 1} (L (p ,q ) ) .

Step 3. For p <q , the relation ( * ) is equivalent to [figure omitted; refer to PDF] Then, for r sufficiently large [figure omitted; refer to PDF] Passing to the upper limit after division by ^{log [q ]} (2r ) we obtain ρ ...4;L (p ,q ) .

By combining (i), (ii), and (iii) we obtain for p ...5;q ...5;1 , ρ ...4; _{P 1} (L (p ,q ) ) . The inequality is obviously true for L (p ,q ) = + ∞ .

Theorem 8.

Let f be an entire function of (K ,p ,q ) -order _{ρ K} (p ,q ) ∈ ] β , ∞ [ . Then f is of finite (K ,p ,q ) -type _{σ K} (p ,q ) if and only if [figure omitted; refer to PDF] and _{σ K} (p ,q ) = γ (p ,q )M (p ,q ) , where β =1 if p =q , β =0 if p >q , C =1 if p =q =2 , C =0 if (p ,q ) ...0; (2,2 ) , _{s k} =deg ( _{A k} ) , and [figure omitted; refer to PDF]

Proof.

Let us first prove that _{σ K} (p ,q ) ...4;M ( p ,q ) · γ (p ,q ) . By the definition of γ = γ (p ,q ) , for every [straight epsilon] >0 there exists k ( [straight epsilon] ) such that for every k ...5;k ( [straight epsilon] ) , [figure omitted; refer to PDF]

Let k (r ) be a positive integer such that [figure omitted; refer to PDF] and k ...5;k (r ) . By the estimate ( 39) from Lemma 6and the (BM) and (BW) inequalities, there exists _{k 0} ∈ ... such that [figure omitted; refer to PDF] Put [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

By repeating the argument used in the proof of Theorem 7one may easily check that [figure omitted; refer to PDF]

For example we will show that _{σ K} (p ,1 ) ...4;M (p ,1 ) γ (p ,1 ) . By the relation ( 69) we have [figure omitted; refer to PDF]

The maximum of the function x ... ω (x ,r ) is reached for x = _{x r} , where _{x r} is the solution of the equation [figure omitted; refer to PDF]

For p =2 the relation ( 75) becomes [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Therefore H (r ) = ω (r , _{x r} ) = _{x r} / ρ and ω (r , _{x r} ) = (1 /e ρ ) ( γ + [straight epsilon] ) ( (1 + [straight epsilon] )r ^{) ρ} and by ( 69) we get [figure omitted; refer to PDF] which gives for r sufficiently large [figure omitted; refer to PDF] Passing to the upper limit when r [arrow right] + ∞ we get [figure omitted; refer to PDF]

For p ...5;3 we have [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] Then, for r sufficiently large, [figure omitted; refer to PDF] We obtain the result after passing to the upper limit.

Remark 9.

If n =1 and (p ,q ) = (2,1 ) , we know that [figure omitted; refer to PDF] Then by using Theorems 7and 8we get [figure omitted; refer to PDF] 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 ^{... n} , ( n ...5;2 ), 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 ^{L p} -Norm

Let f be a bounded function defined on a L -regular compact K of ^{... n} .

The object of this section is to study the relationship between the rate of the best polynomial approximation of f in ^{L p} -norm and the (p ,q ) -growth of an entire function g such that _{g |" K} =f .

To our knowledge, no similar result is known according to polynomial approximation in ^{L p} -norm ( 1 ...4;p ...4; ∞ ) with respect to a measure μ on K in ^{... n} . To prove the aim results we use the results obtained in the second section to give relationship between the general growth of f and the sequence [figure omitted; refer to PDF] which extend the classical results of Reddy and Winiarsk in ^{... n} . We need the following lemmas.

Lemma 11.

If K is compact L -regular in ^{... n} , then every function f ∈ ^{L ...AB; 2} (K , μ ) can be written in the form [figure omitted; refer to PDF] where ^{L ...AB; 2} (K , μ ) is the closed subspace of ^{L 2} (K , μ ) generated by the restrictions to E of polynomials ^{... n} and ( _{A k} ) is the sequence defined by ( 25).

Lemma 12.

Let ( _{A k} ) be the sequence defined by ( 25) and f = _{∑ k ...5;0} ... _{f k A k} an element of ^{L p} (K , μ ) , for p ...5;1 , then [figure omitted; refer to PDF]

Proof of Lemma 12.

The proof is done in two steps ( p ...5;2 and 1 <p <2 ). Let f = _{∑ k ...5;0} ... _{f k} · _{A k} be an element of ^{L p} (K , μ ) .

Step 1. If f ∈ ^{L p} (K , μ ) with p ...5;2 , then f = ^{∑ k =0 + ∞} ... _{f k} · _{A k} with convergence in ^{L 2} (K , μ ) , where _{f k} =1 / ^{ν k 2} _{∫ K} ...f _{A ¯ k} d μ , k ...5;0 and therefore _{f k} = (1 / ^{ν k 2} ) _{∫ K} ... (f - _{P s k -1} ) · _{A ¯ k} d μ (because deg ( _{A k} ) = _{s k} ). Since | _{f k} | ...4; (1 / ^{ν k 2} ) _{∫ K} ... | f - _{P s k -1} | · | _{A ¯ k} |d μ , we obtain easily, using Bernstein-Walsh inequality and de Hölder inequality, that we have for any [straight epsilon] >0 [figure omitted; refer to PDF] for every k ...5;0 .

Step 2. If 1 ...4;p <2 , let p [variant prime] such 1 /p +1 / p [variant prime] =1 , then p [variant prime] ...5;2 . By the Hölder inequality we have [figure omitted; refer to PDF] But || A k || L p [variant prime] (E , μ ) ...4;C || A k || K =C a k (K ) , therefore, by the (BM) inequality, we have [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] Thus in both cases we have [figure omitted; refer to PDF] where A [straight epsilon] is a constant which depends only on [straight epsilon] . After passing to the upper limit ( 96) gives [figure omitted; refer to PDF] To prove the other inequality we consider the polynomial of degree s k [figure omitted; refer to PDF] then [figure omitted; refer to PDF] By the Bernstein-Walsh inequality we have [figure omitted; refer to PDF] for k ...5;0 and p ...5;1 . If we take as a common factor (1 + [straight epsilon] ) s k ν k , the other factor is convergent, thus we have [figure omitted; refer to PDF] and by ( 39) of Lemma 6we have then [figure omitted; refer to PDF] We then deduce that [figure omitted; refer to PDF] This inequality is a direct consequence of ( 102) and the inequality on coefficients | f k | given by [figure omitted; refer to PDF]

Applying the above lemma we get the following main result.

Theorem 13.

Let f be an element of ^{L p} (K , μ ) , then

(1) f is μ -a ·s the restriction to K of an entire function in ^{... n} of finite (K ,p ,q ) -order ρ if and only if [figure omitted; refer to PDF] and ρ =L ( _{ρ 1} (p ,q ) ) .

(2) f is μ -a ·s the restriction to K of an entire function of (K ,p ,q ) -order ρ ( β < ρ < + ∞ ) and of (K ,p ,q ) -type σ (0 < σ < + ∞ ) if and only if [figure omitted; refer to PDF] and σ =M (p ,q ) ( _{σ 1} (p ,q ) ^{) ρ -C} , where C =0 if (p ,q ) = (2,2 ) and C =1 if p =q =2 .

Proof.

Suppose that f is μ -a.s the restriction to K of an entire function g of (K ,p ,q ) -order ρ ( β < ρ < + ∞ ) and show that ρ =L ( _{ρ 1} (p ,q ) ) . We have g ∈ ^{L p} (K , μ ) , p ...5;2 and g = _{∑ k ...5;0} ... _{g k} · _{A k} in ^{L 2} (K , μ ) , where _{g k} = (1 / _{ν k} ) _{∫ K} ...f _{A ¯ k} d μ , k ...5;0 . From ( 40) of Theorem 7we get ρ = _{P 1} (L (p ,q ) ) , where [figure omitted; refer to PDF]

But g =f on K , thus by Lemma 11we have ρ =L ( _{ρ 1} (p ,q ) ) .

Conversely, suppose now that f is a function of ^{L p} (K , μ ) such that the relation ( 105) holds.

(1) Let p ...5;2 , then we have f = _{∑ k ...5;0} ... _{f k A k} because f ∈ ^{L 2} (K , μ ) , ( ^{L p} (K , μ ) ⊂ ^{L 2} (K , μ ) ) and ( _{A k ) k} is a basis of ^{L 2} (K , μ ) as in Section 3. Consider in ^{... n} the series ∑ ... _{f k A k} . By ( 90) of Lemma 12one may easily check that this series converges normally on every compact subset of ^{... n} to an entire function denoted _{f 1} (this result is a direct consequence of the inequality (BM) and the inequality on coefficients | _{f k} | ). We have obviously _{f 1} =f μ -a.s on K , and by Theorem 8, the (K ,p ,q ) -order of _{f 1} is [figure omitted; refer to PDF] By Lemma 12we check that ρ ( _{f 1} ,p ,q ) = ρ so the proof is completed for p ...5;2 .

(2) Now let p ∈ [1,2 [ and f ∈ ^{L p} (K , μ ) , 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 8and the arguments discussed above, hence we omit the details.

Remark 14.

(1) If n =1 and (p ,q ) = (2,1 ) , using the results of Theorem 13we obtain the result of Winiarski (see [ 6]): [figure omitted; refer to PDF]

(2) If n =1 , p >2 , and q =1 , using the results of Theorem 13we obtain the result of Nguyen (see [ 13]): [figure omitted; refer to PDF]

Remark 15.

The above result holds for 0 <p <1 (see [ 14]).

Let 0 <p <1 ; of course, for 0 <p <1 , the ^{L p} -norm does not satisfy the triangle inequality. But our relations ( 92) and relation ( 102) are also satisfied for 0 <p <1 , because by using Holder's inequality we have, for some M >0 and all r >p ( p fixed), [figure omitted; refer to PDF]

Using the inequality [figure omitted; refer to PDF] we get [figure omitted; refer to PDF] We deduce that (E , μ ) satisfies the Bernstein-Markov inequality. For ... >0 there is a constant C =C ( ... ,p ) >0 such that for all (analytic) polynomials P we have [figure omitted; refer to PDF]

Thus if (E , μ ) satisfies the Bernstein-Markov inequality for one p >0 , then ( 92) and ( 95) are satisfied for all p >0 .

References

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