Best Polynomial Approximation in 𝐿^𝑝-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 ℂ^𝑛 by means of the best polynomial approximation in terms of 𝐿^𝑝-norm, with respect to the set Ω_𝑟={𝑧∈𝐂^𝑛;exp𝑉_𝐾(𝑧)≤𝑟}, where 𝑉_𝐾=sup{(1/𝑑)log|𝑃_𝑑|,𝑃_𝑑polynomialofdegree≤𝑑,‖𝑃_𝑑‖_𝐾≤1} is the Siciak's extremal function on an 𝐿-regular nonpluripolar compact 𝐾 is not pluripolar.

1. Introduction

Let 𝑓(𝑧)=∑^+∞_𝑘=0𝑎_𝑘𝑧^𝜆𝑘 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≤𝜌≤+∞): 𝜌=limsup_𝑟→+∞loglog𝑀𝑓,𝑟(𝑟).log(1)

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≤𝜎≤+∞, if 𝜎=limsup_𝑟→+∞log𝑀𝑓,𝑟𝑟^𝜌.(2)

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≥𝑞≥1, they have defined the number 𝜌𝑝,𝑞=limsup_𝑟→+∞log^[𝑝]𝑀𝑓,𝑟log^[𝑞](𝑟),(3)𝑏≤𝜌(𝑝,𝑞)≤+∞, 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≥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 𝜎𝑝,𝑞=limsup_𝑟→+∞log^[𝑝−1]𝑀𝑓,𝑟log^[𝑞−1](𝑟)^{𝜌(𝑝,𝑞)}.(4)

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 ℂ of positive logarithmic capacity and 𝑓 a complex function defined and bounded on 𝐾. For 𝑘∈ℕ, put 𝐸_𝑘𝐾,𝑓=𝑓−𝑇_𝑘𝐾,(5) where the norm ‖⋅‖_𝐾 is the maximum on 𝐾 and 𝑇_𝑘 is the 𝑘th 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 lim_𝑘→+∞𝑘^1/𝜌𝑘𝐸_𝑘𝐾,𝑓(6) 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: lim_𝑘→+∞^𝑘𝐸_𝑘𝐾,𝑓=𝜌𝑒𝜎2^−𝜌(7) 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 ℂ of positive logarithmic capacity, denoted 𝑐=cap(𝐾) as follows.

If 𝐾 is a compact subset of the complex plane ℂ, of positive logarithmic capacity, then lim_𝑘→+∞𝑘𝐸_𝑘𝐾,𝑓^𝜌/𝑘=𝑐^𝜌𝑒𝜌𝜎(8) 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 𝑊_𝑛(𝑧)=^𝑗=𝑛_𝑗=1𝑧−𝜂_𝑛𝑗,𝑛=1,2,…,(9) where 𝜂^(𝑛)=(𝜂_𝑛0,𝜂_𝑛1,…,𝜂_𝑛𝑛) is the 𝑛th extremal points system of 𝐾 (see [6, page 260]).

We remark that the above results suggest that the rate at which the sequence (^𝑘𝐸_𝑘(𝐾,𝑓))_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 ℂ^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 ℂ^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 𝛼(𝑥)=log^𝑝−1(𝑥),𝛽(𝑥)=log^𝑞−1(𝑥)for𝑝,𝑞∈ℕ².(10)

So we establish relationship between the rate at which (𝜋^𝑝_𝑘(𝐾,𝑓))^1/𝑘, for 𝑘∈ℕ, 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 ℂ^𝑛, where 𝐾 is a compact well selected and 𝜋^𝑝_𝑘𝐾,𝑓=inf𝑓−𝑃_{𝐿𝑝(𝐾,𝜇)};𝑃∈𝒫_𝑘ℂ^𝑛,(11) where 𝒫_𝑘(ℂ^𝑛) is the family of all polynomials of degree ≤𝑘 and 𝜇 is the well selected measure (the equilibrium measure 𝜇=(𝑑𝑑^𝑐𝑉_𝐾)^𝑛 associated to a 𝐿-regular compact 𝐾) (see [8]) and 𝐿^𝑝(𝐾,𝜇), 𝑝≥1, is the class of all functions such that 𝑓_{𝐿𝑝(𝐾,𝜇)}=_𝐾𝑓^𝑝𝑑𝜇^1/𝑝<∞.(12)

In this work we give the generalization of these results in 𝐂^𝑛, replacing the circle {𝑧∈ℂ;|𝑧|=𝑟} by the set {𝑧∈ℂ^𝑛;exp(𝑉_𝐾(𝑧))<𝑟}, where 𝑉_𝐾 is the Siciak’s extremal function of 𝐾, a compact of ℂ^𝑛 satisfying some properties (see [9, 10]), and using the development of 𝑓 with respect to the sequence (𝐴_𝑘)_𝑘∈ℕ 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 ℂ^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 ℂ^𝑛, 𝑉_𝐾1=sup𝑑log𝑃_𝑑,𝑃_𝑑polynomialinℂ^𝑛ofdegree≤𝑑,𝑃_𝑑𝐾≤1.(13)

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

It is known (see [10]) that 𝑉_𝐾(𝑧)=sup𝑢(𝑧)∶𝑢∈ℒℂ^𝑛;𝑢≤0on𝐾,(14) where ℒℂ^𝑛=𝑢∈PSHℂ^𝑛(𝑧)𝑧(1)∶𝑢−log(||)≤𝑂as|𝑧|→∞(15) 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)=limsup_𝑤→𝑧𝑉_𝐾(𝑤) is the unique function in the class ℒ(ℂ^𝑛) which vanishes on 𝐾 except perhaps for a pluripolar subset and satisfies the complex Monge-Ampère equation (see [12]): 𝑑𝑑^𝑐𝑉_𝐾^𝑛=0onℂ^𝑛⧵𝐾.(16)

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 𝑉_𝐾1=sup𝑑log𝑃_𝑑,𝑃_𝑑polynomialofdegree𝑃≤𝑑,_𝑑𝐾≤1(17) is called the Siciak’s extremal function of the compact 𝐾.

Definition 2. A compact 𝐾 in ℂ^𝑛 is said to be 𝐿-regular if the extremal function, 𝑉_𝐾, associated to 𝐾 is continuous on ℂ^𝑛.

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

For any 𝜖>0, there exists an open 𝑈⊃𝐾 such that for any polynomial 𝑃‖𝑃‖_𝑈≤𝑒^{𝜖⋅deg(𝑃)}‖𝑃‖_𝐾.(18)

In this case we take 𝑈={𝑧∈ℂ^𝑛;𝑉_𝐾(𝑧)<𝜖}.

Regularity also arises in polynomial approximation. For 𝑓∈𝒞(𝐾), we let 𝜖_𝑘𝐾,𝑓=inf𝑓−𝑃_𝐾,𝑃∈𝒫_𝑘ℂ^𝑛,(19) where 𝒫_𝑘(ℂ^𝑛) is the set of polynomials of degree at most 𝑑. Siciak showed that (see [10]).

If 𝐾 is 𝐿-regular, then limsup_𝑘→+∞𝜀_𝑘𝐾,𝑓^1/𝑘=1𝑅<1(20) if and only if 𝑓 has an analytic continuation to 𝑧∈𝐂^𝑛;𝑉_𝐾(𝑧)1<log𝑅.(21)

It is known that if 𝐾 is a compact 𝐿-regular of ℂ^𝑛, there exists a measure 𝜇, called extremal measure, having interesting properties (see [9, 10]), in particular, we have the following properties. (𝑃₁) Bernstein-Markov inequality: for all 𝜖>0, there exists a constant 𝐶=𝐶_𝜀 such that (BM)∶𝑃_𝑑𝐾=𝐶(1+𝜀)^𝑠𝑘𝑃_{𝑑𝐿2(𝐾,𝜇)},(22) for every polynomial of 𝑛 complex variables of degree at most 𝑑. (𝑃₂) Bernstein-Walsh (BW) inequality: for every set 𝐿-regular 𝐾 and every real 𝑟>1 we have 𝑓_𝐾≤𝑀𝑟^deg(𝑓)_𝐾𝑓^𝑝𝑑𝜇^1/𝑝.(23)

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

Let 𝛼∶ℕ→ℕ^𝑛,𝑘↦𝛼(𝑘)=(𝛼₁(𝑘),…,𝛼_𝑛(𝑘)) be a bijection such that |𝛼(𝑘+1)|≥|𝛼(𝑘)|,where|𝛼(𝑘)|=𝛼₁(𝑘)+⋯+𝛼_𝑛(𝑘).(24)

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 𝐴_𝑘(𝑧)=𝑧^𝛼(𝑘)+^𝑘−1_𝑗=1𝑎_𝑗𝑧^𝛼(𝑗)(25) such that 𝐴_{𝑘𝐿𝑝(𝐾,𝜇)}=inf𝑧^𝛼(𝑘)+^𝑘−1_𝑗=1𝑎_𝑗𝑧^𝛼(𝑗)_{𝐿𝑝(𝐾,𝜇)},𝑎₁,𝑎₂,…,𝑎_𝑛∈ℂ^{𝑛1/𝛼𝑘}.(26) We need the following notations and lemma which will be used in the sequel (see [2]):  (𝑁₁) 𝜈_𝑘=𝜈_𝑘(𝐾)=‖𝐴_𝑘‖_𝐿²_(𝐾,𝜇),  (𝑁₂) 𝑎_𝑘=𝑎_𝑘(𝐾)=‖𝐴_𝑘‖_𝐾=max_𝑧∈𝐾|𝐴_𝑘(𝑧)| and 𝜏_𝑘=(𝑎_𝑘)^1/𝑠𝑘, where 𝑠_𝑘=deg(𝐴_𝑘).

For 𝑝∈ℕ, put, for 𝑝≥1 and 𝑥>0, log^[𝑝](𝑥)=loglog^[𝑝−1](𝑥),exp^[𝑝](𝑥)=expexp^[𝑝−1](𝑥),Λ_[𝑝]=^𝑝_𝑘=1log^[𝑘](𝑥),𝐸_[𝑝](𝑥)=^𝑝_𝑘=0exp^𝑘(𝑥),log^[0](𝑥)=𝑥,exp^[0](𝑥)=𝑥.(27)

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) 𝐸_𝑝⁻¹(𝑥)=𝑥,𝑖𝑓𝑝=0,log^[𝑝−1]log(𝑥)−log^[2](𝑥)+𝑜log_[3](𝑥),𝑖𝑓𝑝=1,2,…,(28)  (𝑅𝑅5) lim_𝑥→+∞exp𝐸_[𝑝−2](𝑥)=𝑒,𝑖𝑓𝑝=2,1,𝑖𝑓𝑝≥3,(29) (𝑅𝑅6) lim_𝑥→+∞exp^[𝑝−1]𝐸⁻¹_𝑝−2(𝑥)^1/𝑥=𝑒,𝑖𝑓𝑝=2,1,𝑖𝑓𝑝≥3.(30)

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

Definition 4. Let 𝐾 be a compact 𝐿-regular and put Ω_𝑟=𝑧∈ℂ^𝑛;exp𝑉_𝐾(𝑧)≤𝑟.(31)
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 𝜌=𝜌_𝐾𝑝,𝑞=limsup_𝑟→+∞log^[𝑝]𝑓_Ω𝑟log^[𝑞]𝑟.(32)
If 𝜌∈[𝛽,+∞[, the (𝐾,𝑝,𝑞)-type is defined by 𝜎=𝜎_𝐾𝑝,𝑞=limsup_𝑟→+∞log^[𝑝−1]𝑓_Ω𝑟log^[𝑞−1]𝑟^𝜌,(33) 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 Ω_𝑟=exp𝑉_𝐾<𝑟(34) and the coefficients of entire function 𝑓 on ℂ^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 𝑓=_𝑘≥1𝑓_𝑘𝐴_𝑘.(35)

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  ℂ^𝑛 and let 𝑓 be an entire function such that 𝑓=∑^+∞_𝑘=0𝑓_𝑘𝐴_𝑘. Then for every 𝜃>1, there exists an integer 𝑁_𝜃≥1 and a constant 𝐶_𝜃 such that 𝜋^𝑝_𝑘𝐾,𝑓≤𝐶_𝜃(𝑟+1)^𝑁𝜃(𝑟−1)^2𝑁−1𝑓_Ω𝑟𝜃𝑟^𝑘,(36)𝑓_𝑘𝜈_𝑘≤𝐶_𝜃(𝑟+1)^𝑁𝜃(𝑟−1)^2𝑁−1𝑓_Ω𝑟𝜃𝑟^𝑠𝑘,(37) where 𝑁_𝜃∈ℕ 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 lim_𝑘→+∞𝐴_𝑘(𝑧)𝜈_𝑘^1/𝑠𝑘=exp𝑉_𝐾(𝑧)(38) for every 𝑧∈ℂ^𝑛, and lim_𝑘→+∞𝐴_𝑘^𝑠𝑘_𝐾𝜈_𝑘^1/𝑠𝑘=1.(39)

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

Theorem 7. Let 𝑓=∑_𝑘≥1𝑓_𝑘𝐴_𝑘 be an entire function. Then 𝑓 is said of a finite (𝐾,𝑝,𝑞)-order 𝜌_𝐾(𝑝,𝑞) if and only if 𝐿𝑝,𝑞=limsup_𝑘→+∞log^[𝑝−1]𝑠_𝑘log^[𝑞−1]−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘<+∞(40) and 𝜌_𝐾(𝑝,𝑞)=𝑃₁(𝐿(𝑝,𝑞)), where 𝑃₁𝐿=𝐿()𝑝,𝑞𝑝,𝑞,𝑖𝑓𝑝>𝑞,1+𝐿2,2,𝑖𝑓𝑝=𝑞=2,max1,𝐿𝑝,𝑞,𝑖𝑓3≤𝑝=𝑞<+∞,+∞,𝑖𝑓𝑝=𝑞=+∞,(41) for (𝑝,𝑞)∈ℕ² with 𝑝≥𝑞.

Proof. Put 𝜌=𝜌_𝐾(𝑝,𝑞). Let us prove that 𝜚≥𝑃₁(𝐿(𝑝,𝑞)). If 𝑓 is of finite (𝑝,𝑞)-order 𝜌, then we have 𝜌=limsup_𝑟→+∞log^[𝑝]𝑓_Ω𝑟log^[𝑞]𝑟=limsup_𝑟→+∞log^[𝑝]𝑓_Ω𝑟𝜃log^[𝑞]().𝑟𝜃(42) Thus for every 𝜀>0 there exists 𝑟(𝜀) such that for every 𝑟>𝑟(𝜀)log𝑓_Ω𝑟𝜃≤exp^[𝑝−2]log^[𝑞−1](𝑟𝜃)^𝜌+𝜀.(43) Using the inequalities (37) of Lemma 5 and (39) of Lemma 6, one has, for every 𝜀>0, there exist 𝑟(𝜀) and 𝑘(𝜀) such that for every 𝑟>𝑟(𝜀) and 𝑘>𝑘(𝜀)log𝑓_𝑘𝜏^𝑠𝑘_𝑘≤𝑠_𝑘log(1+𝜀)+log𝐶_𝜃+𝑁_𝜃()−()()log1+𝑟2𝑁−1log𝑟−1−𝑠_𝑘log(𝑟)+log𝑓_Ω𝑟𝜃(44) for 𝑟>𝑟(𝜀) and 𝑘>𝑘(𝜀). But for 𝑟>𝑟(𝜀) and 𝑘>𝑘(𝜀) we have −1𝑠_𝑘log𝑓_𝑘𝜏^𝑠𝑘_𝑘1≥−𝑠_𝑘𝐶⋅log^𝜃(1+𝜀)⋅log(1+𝑟)^𝑁𝜃(𝑟−1)^(2𝑁−1)+exp^[𝑝−2]log^[𝑞−1](𝑟𝜃)^𝜌+𝜀.(45)
Then, by proceeding to limits as 𝑘→∞, we get for 𝑟 sufficiently large log𝑓_𝑘𝜏^𝑠𝑘_𝑘≤(1+𝑜(1))log(𝑟).(46)
(i) For (𝑝,𝑞)≠(2,2) with 𝑝>𝑞, let 𝑟_𝑘=1𝜃exp^[𝑞−1]log^[𝑝−2]𝑠_𝑘𝜌+𝜀^1/𝜌+𝜀>𝑟(𝜀).(47) Then if we replace in the equality (46) 𝑟 by 𝑟_𝑘, we get easily that for 𝑘 sufficiently large −1𝑠_𝑘log𝑓_𝑘𝜏^𝑠𝑘_𝑘≥exp^[𝑞−2]log^[𝑝−2]𝑠_𝑘𝜌+𝜀^{1/(𝜌+𝜀)}[1+𝑜(1)].(48) After passing to the upper limit, we get for 𝑝>𝑞limsup_𝑘→+∞log^[𝑝−1]𝑠_𝑘log^[𝑞−1]−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘≤𝜌.(49)
(ii) For 3≤𝑝=𝑞<+∞, the inequality (46) gives 𝜌≥max(1,𝐿(𝑝,𝑞)) (because 𝜌≥1 for 𝑝=𝑞).
(iii) For (𝑝,𝑞)=(2,2), choose 𝑟_𝑘=(1/𝜃)exp(𝑠_𝑘/(𝜌+𝜀))^{1/(𝜌−1+𝜀)}>𝑟(𝜀) and in the same way we show that −1𝑠_𝑘log𝑓_𝑘∣𝜏^𝑠𝑘_𝑘≥𝑠_𝑘𝜌+𝜀^{1/(𝜌−1+𝜀)}⋅𝜌−1+𝜀⋅((1𝜌+𝜀1+𝑜))(50) for 𝑘 sufficiently large, thus limsup_𝑘→+∞log𝑠_𝑘log−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘≤𝜌−1.(51)
By combining (i), (ii), and (iii) we have 𝜌≥𝑃₁(𝐿(𝑝,𝑞)). This result holds obviously if 𝜌=+∞.
We prove now reverse inequality 𝜌≤𝑃₁(𝐿(𝑝,𝑞)). By the definition of 𝐿(𝑝,𝑞), for every 𝜀>0 there exists 𝑘(𝜀) such that for every 𝑘≥𝑘(𝜀)𝑓_𝑘𝜏^𝑠𝑘_𝑘≤exp−𝑠_𝑘exp^[𝑞−2]log^[𝑝−2]𝑠_𝑘^{1/(𝐿+𝜀)},(52) where 𝐿=𝐿(𝑝,𝑞), for simplification.
Let 𝑘(𝑟) be a positive integer such that, for 𝑘≥𝑘(𝑟), 𝑠_𝑘(𝑟)≤exp^[𝑝−2]log^[𝑞−1](2(1+𝜀)𝑟)^𝐿+𝜀<𝑠_𝑘(𝑟)+1(53) and 𝑘(𝑟)>𝑘(𝜀), by (39), (37), (BM) and (BW) inequalities, there exists 𝑘₀∈ℕ, such that 𝑓_Ω𝑟≤^{𝑘=𝑘(𝜀)}_𝑘=0𝑓_𝑘𝐴_𝑘(𝑧)+𝐶_𝜀^{𝑘=𝑘(𝑟)}_{(𝜀)𝑘=𝑘+1}𝑓_𝑘𝜏^𝑠𝑘_𝑘𝑟^𝑠𝑘+𝐶^′_𝜀⋅^+∞_{(𝑟)𝑘=𝑘+1}𝑓_𝑘𝜏^𝑠𝑘_𝑘((1+𝜀)𝑟)^𝑠𝑘.(54)
Indeed, 𝑓_Ω𝑟≤^𝑘=+∞_𝑘=0𝑓_𝑘⋅𝐴_𝑘(𝑧)(55) (because 𝑧 satisfies exp(𝑉_𝐾(𝑧))≤𝑟).
By (38) and (39), for 𝑘 sufficiently large we have 𝐴_𝑘≤(1+𝜀)^𝑠𝑘exp𝑉_𝐾(𝑧)^𝑠𝑘,𝜏^𝑠𝑘_𝑘≤(1+𝜀)^𝑠𝑘𝜈_𝑘.(56)
Therefore, for 𝑟 sufficiently large we have 𝑓_Ω𝑟≤𝐴_𝑘(𝜀)+^𝑁(𝑟)_{𝑘=𝑘𝜀+1}𝑓_𝑘𝜏^𝑠𝑘_𝑘1+^+∞_{(𝑟)𝑘=𝑁}2^−𝑠𝑘₂,(57) where 𝐴_𝑘(𝜀) is a polynomial of degree not exceeding 𝑘_𝜀. By using (46) we get 𝑓_Ω𝑟≤𝐴_𝑘(𝜀)+𝑐_𝜀(2(1+𝜀)𝑟)^{𝑠𝑘(𝑟)}×^+∞_𝑘=0exp−𝑠_𝑘exp^[𝑞−2]ln^[𝑝−2]𝑠_𝑘^𝐿+𝜀₁+^+∞_𝑘=02^−𝑠𝑘₂.(58)
By (52) the series (1) is convergent, and (2) is obviously convergent, hence we have for 𝑟 sufficiently large 𝑓_Ω𝑟≤𝐴_𝑘(𝜀)+𝐴₁(2(1+𝜀)𝑟)^{𝑠𝑘(𝑟)}+1,(59) where 𝐴₁ is a constant. Thus, for 𝑟 sufficiently large we obtain log𝑓_Ω𝑟≤𝑠_𝑘log((1+𝜀)𝑟)(1+𝑜(1)).(60) Therefore, for 𝑟 sufficiently large log^[2]𝑓_Ω𝑟≤log𝑠_𝑘+log^[2]((1+𝜀)𝑟)+𝑜(1).(61)
For 𝑟 sufficiently large let 𝑠_𝑘=𝐸exp^[𝑝−2]log^[𝑞−1](2𝑟)^𝐿+𝜖,(62) where 𝐸(𝑥) is the integer part of 𝑥. Replacing 𝑠_𝑘 in the inequality (61) we get log^[2]𝑓_Ω𝑟≤exp^[𝑝−3]log^[𝑞−1](2𝑟)^𝐿+𝜀+log^[2])𝑟)(1).(∗)((1+𝜀+𝑜
To prove the result we proceed in three steps.
Step 1. For (𝑝,𝑞)=(2,2), we have log^[2]𝑓_Ω𝑟≤(𝐿+𝜀)log^[2](2𝑟)+log^[2]((1+𝜀)𝑟)+𝑜(1).(63) Then log^[2]𝑓_Ω𝑟log^[2]()≤()(1).2𝑟𝐿+𝜀+1+𝑜(64) Proceeding to the upper limit we get 𝜌≤1+𝐿(2,2)=𝑃₁(𝐿(2,2)).
Step 2. For 3≤𝑝=𝑞, since 𝜌≥1, we get 𝜌≤max(1(𝐿(𝑝,𝑝))=𝑃₁(𝐿(𝑝,𝑞)).
Step 3. For 𝑝<𝑞, the relation (∗) is equivalent to log^[2]𝑓_Ω𝑟≤exp^[𝑝−3]log^[𝑞−1](2𝑟)^𝐿+𝜀(1++𝑜(1)).(65) Then, for 𝑟 sufficiently large log^[𝑝]𝑓_Ω𝑟≤loglog^[𝑞−1](2𝑟)^𝐿+𝜀((1=(1+𝑜))𝐿+𝜀)log^[𝑞](2𝑟)(1+𝑜(1)).(66) Passing to the upper limit after division by log^[𝑞](2𝑟) we obtain 𝜌≤𝐿(𝑝,𝑞).
By combining (i), (ii), and (iii) we obtain for 𝑝≥𝑞≥1, 𝜌≤𝑃₁(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 𝛾𝑝,𝑞=limsup_𝑘→+∞log^[𝑝−2]𝑠_𝑘log^[𝑞−2]−1/𝑠_𝑘log𝑓_𝑘𝜏^𝑠𝑘_𝑘^𝜌−𝐶<+∞(67) 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)≠(2,2), 𝑠_𝑘=deg(𝐴_𝑘), and 𝑀=1𝑝,𝑞1,𝑖𝑓𝑝≥3,()=(),𝜌()𝑒⋅𝜌2,1,𝑖𝑓𝑝,𝑞2,12,2−1^{𝜌(2,2)−1}𝜌(2,2)^𝜌(2,2),𝑝,𝑞=(2,2).(68)

Proof. Let us first prove that 𝜎_𝐾(𝑝,𝑞)≤𝑀(𝑝,𝑞)⋅𝛾(𝑝,𝑞). By the definition of 𝛾=𝛾(𝑝,𝑞), for every 𝜀>0 there exists 𝑘(𝜀) such that for every 𝑘≥𝑘(𝜀), 𝑓_𝑘𝜏^𝑠𝑘_𝑘≤exp−𝑠_𝑘exp^[𝑞−2]log^[𝑝−2]𝑠_𝑘𝛾+𝜀^{1/(𝜌−𝐶)}.(69)
Let 𝑘(𝑟) be a positive integer such that 𝑘(𝑟)≤exp^[𝑝−2]𝛾+𝜀log^[𝑞−1](2(1+𝜀)𝑟)^𝜌−𝐶<𝑘(𝑟)+1(70) and 𝑘≥𝑘(𝑟). By the estimate (39) from Lemma 6 and the (BM) and (BW) inequalities, there exists 𝑘₀∈ℕ such that 𝑓_Ω𝑟≤𝑘(𝑟)+max_𝑘≥0𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘((1+𝜀)𝑟)^𝑠𝑘+_𝑘=02^−𝑠𝑘.(71) Put 𝐻(𝑟)=max_𝑥≥0{𝜔(𝑟,𝑥)},(72) where 𝜔(𝑟,𝑥)=−𝑥exp^[𝑞−2]log^[𝑝−2](𝑥)𝛾+𝜀^{1/𝜌−𝐶}+𝑥log((1+𝜀)𝑟).(73)
By repeating the argument used in the proof of Theorem 7 one may easily check that 𝜎_𝐾𝑝,1≤𝑀𝑝,1𝛾𝑝,1.(74)
For example we will show that 𝜎_𝐾(𝑝,1)≤𝑀(𝑝,1)𝛾(𝑝,1). By the relation (69) we have 𝑓_𝑘𝜏^𝑠𝑘_𝑘≤𝛾+𝜀log^[𝑝−2]𝑠_𝑘^{𝑠𝑘/𝜌}.(75)
The maximum of the function 𝑥↦𝜔(𝑥,𝑟) is reached for 𝑥=𝑥_𝑟, where 𝑥_𝑟 is the solution of the equation −loglog^[𝑝−2](𝑥)𝛾+𝜀^1/𝜌)𝑟)=𝐸+log((1+𝜀^[1−𝑝](𝑥)𝜌.(76)
For 𝑝=2 the relation (75) becomes 𝑥−log𝛾+𝜀^1/𝜌)𝑟)=1+log((1+𝜀𝜌.(77) Thus 𝑥_𝑟=1𝑒𝛾+𝜀((1+𝜀)𝑟)^𝜌.(78) Therefore 𝐻(𝑟)=𝜔(𝑟,𝑥_𝑟)=𝑥_𝑟/𝜌 and 𝜔(𝑟,𝑥_𝑟)=(1/𝑒𝜌)(𝛾+𝜀)((1+𝜀)𝑟)^𝜌 and by (69) we get 𝑓_Ω𝑟≤𝐶₀+𝛾+𝜀((1+𝜀)𝑟)^𝜌exp(𝐻(𝑟))(79) which gives for 𝑟 sufficiently large log𝑓_Ω𝑟≤𝐻(𝑟)+𝜌log((1+𝜀)𝑟)(1+𝑜(1)).(80) Passing to the upper limit when 𝑟→+∞ we get 𝜎_𝐾(2,1)≤𝑀(2,1)𝛾(2,1).(81)
For 𝑝≥3 we have 𝜔𝑟,𝑥_𝑟=𝑥_𝑟𝜌𝐾_{[1−𝑝](𝑥)}=𝑥²_𝑟𝜌⋀_[𝑝−2]𝑥_𝑟.(82) Therefore log𝜔𝑟,𝑥_𝑟=2log𝑥_𝑟−log𝜌−log_[𝑝−2]𝑥_𝑟=2log𝑥_𝑟−^𝑝−1_𝑘=1log^𝑘𝑥_𝑟.(83) Hence log𝜔𝑟,𝑥_𝑟log𝑥_𝑟=1−^𝑝−1_𝑘=1log^𝑘𝑥_𝑟log𝑥_𝑟.(84) Then, for 𝑟 sufficiently large, log𝜔𝑟,𝑥_𝑟∼log𝑥_𝑟∼exp^[𝑝−2]𝛾+𝜀((1+𝜀)𝑟)^𝜌,𝑓_Ω𝑟≤exp^[𝑝−2]𝛾+𝜀((1+𝜀)𝑟)^𝜌.(85) We obtain the result after passing to the upper limit.

Remark 9. If 𝑛=1 and (𝑝,𝑞)=(2,1), we know that lim_𝑘→+∞𝜏_𝑘^1/𝑠𝑘=𝜏(𝐾)=cap(𝐾).(86) Then by using Theorems 7 and 8 we get 𝜌_𝐾(2,1)=lim_𝑘→+∞−𝑠_𝑘log𝑠_𝑘log𝑓_𝑘(),𝜎=𝜌2,1_𝐾(2,1)=limsup_𝑟→+∞𝑠_𝑘⋅𝑓_𝑘⋅𝜏_𝑘^{1/𝑠𝑘𝜌𝐾(2,1)}=𝜎(2,1)cap(𝐾)^{𝜌𝐾(2,1)}(87) which gives the result of Winiarski.

Remark 10. The notion of the type associated to a compact in ℂ 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 ℂ^𝑛, (𝑛≥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 ℂ^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≤𝑝≤∞) with respect to a measure 𝜇 on 𝐾 in ℂ^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 𝜋^𝑝_𝑘𝐾,𝑓=inf𝑓−𝑃_{𝐿𝑝(𝐾,𝜇)},𝑃∈𝒫_𝑘ℂ^𝑛(88) which extend the classical results of Reddy and Winiarsk in ℂ^𝑛. We need the following lemmas.

Lemma 11. If 𝐾 is compact 𝐿-regular in ℂ^𝑛, then every function 𝑓∈𝐿²_𝒫(𝐾,𝜇) can be written in the form 𝑓=^+∞_𝑘=0𝑓_𝑘⋅𝐴_𝑘,(89) where 𝐿²_𝒫(𝐾,𝜇) is the closed subspace of 𝐿²(𝐾,𝜇) generated by the restrictions to 𝐸 of polynomials ℂ^𝑛 and (𝐴_𝑘) is the sequence defined by (25).

Lemma 12. Let (𝐴_𝑘) be the sequence defined by (25) and 𝑓=∑_𝑘≥0𝑓_𝑘𝐴_𝑘 an element of 𝐿^𝑝(𝐾,𝜇), for 𝑝≥1, then limsup_𝑘→+∞log^[𝑝−1](𝑘)log^[𝑞−1](−1/𝑘)log𝜋^𝑝_𝑘𝐾,𝑓=limsup_𝑘→+∞log^[𝑝−1]𝑠_𝑘log^[𝑞−1]−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘,(90)limsup_𝑘→+∞log^[𝑝−2](𝑘)log^[𝑞−2](−1/𝑘)log𝜋^𝑝_𝑘𝐾,𝑓^𝜌−𝐶=limsup_𝑘→+∞log^[𝑝−2]𝑠_𝑘log^[𝑞−2]−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘^𝜌−𝐶.(91)

Proof of Lemma 12. The proof is done in two steps (𝑝≥2 and 1<𝑝<2). Let 𝑓=∑_𝑘≥0𝑓_𝑘⋅𝐴_𝑘 be an element of 𝐿^𝑝(𝐾,𝜇).
Step 1. If 𝑓∈𝐿^𝑝(𝐾,𝜇) with 𝑝≥2, then 𝑓=∑^+∞_𝑘=0𝑓_𝑘⋅𝐴_𝑘 with convergence in 𝐿²(𝐾,𝜇), where 𝑓_𝑘=1/𝜈²_𝑘∫_𝐾𝑓𝐴_𝑘𝑑𝜇, 𝑘≥0 and therefore 𝑓_𝑘=(1/𝜈²_𝑘)∫_𝐾(𝑓−𝑃_𝑠𝑘−1)⋅𝐴_𝑘𝑑𝜇 (because deg(𝐴_𝑘)=𝑠_𝑘). Since |𝑓_𝑘|≤(1/𝜈²_𝑘)∫_𝐾|𝑓−𝑃_𝑠𝑘−1|⋅|𝐴_𝑘|𝑑𝜇, we obtain easily, using Bernstein-Walsh inequality and de Hölder inequality, that we have for any 𝜀>0𝑓_𝑘⋅𝜈_𝑘≤𝐶_𝜀⋅(1+𝜀)^𝑠𝑘𝜋^𝑝_𝑠𝑘−1𝐾,𝑓(92) for every 𝑘≥0.
Step 2. If 1≤𝑝<2, let 𝑝^′ such 1/𝑝+1/𝑝^′=1, then 𝑝^′≥2. By the Hölder inequality we have 𝑓_𝑘𝜈²_𝑘≤𝑓−𝑃_{𝑘−1𝐿𝑝(𝐾,𝜇)}𝐴_{𝑘𝐿𝑝′(𝐾,𝜇)}.(93) But ‖𝐴_𝑘‖_{𝐿𝑝′(𝐸,𝜇)}≤𝐶‖𝐴_𝑘‖_𝐾=𝐶𝑎_𝑘(𝐾), therefore, by the (BM) inequality, we have 𝑓_𝑘𝜈²_𝑘≤𝐶𝐶_𝜀(1+𝜀)^𝑠𝑘𝑓−𝑃_{𝑠𝑘−1𝐿𝑝(𝐾,𝜇)}.(94) Hence 𝑓_𝑘𝜈²_𝑘≤𝐶^′_𝜀(1+𝜀)^𝑠𝑘𝜋^𝑝_𝑠𝑘𝐾,𝑓.(95) Thus in both cases we have 𝑓_𝑘𝜈_𝑘≤𝐴_𝜀(1+𝜀)^𝑠𝑘𝜋^𝑝_𝑠𝑘𝐾,𝑓,(96) where 𝐴_𝜀 is a constant which depends only on 𝜀. After passing to the upper limit (96) gives limsup_𝑘→+∞log^[𝑝−1](𝑘)log^[𝑞−1](−1/𝑘)log𝜋^𝑝_𝑘𝐾,𝑓≥limsup_𝑘→+∞log^[𝑝−1]𝑠_𝑘log^[𝑞−1]−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘.(97) To prove the other inequality we consider the polynomial of degree 𝑠_𝑘𝑃_𝑠𝑘(𝑧)=^𝑗=𝑘_𝑗=0𝑓_𝑗𝐴_𝑗,(98) then 𝜋^𝑝_𝑠𝑘−1𝐾,𝑓≤^+∞_{𝑠𝑗=𝑠𝑘}𝑓_𝑗𝐴_{𝑗𝐿𝑝(𝐾,𝜇)}≤𝐶₀^+∞_{𝑠𝑗=𝑠𝑘+1}𝑓_𝑗𝐴_𝑗𝐾.(99) By the Bernstein-Walsh inequality we have 𝜋^𝑝_𝑘𝐾,𝑓≤𝐶_𝜀^+∞_{𝑠𝑗=𝑠𝑘}(1+𝜀)^𝑠𝑗𝑓_𝑗𝜈_𝑗(100) for 𝑘≥0 and 𝑝≥1. If we take as a common factor (1+𝜀)^𝑠𝑘𝜈_𝑘, the other factor is convergent, thus we have 𝜋^𝑝_𝑘𝐾,𝑓≤𝐶(1+𝜀)^𝑠𝑘⋅𝜈_𝑘(101) and by (39) of Lemma 6 we have then 𝜋^𝑝_𝑘𝐾,𝑓≤𝐶(1+𝜀)^𝑠𝑘⋅𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘.(102) We then deduce that limsup_𝑘→+∞log^[𝑝−1](𝑘)log^[𝑞−1](−1/𝑘)log𝜋^𝑝_𝑘𝐾,𝑓≤limsup_𝑘→+∞log^[𝑝−1]𝑠_𝑘log^[𝑞−1]−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘.(103) This inequality is a direct consequence of (102) and the inequality on coefficients |𝑓_𝑘| given by 𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘≤exp−𝑠_𝑘exp^[𝑞−2]ln^[𝑝−2]𝑠_𝑘^{1/(𝜌+𝜀)}.(104)

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 ℂ^𝑛 of finite (𝐾,𝑝,𝑞)-order 𝜌 if and only if 𝜌₁𝑝,𝑞=limsup_𝑘→+∞log^[𝑝−1](𝑘)log^[𝑞−1](−1/𝑘)log𝜋^𝑝_𝑘𝐾,𝑓<+∞(105) and 𝜌=𝐿(𝜌₁(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 𝜎₁𝑝,𝑞=limsup_𝑘→+∞log^[𝑝−2](𝑘)log^[𝑞−2](−1/𝑘)log𝜋^𝑝_𝑘𝐾,𝑓^𝜌−𝐶<+∞(106) and 𝜎=𝑀(𝑝,𝑞)(𝜎₁(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(𝜌₁(𝑝,𝑞)). We have 𝑔∈𝐿^𝑝(𝐾,𝜇), 𝑝≥2 and 𝑔 = ∑_𝑘≥0𝑔_𝑘⋅𝐴_𝑘 in 𝐿²(𝐾,𝜇), where 𝑔_𝑘=(1/𝜈_𝑘)∫_𝐾𝑓𝐴_𝑘𝑑𝜇, 𝑘≥0. From (40) of Theorem 7 we get 𝜌=𝑃₁(𝐿(𝑝,𝑞)), where 𝐿𝑝,𝑞=limsup_𝑘→+∞log^[𝑝−1]𝑠_𝑘log^[𝑞−1]−1/𝑠_𝑘log𝑔_𝑘⋅𝜏^𝑠𝑘_𝑘(𝐸).(107)
But 𝑔=𝑓 on 𝐾, thus by Lemma 11 we have 𝜌=𝐿(𝜌₁(𝑝,𝑞)).
Conversely, suppose now that 𝑓 is a function of 𝐿^𝑝(𝐾,𝜇) such that the relation (105) holds.
(1) Let 𝑝≥2, then we have 𝑓=∑_𝑘≥0𝑓_𝑘𝐴_𝑘 because 𝑓∈𝐿²(𝐾,𝜇), (𝐿^𝑝(𝐾,𝜇)⊂𝐿²(𝐾,𝜇)) and (𝐴_𝑘)_𝑘 is a basis of 𝐿²(𝐾,𝜇) as in Section 3. Consider in ℂ^𝑛 the series ∑𝑓_𝑘𝐴_𝑘. By (90) of Lemma 12 one may easily check that this series converges normally on every compact subset of ℂ^𝑛 to an entire function denoted 𝑓₁ (this result is a direct consequence of the inequality (BM) and the inequality on coefficients |𝑓_𝑘|). We have obviously 𝑓₁=𝑓 𝜇-a.s on 𝐾, and by Theorem 8, the (𝐾,𝑝,𝑞)-order of 𝑓₁ is 𝜌𝑓₁,𝑝,𝑞=limsup_𝑘→+∞log^[𝑝−1]𝑠_𝑘log^[𝑞−1]−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘.(108) By Lemma 12 we check that 𝜌(𝑓₁,𝑝,𝑞)=𝜌 so the proof is completed for 𝑝≥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 8 and 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 13 we obtain the result of Winiarski (see [6]): limsup_𝑘→+∞𝑘𝜋^𝑝_𝑘𝐾,𝑓^𝜌/𝑘=𝑒𝜌𝜎cap(𝐾)^𝜌.(109)
(2) If 𝑛=1, 𝑝>2, and 𝑞=1, using the results of Theorem 13 we obtain the result of Nguyen (see [13]): limsup_𝑘→+∞log^[𝑝−2](𝑘)𝜋^𝑝_𝑘𝐾,𝑓^𝜌/𝑘=𝜎cap(𝐾)^𝜌.(110)

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), 𝑓_{𝐿𝑝(𝐸,𝜇)}≤𝑀𝑓_{𝐿𝑟(𝐸,𝜇)}.(111)
Using the inequality _𝐸𝑓^𝑝𝑑𝜇≤𝑓_𝐸^𝑝−𝑟_𝐸𝑓^𝑟𝑑𝜇(112) we get 𝑓_{𝐿𝑝(𝐸,𝜇)}≤𝑓_𝐸^{1−(𝑟/𝑝)}𝑓_𝐿^𝑟/𝑝_{𝑟𝐸,𝜇}.(113) We deduce that (𝐸,𝜇) satisfies the Bernstein-Markov inequality. For 𝜖>0 there is a constant 𝐶=𝐶(𝜖,𝑝)>0 such that for all (analytic) polynomials 𝑃 we have ‖𝑃‖_𝐸≤𝐶(1+𝜖)_deg(𝑃)‖𝑃‖_𝐿^𝑝_(𝐸,𝜇).(114)
Thus if (𝐸,𝜇) satisfies the Bernstein-Markov inequality for one 𝑝>0, then (92) and (95) are satisfied for all 𝑝>0.