Generalized Order and Best Approximation of Entire Function in 𝐿^𝑝-Norm

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 𝐾 with respect to the set Ω_𝑟={𝑧∈𝐂^𝑛;exp𝑉_𝐾(𝑧)≤𝑟}, where 𝑉_𝐾=sup{(1/𝑑)ln|𝑃_𝑑|,𝑃_𝑑polynomialofdegree≤𝑑,‖𝑃_𝑑‖_𝐾≤1} is the Siciak extremal function of a 𝐿-regular compact 𝐾.

1. Introduction

Let 𝑓(𝑧)=∑^+∞_𝑘=0𝑎_𝑘⋅𝑧^𝜆𝑘 be a no constant entire function in complex plane ℂ and let 𝑀𝑓,𝑟=sup𝑓(𝑧),|𝑧|=𝑟,𝑟>0. (1.1) It is well known that the function 𝑟→ln(𝑀(𝑓,𝑟)) is convex and decreasing of ln(𝑟). To estimate the growth of 𝑓, the concept of order, defined by the number 𝜌 (0≤𝜌≤+∞), such that 𝜌=limsup_𝑟→+∞lnln𝑀𝑓,𝑟ln(𝑟) (1.2) 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 ℂ, of order 𝜌 (0<𝜌<+∞), is said to be of type 𝜎 (0≤𝜎≤+∞) if𝜎=limsup_𝑟→+∞ln𝑀𝑓,𝑟𝑟^𝜌. (1.3)

If 𝑓 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 𝑝≥𝑞≥1, they have defined the number 𝜌𝑝,𝑞=limsup_𝑟→+∞log^[𝑝]𝑀𝑓,𝑟log^[𝑞](𝑟). (1.4)

It is easy to show that 𝑏≤𝜌(𝑝,𝑞)≤+∞ where 𝑏=0 if 𝑝>𝑞 and 𝑏=1 if 𝑝=𝑞.

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 and Juneja 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)}. (1.5)

In their works, the authors have established the relationship of (𝑝,𝑞)-growth of 𝑓 with respect to the coefficients 𝑎_𝑘 in the Maclaurin series of 𝑓 in complex plane ℂ (for (𝑝,𝑞)=(2,1) we obtain the classical case).

We have also many results in terms in polynomial approximation in classical case. Let 𝐾 be a compact subset of the complex plane ℂ, of positive logarithmic capacity and 𝑓 be a complex function defined and bounded on 𝐾. For 𝑘∈ℕ put𝐸_𝑘𝐾,𝑓=𝑓−𝑇_𝑘𝐾, (1.6) 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.

It is known (see [3]) thatlim_𝑘→+∞^𝑘𝐸_𝑘𝐾,𝑓=0 (1.7) if and only if 𝑓 is the restriction to 𝐾 of an entire function 𝑔 in ℂ.

This result has been generalized by Reddy (see [4, 5]) as follows.lim_𝑘→+∞^𝑘𝐸_𝑘𝐾,𝑓=𝜌⋅𝑒⋅𝜎2^−𝜌 (1.8) 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]) has generalized this result for a compact 𝐾 of the complex plane ℂ, of positive logarithmic capacity noted 𝑐=cap(𝐾) as follows:

If 𝐾 be a compact subset of the complex plane ℂ, of positive logarithmic capacity then lim_𝑘→+∞𝑘^(1/𝜌)𝑘𝐸_𝑘𝐾,𝑓=𝑐𝑒𝜌𝜎^1/𝜌 (1.9) if and only if 𝑓 is the restriction to 𝐾 of an entire function of order 𝜌 (0<𝜌<+∞) and type 𝜎.

Recall that cap([−1,1])=1/2 and capacity of a disk unit is cap(𝐷(𝑂,1))=1.

The authors considered 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 by𝑊_𝑛(𝑧)=^𝑗=𝑛_𝑗=1𝑧−𝜂_𝑛𝑗, (1.10) where (𝑎_𝑛𝑗)_𝑛 is the th𝑛th extremal points system of 𝐾.

The aim of this paper is to establish relationship between the rate at which (𝜋^𝑝_𝑘(𝐾,𝑓))^1/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 ℂ^𝑛, where 𝐾 is a compact well-selected. 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_𝐸(𝑧))<𝑟}, where 𝑉_𝐸 is the Siciak's extremal function of 𝐸 a compact of ℂ^𝑛 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 𝐾 be a compact set in ℂ^𝑛 and let ‖⋅‖_𝐾 denote the maximum norm on 𝐾. The function 𝑉_𝐾1=sup𝑑log𝑃_𝑑,𝑃_𝑑polynomialofdegree≤𝑑,𝑃_𝑑𝐾≤1,𝑑∈ℕ (2.1) is called the Siciak's extremal function of the compact 𝐾.

Definition 2.2. 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 [8]).

For any 𝜖>0, there exists an open 𝑈⊃𝐾 such that for any polynomial 𝑃‖𝑃‖_𝑈≤𝑒^{𝜖⋅deg(𝑃)}‖𝑃‖_𝐾. (2.2)

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

Regularity also arises in polynomial approximation. For 𝑓∈𝒞(𝐾), we let𝜖_𝑑𝐾,𝑓=inf𝑓−𝑃_𝐾,𝑃∈𝒫_𝑘ℂ^𝑛, (2.3)

where 𝒫_𝑘(ℂ^𝑛) is the set of polynomials of degree at most 𝑑. Siciak (see [7]) showed the following

If 𝐾 is 𝐿-regular, then limsup_𝑑→+∞𝜀_𝑑𝐾,𝑓^1/𝑑=1𝑟<1 (2.4) if and only if 𝑓 has an analytic continuation to {𝑧∈𝐂^𝑛;𝑉_𝐾(𝑧)<log(1/𝑟)}.

Let 𝑓 be a function defined and bounded on 𝐾. For 𝑘∈ℕ put𝜋^𝑝_𝑘𝐾,𝑓=inf𝑓−𝑃_{𝐿𝑝(𝐾,𝜇)},𝑃∈𝒫_𝑘ℂ^𝑛, (2.5)

where 𝒫_𝑘(ℂ^𝑛) is the family of all polynomial of degree ≤𝑘 and μ the well-selected measure (The equilibrium measure 𝜇=(𝑑𝑑^𝑐𝑉_𝐾)^𝑛 associated to a 𝐿-regular compact 𝐸) (see [9]) and 𝐿^𝑝(𝐾,𝜇), 𝑝≥1, is the class of all function such that:𝑓_{𝐿𝑝(𝐾,𝜇)}=_𝐾𝑓^𝑝𝑑𝜇^1/𝑝<∞. (2.6)

For an entire function 𝑓∈ℂ^𝑛 we establish a precise relationship between the general growth with respect to the set:Ω_𝑟=exp𝑉_𝐾<𝑟 (2.7) and the coefficients of the development of 𝑓 with respect to the sequence (𝐴_𝑘)_𝑘, called extremal polynomial (see [10]). Therefore, we use these results to give the relationship between the generalized growth of 𝑓 and the sequence (𝜋^𝑝_𝑘(𝐾,𝑓))_𝑘.

Recall that the subset of ℂ^𝑛, Ω_𝑟={exp(𝑉_𝐾)<𝑟}, replaces the unit disc 𝐷(𝑂,𝑟)={|𝑧|<𝑟} in the classical case.

It is known that if 𝐾 is an compact 𝐿-regular of ℂ^𝑛, there exists a measure μ, called extremal measure, having interesting properties (see [7, 8]), in particular, we have:(𝑃₁)Bernstein-Markov Inequality. For all 𝜖>0, there exists 𝐶=𝐶_𝜀 is a constant such that (BM)∶𝑃_𝑑𝐾=𝐶(1+𝜀)^𝑠𝑘𝑃_{𝑑𝐿2(𝐾,𝜇)}, (2.8) for every polynomial of 𝑛 complex variables of degree at most 𝑑.(𝑃₂)Bernstein-Waish (B.W) Inequality. For every set 𝐿-regular 𝐾 and every real 𝑟>1, we have: 𝑓_𝐾≤𝑀⋅𝑟^deg(𝑓)_𝐾𝑓^𝑝⋅𝑑𝜇^1/𝑝. (2.9)

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). (2.10)

Zériahi (see [10]) has constructed according to the Hilbert-Shmidt method a sequence of monic orthogonal polynomial according to a extremal measure (see [8]), (𝐴_𝑘)_𝑘, called extremal polynomial, defined by𝐴_𝑘(𝑧)=𝑧^𝛼(𝑘)+^𝑘−1_𝑗=1𝑎_𝑗𝑧^𝛼(𝑗) (2.11) such that𝐴_{𝑘𝐿𝑝(𝐾,𝜇)}=inf𝑧^𝛼(𝑘)+^𝑘−1_𝑗=1𝑎_𝑗𝑧^𝛼(𝑗)_{𝐿2(𝐾,𝜇)},𝑎₁,𝑎₂,…,𝑎_𝑛∈ℂ^{𝑛1/𝛼𝑘}. (2.12)

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

With that notations and (B.W) inequality, we have𝐴_𝑘Ω𝑟≤𝑎_𝑘⋅𝑟^𝑠𝑘, (2.13) where 𝑠_𝑘=deg(𝐴_𝑘). For more details (see [9]).

Let α and β be two positives, strictly increasing to infinity differentiable functions ]0,+∞[ to ]0,+∞[ such that for every 𝑐>0:lim_𝑥→+∞𝛼(𝑐𝑥)𝛼(𝑥)=1,lim_𝑥→+∞𝛽(1+𝑥𝜔(𝑥))𝛽(𝑥)=1, (2.14) where 𝜔 a function such that lim_𝑥→+∞𝜔(𝑥)=0.

Assume that, for every 𝑐>0, there exists two constants 𝑎 and 𝑏 such that for every 𝑥≥𝑎: 𝑑𝛽⁻¹(𝑐𝛼(𝑥))𝛼log(𝑥)≤𝑏, (2.15) where 𝑑(𝑢) means the differential of 𝑢.

Definition 2.3. Let 𝐾 be a compact 𝐿-regular, we put Ω_𝑟=𝑧∈ℂ^𝑛;exp𝑉_𝐾(𝑧)≤𝑟. (2.16)
If 𝑓 is an entire function we define the (𝛼,𝛽)-order and the (𝛼,𝛽)-type of 𝑓 (or generalized order and generalized type), respectively, by𝜌𝛼,𝛽=limsup_𝑟→+∞𝛼log‖𝑓‖_Ω𝑟𝛽(𝑟,𝜎log)𝛼,𝛽=limsup_𝑟→+∞𝛼𝑓_Ω𝑟𝛽(𝑟)^{𝜌(𝛼,𝛽)}, (2.17) where ‖𝑓‖_Ω𝑟=sup_Ω𝑟|𝑓(𝑧)|.

Note that in the classical case 𝛼(𝑥)=log(𝑥) and 𝛽(𝑥)=𝑥.

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 𝐾 be a compact 𝐿-regular subset of ℂ^𝑛. 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. (2.18) Let 𝑓=∑^+∞_𝑘=0𝑓_𝑘⋅𝐴_𝑘 be an entire function. Then for every 𝜃>1, there exists 𝑁_𝜃∈ℕ and 𝐶_𝜃>0 such that 𝑓_𝑘𝜈_𝑘≤𝐶_𝜃(𝑟+1)^𝑁𝜃(𝑟−1)^2𝑁−1𝑓_Ω𝑟𝜃𝑟^𝑠𝑘, (2.19) for every 𝑘≥1 and 𝑟>1.

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 ℂ.

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 setΩ_𝑟=exp𝑉_𝐾<𝑟 (3.1) and the coefficient of entire function 𝑓∈ℂ^𝑛 of the development with respect to the sequence of extremal polynomials.

Let (𝐴_𝑘)_𝑘 be a basis of extremal polynomial associated to the set 𝐾 defined the relation (2.11). We recall that (𝐴_𝑘)_𝑘 is a basis of 𝒪(ℂ^𝑛) (the set of entire functions on ℂ^𝑛). So if 𝑓 is an entire function then 𝑓=_𝑘≥1𝑓_𝑘⋅𝐴_𝑘 (3.2) and we have the following results.

Theorem 3.1. If 𝑓=∑_𝑘≥1𝑓_𝑘⋅𝐴_𝑘 then the (𝛼,𝛽)-order 𝜌(𝛼,𝛽) of 𝑓 is given by formula 𝜌𝛼,𝛽=limsup_𝑘→+∞𝛼𝑠_𝑘𝛽−1/𝑠_𝑘ln𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘<+∞. (3.3)

To prove theorem we need the following lemmas.

Lemma 3.2. Let 𝐾 be a compact 𝐿-regular subset of ℂ^𝑛. Then lim_𝑘→+∞𝐴_𝑘(𝑧)𝜈_𝑘^1/𝑠𝑘=exp𝑉_𝐾(𝑧), (3.4) for every 𝑧∈ℂ^𝑛⧵𝐾 the connected component of ℂ^𝑛⧵𝐾, lim_𝑘→+∞𝐴_𝑘𝐾𝜈_𝑘^1/𝑠𝑘=1. (3.5)

Lemma 3.3. For every 𝑟>1 the maximum of the function 𝑥⟶𝜔(𝑥,𝑟)=𝑟^𝑥exp−𝑥𝛽⁻¹1𝜇𝛼(𝑥) (3.6) is reached for 𝑥=𝑥_𝑟 solution of the equation 𝑥=𝛼⁻¹(𝑟)−𝑑𝛽𝜇𝛽log⁻¹1/𝜇𝛼(𝑥)𝑑(𝑥).log (3.7)

Consequence 3. This relation is equivalent to 𝑥≤𝛼⁻¹𝛽log(𝑟)+𝑏. (3.8)
Indeed, the relations𝑑𝛽⁻¹(𝑐𝛼(𝑥))𝑑(𝑥)𝛽log≤𝑏,₋₁𝛼(𝑥)𝜇(𝑟)−𝑑𝛽=log⁻¹(𝑐𝛼(𝑥))𝑑log(𝑥) (3.9) give (𝑟)(𝑟)−𝑑𝛽log−𝑏≤log⁻¹(𝑐𝛼(𝑥))𝑑(𝑥)(𝑟)log≤log+𝑏. (3.10)
Then𝛽⁻¹𝛼(𝑥)𝜇≤log(𝑟)+𝑏,𝑥≤𝛼⁻¹𝛽log(𝑟)+𝑏. (3.11)
We verify easily with the relation 𝑑𝛽⁻¹(𝑐𝛼(𝑥))𝛼log(𝑥)≤𝑏, (3.12) that 𝑟^𝑛exp−𝑛𝛽⁻¹1𝜇𝛼(𝑛)≤exp𝑏𝛼⁻¹𝜇⋅𝛽log(𝑟)+𝑏 (3.13) for every 𝑛>𝑛₀ and 𝑟>0.

Proof of Theorem 3.1. Put 𝛾=limsup_𝑘→+∞(𝛼(𝑠_𝑘)/𝛽(−(1/𝑠_𝑘)ln|𝑓_𝑘|⋅𝜏^𝑠𝑘_𝑘)) and show that 𝛾=𝜌(𝛼,𝛽).
(1) Show that 𝛾≤𝜌(𝛼,𝛽).
By definition of 𝛾, we have, for all 𝜖>0, ∃𝑘_𝜖 such that for all 𝑘>𝑘_𝜖𝛼𝑠_𝑘𝛽−1/𝑠_𝑘ln𝑓_𝑘⋅𝜈^𝑠𝑘_𝑘<𝛾+𝜖. (3.14)
From the second assertion of the Lemma 3.2, we have for every 𝜃>1, there exists 𝑁_𝜃∈ℕ and 𝐶_𝜃>0 such thatlog𝑓_𝑘𝜏^𝑠𝑘_𝑘≤log𝐶_𝜃()+log𝑟+1^𝑁𝜃(𝑟−1)^2𝑁−1−𝑠_𝑘log(𝑟)+𝛼⁻¹𝜌+𝜖⋅𝛽log(𝑟𝜃)=𝜑(𝑟) (3.15) or −1𝑠_𝑘log𝑓_𝑘𝜏^𝑠𝑘_𝑘1≥−𝑠_𝑘log𝐶_𝜃−𝑁_𝜃𝑠_𝑘log(𝑟+1)−2𝑁−1𝑠_𝑘()(𝑟)−1log𝑟−1+log𝑠_𝑘𝛼⁻¹𝜌+𝜖𝛽log(𝑟𝜃)=𝜑(𝑟,𝑘). (3.16)
But lim_𝑘→+∞𝜑(𝑟,𝑘)=log(𝑟) so 𝜑(𝑟,𝑘)∼log(𝑟) for 𝑟 sufficiently large.
Then−1𝑠_𝑘log𝑓_𝑘𝜏^𝑠𝑘_𝑘≥(1+𝑜(1))log(𝑟). (3.17)
Let 𝑟_𝑘 be a real satisfying𝑟_𝑘=exp𝛽⁻¹1𝛼𝑠𝜌+𝜖^𝑘. (3.18)
Then−1𝑠_𝑘log𝑓_𝑘𝜏^𝑠𝑘_𝑘≥𝛽⁻¹1𝑠𝜌+𝜖⋅𝛼_𝑘(1+𝑜(1)). (3.19)
This is equivalent to𝛽−1/𝑠_𝑘log𝑓_𝑘𝜏^𝑠𝑘_𝑘𝛼𝑠_𝑘≥1,𝜌+𝜖 (3.20) or 𝛼𝑠_𝑘𝛽−1/𝑠_𝑘log𝑓_𝑘𝜏^𝑠𝑘_𝑘≤𝜌+𝜖, (3.21) hence 𝛾≤𝜌𝛼,𝛽. (3.22)
 (2) Show that 𝛾≥𝜌(𝛼,𝛽). According to the definition of 𝛾, we have for every 𝜖>0 there exists 𝑘_𝜖 such that for all 𝑘>𝑘_𝜖𝑓_𝑘𝜏^𝑠𝑘_𝑘≤exp−𝑠_𝑘𝛽⁻¹𝛼𝑠_𝑘𝛾+𝜖 (3.23) for 𝑘≥𝑘_𝜖 sufficiently large.
According to the first assertion of Lemma 3.2 and (BM) and (BW) inequalities, we have𝐴_𝑘Ω𝑟≤𝑎_𝑘⋅𝑟^𝑠𝑘 (3.24) and for every entire function 𝑓(𝑧)=_𝑘≥0𝑓_𝑘⋅𝐴_𝑘(𝑧),𝑓_Ω𝑟≤^𝑘𝜖_𝑘=0𝑎_𝑘⋅𝑟^𝑠𝑘₍₁₎+𝐶_𝜖^𝑘𝑟_{𝑘=𝑘𝜖+1}|𝑓_𝑘|⋅𝜈^𝑠𝑘⋅(1+𝜖)𝑟^𝑠𝑘₍₂₎+𝐶_𝜖^+∞_{𝑘=𝑘𝑟+1}|𝑓_𝑘|⋅𝜏^𝑠𝑘_𝑘⋅𝑟^𝑠𝑘₍₃₎. (3.25)
The term (1) is a constant denoted 𝐶₀, and^𝑘𝑟_{𝑘=𝑘𝜖+1}𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘⋅(1+𝜖)𝑟^𝑠𝑘≤((1+𝜖)𝑟)^{𝑠𝑘(𝑟)+∞}_𝑘=0𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘(4). (3.26)
The series (4) is convergent.
Let, for 𝑟 sufficiently,𝑁(𝑟)=𝐸𝛼⁻¹𝛾+𝜖𝛽log2(1+𝜖)𝑟, (3.27) where 𝐸(𝑥) means the integer part of 𝑥.
Then𝑓_𝑘⋅𝜈_𝑘⋅((1+𝜖)𝑟)^𝑠𝑘≤exp−𝑠_𝑘⋅𝛽⁻¹𝛼𝑠_𝑘)𝑟)𝛾+𝜖((1+𝜀^𝑠𝑘. (3.28)
Applying the the relation (3.13) with 𝜇=𝛾+𝜖 and if we replace 𝑟 by (1+𝜀)𝑟, we obtain:𝑓_Ω𝑟≤exp𝑏⋅𝛼⁻¹𝛾+𝜖⋅𝛽log((1+𝜀)𝑟)+𝑏. (3.29) And so 𝑓_Ω𝑟≤𝐶⋅((1+𝜀)𝑟)^𝑠𝑘⋅exp−𝑠_𝑘⋅𝛽⁻¹𝛼𝑠_𝑘,𝛾+𝜖 (3.30) or 𝑓_Ω𝑟≤𝐶⋅exp𝑏⋅𝛼⁻¹𝛾+𝜖⋅𝛽log((1+𝜀)𝑟)+𝑏. (3.31)
Thuslog𝑓_Ω𝑟≤log(𝐶)+𝑏⋅𝛼⁻¹𝛾+𝜖⋅𝛽log((1+𝜀)𝑟)+𝑏 (3.32) and for 𝑟 sufficiently large, we have 𝛼𝑏⁻¹log𝑓_Ω𝑟𝛽log((1+𝜀)𝑟)+𝑏≤𝛾+𝜖. (3.33)
But 𝛼(𝑐⋅𝑥)∼𝛼(𝑥) near to the infinity, thus𝛼𝑏⁻¹log𝑓_Ω𝑟∼𝛼log𝑓_Ω𝑟. (3.34)
If we put 𝑥=log((1+𝜀)𝑟), then 𝛽(log((1+𝜖)𝑟)+𝑏)∼𝛽(log((1+𝜖)𝑟)) and,𝛼log𝑓_Ω𝑟𝛽log(𝑟)≤𝛾+𝜖. (3.35)
This is true for every 𝜖>0 hence 𝜌(𝛼,𝛽)≤𝛾. Thus the assertion is proved.

4. Best Approximation Polynomial in 𝐿^𝑝-Norm

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.

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 𝑓=∑_𝑘≥0𝑓_𝑘⋅𝐴_𝑘 an element of 𝐿^𝑝(𝐾,𝜇), for 𝑝≥1, then limsup_𝑘→+∞𝛼(𝑘)𝛽−(1/𝑘)log𝜋^𝑝_𝑘𝐾,𝑓=limsup_𝑘→+∞𝛼𝑠_𝑘𝛽−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘. (4.1)

Proof of Lemma 4.1. The proof is done in two steps 𝑝≥2 and 1<𝑝<2.Step 1. If 𝑓∈𝐿^𝑝(𝐾,𝜇) where 𝑝≥2, then 𝑓=∑^+∞_𝑘=0𝑓_𝑘⋅𝐴_𝑘 with convergence in 𝐿²(𝐾,𝜇), hence for 𝑘≥0𝑓_𝑘=1𝜈²_𝑘𝐾𝑓⋅𝐴_𝑘𝑑𝜇 (4.2) and therefore 𝑓_𝑘=1𝜈²_𝑘𝐾𝑓−𝑃_𝑘−1⋅𝐴_𝑘𝑑𝜇 (4.3) (because deg(𝐴_𝑘)=𝑠_𝑘).
Since the relation,𝑓_𝑘≤1𝜈²_𝑘𝐾𝑓−𝑃_𝑘−1⋅𝐴_𝑘𝜇 (4.4) satisfied, is easily verified by using inequalities Bernstein-walsh and Holder that, we have for all 𝜀>0𝑓_𝑘⋅𝜈_𝑘≤𝐶_𝜀⋅(1+𝜀)^𝑠𝑘⋅𝜋^𝑝_𝑠𝑘−1𝐾,𝑓. (4.5) for all 𝑘≥0.Step 2. If 1≤𝑝<2, let 𝑝^′ such that 1/𝑝+1/𝑝^′=1, we have 𝑝^′≥2. According to the inequality of Hölder, we have: 𝑓_𝑘⋅𝜈²_𝑘≤𝑓−𝑃_{𝑘−1𝐿𝑝(𝐾,𝜇)}⋅𝐴_{𝑘𝐿𝑝′(𝐾,𝜇)}. (4.6)
But,𝐴_{𝑘𝐿𝑝′(𝐾,𝜇)}≤𝐶⋅𝐴_𝑘𝐾=𝐶⋅𝑎_𝑘(𝐾). (4.7)
This shows, according to inequality (BM), that:𝑓_𝑘⋅𝜈²_𝑘≤𝐶⋅𝐶_𝜀⋅(1+𝜀)^𝑠𝑘⋅𝑓−𝑃_{𝑠𝑘−1𝐿𝑝(𝐾,𝜇)}. (4.8)
Hence the result 𝑓_𝑘⋅𝜈²_𝑘≤𝐶^′_𝜀⋅(1+𝜀)^𝑠𝑘⋅𝜋^𝑝_𝑠𝑘𝐾,𝑓. (4.9)
In both cases, we have therefore𝑓_𝑘⋅𝜈²_𝑘≤𝐴_𝜀⋅(1+𝜀)^𝑠𝑘⋅𝜋^𝑝_𝑠𝑘𝐾,𝑓, (4.10) where 𝐴_𝜀 is a constant which depends only on 𝜀.
After passing to the upper limit in the relation (4.10) and Applying the relation (3.5) of the Lemma 3.2 we getlimsup_𝑘→+∞𝛼(𝑘)𝛽−(1/𝑘)log𝜋^𝑝_𝑘𝐾,𝑓≥limsup_𝑘→+∞𝛼𝑠_𝑘𝛽−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘. (4.11)
To prove the other inequality we consider the polynomial of degree 𝑠_𝑘,𝑃_𝑘(𝑧)=^𝑘_𝑠𝑗=0𝑓_𝑗⋅𝐴_𝑗 (4.12) then 𝜋^𝑝_𝑠𝑘−1𝐾,𝑓≤^+∞_{𝑠𝑗=𝑠𝑘}𝑓_𝑗⋅𝐴_{𝑗𝐿𝑝(𝐾,𝜇)}≤𝐶₀^+∞_{𝑠𝑗=𝑠𝑘}𝑓_𝑗⋅𝐴_𝑗𝐾. (4.13)
By Bernstein-Walsh inequality, we have𝜋^𝑝_𝑘𝐾,𝑓≤𝐶_𝜖^+∞_{𝑠𝑗=𝑠𝑘}(1+𝜖)^𝑠𝑗𝑓_𝑗⋅𝜈_𝑗 (4.14) for 𝑘≥0 and 𝑝≥1. If we take as a common factor (1+𝜀)^𝑠𝑘⋅|𝑓_𝑘|⋅𝜈_𝑘 the other factor is convergent thus, we have 𝜋^𝑝_𝑘𝐾,𝑓≤𝐶(1+𝜖)^𝑠𝑘⋅𝑓_𝑘⋅𝜈_𝑘 (4.15) and by (3.5) of Lemma 3.2, we have, then 𝜋^𝑝_𝑘𝐾,𝑓≤𝐶(1+𝜖)^2𝑠𝑘⋅𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘. (4.16)
We deducelimsup_𝑘→+∞𝛼(𝑘)𝛽−(1/𝑘)log𝜋^𝑝_𝑘𝐾,𝑓=limsup_𝑘→+∞𝛼𝑠_𝑘𝛽−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘. (4.17)
This inequality is a direct consequence of the relation (4.10) and the inequality on coefficients |𝑓_𝑘| given by 𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘≤exp−𝑠_𝑘𝛽⁻¹𝛼𝑠_𝑘^{1/(𝜌+𝜖)},𝑓_𝑘⋅𝜈²≤𝐶_𝜀⋅(1+𝜀)^𝑠𝑘⋅𝜋^𝑝_𝑠𝑘𝐾,𝑓. (4.18)

Applying this lemma we get the following main result:

Theorem 4.2. Let 𝑓∈𝐿^𝑝(𝐾,𝜇), then 𝑓 is μ-almost-surely the restriction to 𝐾 of an entire function in ℂ^𝑛 of finite generalized order 𝜌(𝛼,𝛽) finite if and only if 𝜌𝛼,𝛽=limsup_𝑘→+∞𝛼(𝑘)𝛽−(1/𝑘)log𝜋^𝑝_𝑘𝐾,𝑓<+∞. (4.19)

Proof. Suppose that 𝑓 is μ-almost-surely the restriction to 𝐾 of an entire function 𝑔 of general order 𝜌 (0<𝜌<+∞) and show that 𝜌=𝜌(𝛼,𝛽).
We have 𝑔∈𝐿^𝑝(𝐾,𝜇), 𝑝≥2 and 𝑔=∑_𝑘≥0𝑔_𝑘⋅𝐴_𝑘 in 𝐿²(𝐾,𝜇) Since 𝑔 is an element of 𝐿²(𝐾,𝜇) then 𝑔=∑^+∞_𝑘=0𝑔_𝑘⋅𝐴_𝑘 and according to the Theorem 3.1.𝜌𝑔,𝛼,𝛽=limsup_𝑘→+∞𝛼𝑠_𝑘𝛽−1/𝑠_𝑘log𝑔_𝑘⋅𝜏^𝑠𝑘_𝑘 (4.20) and with the Lemma 4.1, we have limsup_𝑘→+∞𝛼(𝑘)𝛽−(1/𝑘)log𝜋^𝑝_𝑘𝐾,𝑔=limsup_𝑘→+∞𝛼𝑠_𝑘𝛽−1/𝑠_𝑘log𝑔_𝑘⋅𝜏^𝑠𝑘_𝑘. (4.21)
But 𝑔=𝑓 on 𝐾 hence𝜌=limsup_𝑘→+∞𝛼(𝑘)𝛽−(1/𝑘)log𝜋^𝑡_𝑘𝐾,𝑓<+∞. (4.22)
Suppose now that 𝑓 is a function of 𝐿^𝑝(𝐾,𝜇) such that the relation (4.19) is verified.
(1) Let 𝑝≥2, then 𝑓=∑^+∞_𝑘=0𝑓_𝑘⋅𝐴_𝑘, because 𝑓 is an element of 𝐿²(𝐾,𝜇) ((𝐿^𝑝(𝐾,𝜇))_𝑝≥1 is decreasing sequence).
Consider in ℂ^𝑛 the series ∑𝑓_𝑘⋅𝐴_𝑘, 𝑘≥0, we verify easily that this series converges normally on all compact of ℂ^𝑛 to an entire function denoted 𝑓₁. We have 𝑓₁=𝑓, obviously, μ-almost surly on 𝐾, and by Theorem 3.1, we have𝜌𝑓₁=limsup_𝑘→+∞𝛼𝑠_𝑘𝛽−1/𝑠_𝑘log𝑓_𝑘⋅𝜏^𝑠𝑘_𝑘=limsup_𝑘→+∞𝛼(𝑘)𝛽−(1/𝑘)log𝜋^𝑡_𝑘𝐾,𝑓<+∞. (4.23) Applying the Lemma 4.1, we find 𝜌𝑓₁=limsup_𝑘→+∞𝛼(𝑘)𝛽−(1/𝑘)log𝜋^𝑡_𝑘𝐾,𝑓<+∞. (4.24)
Consider the function 𝑓₁=∑_𝑘≥0𝑓_𝑘⋅𝐴_𝑘, we have 𝑓₁(𝑧)=𝑓(𝑧)μ-almost surely for every 𝑧 in 𝐾. Therefore the (𝛼,𝛽)-order of 𝑓₁ is:𝜌𝑓₁,𝛼,𝛽=limsup_𝑘→+∞𝛼(𝑘)𝛽−(1/𝑘)log𝜋^𝑡_𝑘𝐾,𝑓<+∞ (4.25) (see Theorem 3.1).
By Lemma 4.1, we check 𝜌(𝑓₁)=𝜌 so the proof is completed.

(2) Now let 𝑝∈[1,2[ and 𝑓∈𝐿^𝑝(𝐾,𝜇).

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