Generalized Order and Best Approximation of Entire Function in Lp-Norm

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 Ω𝑟={𝑧∈𝐂𝑛;exp𝑉𝐾(𝑧)≤𝑟}, where 𝑉𝐾=sup{(1/𝑑)ln|𝑃𝑑|,𝑃𝑑polynomialofdegree≤𝑑,‖𝑃𝑑‖𝐾≤1} is the Siciak extremal function of a 𝐿-regular compact 𝐾.


Introduction
Let f z ∞ k 0 a k • z λ k be a no constant entire function in complex plane C and let M f, r sup f 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 lim sup r → ∞ ln ln M f, r ln 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, International Journal of Mathematics and Mathematical Sciences of order ρ 0 < ρ < ∞ , is said to be of type σ 0 ≤ σ ≤ ∞ if 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 ρ p, q lim sup r → ∞ log p M f, r 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 σ p, q lim sup r → ∞ log p−1 M f, r log 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 put 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.
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: International Journal of Mathematics and Mathematical Sciences 3 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 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 by 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.

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 Regularity is equivalent to the following Bernstein-Markov inequality see 8 .
For any > 0, there exists an open U ⊃ K such that for any polynomial P P U ≤ e •deg P P K .

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In this case we take U {z ∈ C n ; V K z < }.Regularity also arises in polynomial approximation.For f ∈ C K , we let where P k C n is the set of polynomials of degree at most d.Siciak see 7 showed the following.
Let f be a function defined and bounded on K.For k ∈ N put 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 c V K n associated to a L-regular compact E see 9 and is the class of all function such that:

2.6
For an entire function f ∈ C n we establish a precise relationship between the general growth with respect to the set: 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 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: For all > 0, there exists C C ε is a constant such that BM : for every polynomial of n complex variables of degree at most d.
International Journal of Mathematics and Mathematical Sciences 5 For every set L-regular K and every real r > 1, we have: Note that the regularity is equivalent to the Bernstein-Markov inequality. Let 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 by

2.12
We need the following notations which will be used in the sequel: With that notations and B.W inequality, we have 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:

2.14
where ω a function such that lim x → ∞ ω x 0.

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Assume that, for every c > 0, there exists two constants a and b such that for every x ≥ a: where d u means the differential of u.
Definition 2.3.Let K be a compact L-regular, we put 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, by 2.17 where 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: 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.

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 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 and we have the following results.

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

3.5
Lemma 3.3.For every r > 1 the maximum of the function is reached for x x r solution of the equation

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Consequence.This relation is equivalent to Indeed, the relations

3.11
We verify easily with the relation for every n > n 0 and r > 0.
1 Show that γ ≤ ρ α, β .By definition of γ, we have, for all > 0, ∃k such that for all k > k From the second assertion of the Lemma 3.2, we have for every θ > 1, there exists

3.16
But lim k → ∞ ϕ r, k log r so ϕ r, k ∼ log r for r sufficiently large.Then

3.17
Let r k be a real satisfying

3.19
This is equivalent to

3.22
International Journal of Mathematics and Mathematical Sciences 2 Show that γ ≥ ρ α, β .According to the definition of γ, we have for every > 0 there exists k such that for all k > k According to the first assertion of Lemma 3.2 and BM and BW inequalities, we have and for every entire function

3.25
The term 1 is a constant denoted C 0 , and

3.26
The series 4 is convergent.Let, for r sufficiently, where E x means the integer part of x.Then

3.28
Applying the the relation 3.13 with μ γ and if we replace r by 1 ε r, we obtain:

3.29
And so and for r sufficiently large, we have

3.34
If we put x log 1 ε r , then β log 1 r b ∼ β log 1 r and, This is true for every > 0 hence ρ α, β ≤ γ.Thus the assertion is proved.

Best Approximation Polynomial in L p -Norm
To our knowledge, no similar result is known according to polynomial approximation in L pnorm 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. 4.1

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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 and therefore satisfied, is easily verified by using inequalities Bernstein-walsh and Holder that, we have for all ε > 0 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 H ölder, we have: This shows, according to inequality BM , that: Hence the result International Journal of Mathematics and Mathematical Sciences 13 In both cases, we have therefore 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 get lim sup

4.11
To prove the other inequality we consider the polynomial of degree s k ,

4.13
By Bernstein-Walsh inequality, we have for k ≥ 0 and p ≥ 1.If we take as a common factor 1 ε s k •|f k |•ν k the other factor is convergent thus, we have and by 3.5 of Lemma 3.2, we have, then 4.17

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This inequality is a direct consequence of the relation 4.10 and the inequality on coefficients |f k | given by

4.18
Applying this lemma we get the following main result: 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

4.21
But g f on K hence

4.22
Suppose now that f is a function of L p K, μ such that the relation 4.19 is verified.
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 f 1 .We have f 1 f, obviously, μ-almost surly on K, and by Theorem 3.1, we have Consider the function f 1 k≥0 f k • A k , we have f 1 z f z μ-almost surely for every z in K. Therefore the α, β -order of f 1 is:

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 and according to the Theorem 3.1.ρg, α, β lim sup k → ∞ α s k β − 1/s k log g k • τ s k k β − 1/s k log g k • τ s k k .
25 see Theorem 3.1 .By Lemma 4.1, we check ρ f 1 ρ so the proof is completed.2 Now let p ∈ 1, 2 and f ∈ L p K, μ .By BM inequality and H ölder inequality, we have again the inequality the relation 4.10 and by the previous arguments we obtain the result.