# Generalized Order and Best Approximation of Entire Function inLp-Norm

Authors: Mohammed Harfaoui
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## 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 compactK with respect to the set Ωr={z∈Cn;expVK(z)≤r}, where VK=sup{(1/d)ln|Pd|,Pdpolynomial of degree≤d,∥Pd∥K≤1} is the Siciak extremal function of a L-regular compact K.


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## Body

## 1. Introduction

Letf(z)=∑k=0+∞ak·zλk be a no constant entire function in complex plane ℂ and let (1.1)M(f,r)=sup⁡{|f(z)|,|z|=r,r>0}.
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
(1.2)ρ=limsup⁡r→+∞ln⁡ln⁡M(f,r)ln⁡(r)
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 planeℂ, of order ρ (0<ρ<+∞), is said to be of type σ (0≤σ≤+∞) if(1.3)σ=limsup⁡r→+∞ln⁡M(f,r)rρ.Iff 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 (1.4)ρ(p,q)=limsup⁡r→+∞log⁡[p](M(f,r))log⁡[q](r).It is easy to show thatb≤ρ(p,q)≤+∞ where b=0 if p>q and b=1 if p=q.The functionf 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 (1.5)σ(p,q)=limsup⁡r→+∞log⁡[p-1](M(f,r))(log⁡[q-1](r))ρ(p,q).In their works, the authors have established the relationship of(p,q)-growth of f with respect to the coefficients ak 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. LetK 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(1.6)Ek(K,f)=‖f-Tk‖K,
where the norm ∥·∥K is the maximum on K and Tk is the kth Chebytchev polynomial of the best approximation to f on K.It is known (see [3]) that(1.7)lim⁡k→+∞Ek(K,f)k=0
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.(1.8)lim⁡k→+∞Ek(K,f)k=(ρ⋅e⋅σ)2-ρ
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:IfK be a compact subset of the complex plane ℂ, of positive logarithmic capacity then (1.9)lim⁡k→+∞k(1/ρ)Ek(K,f)k=c(eρσ)1/ρ
if and only if f is the restriction to K of an entire function of order ρ (0<ρ<+∞) and type σ.Recall thatcap([-1,1])=1/2 and capacity of a disk unit is cap(D(O,1))=1.The authors considered the Taylor development off with respect to the sequence (zn)n and the development of f with respect to the sequence (Wn)n defined by(1.10)Wn(z)=∏j=1j=n(z-ηnj),
where (anj)n is the thnth extremal points system of K.The aim of this paper is to establish relationship between the rate at which(πkp(K,f))1/k tends to zero in terms of best approximation in Lp-norm, and the generalized growth of entire functions of several complex variables for a compact subset K of ℂn, where K is a compact well-selected. In this work we give the generalization of these results in ℂn, replacing the circle {z∈ℂ;|z|=r} by the set {z∈ℂn;exp(VE(z))<r}, where VE is the Siciak's extremal function of E a compact of ℂn 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]).
LetK be a compact set in ℂn and let ∥·∥K denote the maximum norm on K. The function
(2.1)VK=sup⁡{1dlog⁡|Pd|,Pdpolynomialofdegree≤d,‖Pd‖K≤1,d∈N}
is called the Siciak's extremal function of the compact K.Definition 2.2.
A compactK in ℂn is said to be L-regular if the extremal function, VK, associated to K is continuous on ℂn.Regularity is equivalent to the following Bernstein-Markov inequality (see [8]).For anyϵ>0, there exists an open U⊃K such that for any polynomial P(2.2)‖P‖U≤eϵ⋅deg⁡(P)‖P‖K.In this case we takeU={z∈ℂn;VK(z)<ϵ}.Regularity also arises in polynomial approximation. Forf∈𝒞(K), we let(2.3)ϵd(K,f)=inf⁡{‖f-P‖K,P∈Pk(Cn)},where𝒫k(ℂn) is the set of polynomials of degree at most d. Siciak (see [7]) showed the followingIfK is L-regular, then (2.4)limsup⁡d→+∞(ɛd(K,f))1/d=1r<1
if and only if f has an analytic continuation to {z∈Cn;VK(z)<log(1/r)}.Letf be a function defined and bounded on K. For k∈ℕ put(2.5)πkp(K,f)=inf⁡{‖f-P‖Lp(K,μ),P∈Pk(Cn)},where𝒫k(ℂn) is the family of all polynomial of degree ≤k and μ the well-selected measure (The equilibrium measure μ=(ddcVK)n associated to a L-regular compact E) (see [9]) and Lp(K,μ), p≥1, is the class of all function such that:(2.6)‖f‖Lp(K,μ)=(∫K|f|pdμ)1/p<∞.For an entire functionf∈ℂn we establish a precise relationship between the general growth with respect to the set:(2.7)Ωr={exp⁡(VK)<r}
and the coefficients of the development of f with respect to the sequence (Ak)k, called extremal polynomial (see [10]). Therefore, we use these results to give the relationship between the generalized growth of f and the sequence (πkp(K,f))k.Recall that the subset ofℂn, Ωr={exp(VK)<r}, replaces the unit disc D(O,r)={|z|<r} in the classical case.It is known that ifK is an compact L-regular of ℂn, there exists a measure μ, called extremal measure, having interesting properties (see [7, 8]), in particular, we have:(P1)
Bernstein-Markov Inequality.For allϵ>0, there exists C=Cɛ is a constant such that
(2.8)(BM):‖Pd‖K=C(1+ɛ)sk‖Pd‖L2(K,μ),
for every polynomial of n complex variables of degree at most d.(P2)
Bernstein-Waish (B.W) Inequality.For every setL-regular K and every real r>1, we have:
(2.9)‖f‖K≤M⋅rdeg⁡(f)(∫K|f|p⋅dμ)1/p.Note that the regularity is equivalent to the Bernstein-Markov inequality.Letα:ℕ→ℕn,k→α(k)=(α1(k),…,αn(k)) be a bijection such that (2.10)|α(k+1)|≥|α(k)|where|α(k)|=α1(k)+⋯+αn(k).Zériahi (see [10]) has constructed according to the Hilbert-Shmidt method a sequence of monic orthogonal polynomial according to a extremal measure (see [8]), (Ak)k, called extremal polynomial, defined by(2.11)Ak(z)=zα(k)+∑j=1k-1ajzα(j)
such that(2.12)‖Ak‖Lp(K,μ)=[inf⁡{‖zα(k)+∑j=1k-1ajzα(j)‖L(K,μ)2,(a1,a2,…,an)∈Cn}]1/αk.We need the following notations which will be used in the sequel:(N1)
νk=νk(K)=∥Ak∥L2(K,μ).(N2)
ak=ak(K)=∥Ak∥K=maxz∈K|Ak(z)| and τk=(ak)1/sk, where sk=deg(Ak).With that notations and (B.W) inequality, we have(2.13)‖Ak‖Ω¯r≤ak⋅rsk,
where sk=deg(Ak). For more details (see [9]).Letα and β be two positives, strictly increasing to infinity differentiable functions ]0,+∞[ to ]0,+∞[ such that for every c>0:(2.14)lim⁡x→+∞α(cx)α(x)=1,lim⁡x→+∞β(1+xω(x))β(x)=1,
where ω a function such that limx→+∞ω(x)=0.Assume that, for everyc>0, there exists two constants a and b such that for every x≥a:
(2.15)|d(β-1(cα(x)))α(log⁡(x))|≤b,
where d(u) means the differential of u.Definition 2.3.
LetK be a compact L-regular, we put
(2.16)Ωr={z∈Cn;exp⁡VK(z)≤r}.
Iff 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)ρ(α,β)=limsup⁡r→+∞α(log⁡(∥f∥Ω¯r))β(log⁡(r)),σ(α,β)=limsup⁡r→+∞α(‖f‖Ω¯r)[β(r)]ρ(α,β),
where ∥f∥Ω¯r=supΩ¯r|f(z)|.Note that in the classical caseα(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.
LetK be a compact L-regular subset of ℂn. Then for every θ>1, there exists an integer Nθ≥1 and a constant Cθ such that:
(2.18)πkp(K,f)≤Cθ(r+1)Nθ(r-1)2N-1‖f‖Ω¯rθrk.
Let f=∑k=0+∞fk·Ak be an entire function. Then for every θ>1, there exists Nθ∈ℕ and Cθ>0 such that
(2.19)|fk|νk≤Cθ(r+1)Nθ(r-1)2N-1‖f‖Ω¯rθrsk,
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 planeℂ.


## 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 set(3.1)Ωr={exp⁡(VK)<r}
and the coefficient of entire function f∈ℂn of the development with respect to the sequence of extremal polynomials.Let(Ak)k be a basis of extremal polynomial associated to the set K defined the relation (2.11). We recall that (Ak)k is a basis of 𝒪(ℂn) (the set of entire functions on ℂn). So if f is an entire function then (3.2)f=∑k≥1fk⋅Ak
and we have the following results.Theorem 3.1.
Iff=∑k≥1fk·Ak then the (α,β)-order ρ(α,β) of f is given by formula
(3.3)ρ(α,β)=limsup⁡k→+∞α(sk)β(-(1/sk)ln⁡|fk|⋅τksk)<+∞.To prove theorem we need the following lemmas.Lemma 3.2.
LetK be a compact L-regular subset of ℂn. Then
(3.4)lim⁡k→+∞[|Ak(z)|νk]1/sk=exp⁡(VK(z)),
for every z∈ℂn∖K̂ the connected component of ℂn∖K,
(3.5)lim⁡k→+∞[‖Ak‖Kνk]1/sk=1.Lemma 3.3.
For everyr>1 the maximum of the function
(3.6)x⟶ω(x,r)=rxexp⁡[-xβ-1(1μα(x))]
is reached for x=xr solution of the equation
(3.7)x=α-1{μβ[log⁡(r)-d(β-1((1/μ)α(x)))d(log⁡(x))]}.Consequence 3.
This relation is equivalent to(3.8)x≤α-1(β(log⁡(r)+b)).
Indeed, the relations(3.9)|d(β-1(cα(x)))d(log⁡(x))|≤b,β-1(α(x)μ)=log⁡(r)-d(β-1(cα(x)))d(log⁡(x))
give
(3.10)log⁡(r)-b≤log⁡(r)-d(β-1(cα(x)))d(log⁡(x))≤log⁡(r)+b.
Then(3.11)β-1(α(x)μ)≤log⁡(r)+b,x≤α-1(β(log⁡(r)+b)).
We verify easily with the relation(3.12)|d(β-1(cα(x)))α(log⁡(x))|≤b,
that
(3.13)rnexp⁡[-nβ-1(1μα(n))]≤exp⁡{bα-1(μ⋅β(log⁡(r)+b))}
for every n>n0 and r>0.Proof of Theorem3.1.
Putγ=limsupk→+∞(α(sk)/β(-(1/sk)ln|fk|·τksk)) and show that γ=ρ(α,β).
(1) Show   thatγ≤ρ(α,β).
By definition ofγ, we have, for all ϵ>0, ∃kϵ such that for all k>kϵ(3.14)α(sk)β(-(1/sk)ln⁡|fk|⋅νksk)<γ+ϵ.
From the second assertion of the Lemma3.2, we have for every θ>1, there exists Nθ∈ℕ and Cθ>0 such that(3.15)log⁡(|fk|τksk)≤log⁡(Cθ)+log⁡((r+1)Nθ(r-1)2N-1)-sklog⁡(r)+α-1[(ρ+ϵ)⋅β(log⁡(rθ))]=φ(r)
or
(3.16)-1sklog⁡(|fk|τksk)≥-1sklog⁡(Cθ)-Nθsklog⁡(r+1)-2N-1sklog⁡(r-1)+log⁡(r)-1skα-1((ρ+ϵ)β(log⁡(rθ)))=φ(r,k).
Butlimk→+∞φ(r,k)=log(r) so φ(r,k)~log(r) for r sufficiently large.
Then(3.17)-1sklog⁡(|fk|τksk)≥(1+o(1))log⁡(r).
Letrk be a real satisfying(3.18)rk=exp⁡[β-1(1ρ+ϵα(sk))].
Then(3.19)-1sklog⁡(|fk|τksk)≥β-1(1ρ+ϵ⋅α(sk))(1+o(1)).
This is equivalent to(3.20)β[-(1/sk)log⁡(|fk|τksk)]α(sk)≥1ρ+ϵ,
or
(3.21)α(sk)β[-(1/sk)log⁡(|fk|τksk)]≤ρ+ϵ,
hence
(3.22)γ≤ρ(α,β).
(2) Show thatγ≥ρ(α,β). According to the definition of γ, we have for every ϵ>0 there exists kϵ such that for all k>kϵ(3.23)|fk|τksk≤exp⁡[-skβ-1(α(sk)γ+ϵ)]
for k≥kϵ sufficiently large.
According to the first assertion of Lemma3.2 and (BM) and (BW) inequalities, we have(3.24)‖Ak‖Ω¯r≤ak⋅rsk
and for every entire function
(3.25)f(z)=∑k≥0fk⋅Ak(z),‖f‖Ω¯r≤∑k=0kϵak⋅rsk︸(1)+Cϵ∑k=kϵ+1kr|fk|⋅νsk⋅(1+ϵ)rsk︸(2)+Cϵ∑k=kr+1+∞|fk|⋅τksk⋅rsk︸(3).
The term (1) is a constant denotedC0, and(3.26)∑k=kϵ+1kr|fk|⋅τksk⋅(1+ϵ)rsk≤((1+ϵ)r)sk(r)∑k=0+∞|fk|⋅τksk︸(4).
The series    (4) is convergent.
Let, forr sufficiently,(3.27)N(r)=E[α-1((γ+ϵ)β(log⁡2(1+ϵ)r))],
where E(x) means the integer part of x.
Then(3.28)|fk|⋅νk⋅((1+ϵ)r)sk≤exp⁡[-sk⋅β-1(α(sk)γ+ϵ)]((1+ɛ)r)sk.
Applying the the relation (3.13) with μ=γ+ϵ and if we replace r by (1+ɛ)r, we obtain:(3.29)‖f‖Ω¯r≤exp⁡[b⋅α-1((γ+ϵ)⋅β(log⁡((1+ɛ)r)+b))].
And so
(3.30)‖f‖Ω¯r≤C⋅((1+ɛ)r)sk⋅exp⁡[-sk⋅β-1(α(sk)γ+ϵ)],
or
(3.31)‖f‖Ω¯r≤C⋅exp⁡b⋅α-1((γ+ϵ)⋅β(log⁡((1+ɛ)r)+b)).
Thus(3.32)log⁡(‖f‖Ω¯r)≤log⁡(C)+b⋅α-1((γ+ϵ)⋅β(log⁡((1+ɛ)r)+b))
and for r sufficiently large, we have
(3.33)α(b-1log⁡(‖f‖Ω¯r))β(log⁡((1+ɛ)r)+b)≤γ+ϵ.
Butα(c·x)~α(x) near to the infinity, thus(3.34)α(b-1log⁡(|f|Ω¯r))~α(log⁡(‖f‖Ω¯r)).
If we putx=log((1+ɛ)r), then β(log((1+ϵ)r)+b)~β(log((1+ϵ)r)) and,(3.35)α(log⁡(‖f‖Ω¯r))β(log⁡(r))≤γ+ϵ.
This is true for everyϵ>0 hence ρ(α,β)≤γ. Thus the assertion is proved.


## 4. Best Approximation Polynomial inLp-Norm

To our knowledge, no similar result is known according to polynomial approximation inLp-norm (1≤p≤∞) with respect to a measure μ on K in ℂn.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.
Letf=∑k≥0fk·Ak an element of Lp(K,μ), for p≥1, then
(4.1)limsup⁡k→+∞α(k)β[-(1/k)log⁡(πkp(K,f))]=limsup⁡k→+∞α(sk)β[-(1/sk)log⁡(|fk|⋅τksk)].Proof of Lemma4.1.
The proof is done in two stepsp≥2 and 1<p<2.Step 1.
Iff∈Lp(K,μ) where p≥2, then f=∑k=0+∞fk·Ak with convergence in L2(K,μ), hence for k≥0(4.2)fk=1νk2∫Kf⋅A¯kdμ
and therefore
(4.3)fk=1νk2∫K(f-Pk-1)⋅A¯kdμ
(because deg(Ak)=sk).
Since the relation,(4.4)|fk|≤1νk2∫K|f-Pk-1|⋅|A¯k|μ
satisfied, is easily verified by using inequalities Bernstein-walsh and Holder that, we have for all ɛ>0(4.5)|fk|⋅νk≤Cɛ⋅(1+ɛ)sk⋅πsk-1p(K,f).
for all k≥0.Step 2.
If1≤p<2, let p′ such that 1/p+1/p′=1, we have p′≥2. According  to the inequality of Hölder, we have:
(4.6)|fk|⋅νk2≤‖f-Pk-1‖Lp(K,μ)⋅‖Ak‖Lp′(K,μ).
But,(4.7)‖Ak‖Lp′(K,μ)≤C⋅‖Ak‖K=C⋅ak(K).
This shows, according to inequality (BM), that:(4.8)|fk|⋅νk2≤C⋅Cɛ⋅(1+ɛ)sk⋅‖f-Psk-1‖Lp(K,μ).
Hence the result(4.9)|fk|⋅νk2≤Cɛ′⋅(1+ɛ)sk⋅πskp(K,f).
In both cases, we have therefore(4.10)|fk|⋅νk2≤Aɛ⋅(1+ɛ)sk⋅πskp(K,f),
where Aɛ is a constant which depends only on ɛ.
After passing to the upper limit in the relation (4.10) and Applying the relation (3.5) of the Lemma 3.2 we get(4.11)limsup⁡k→+∞α(k)β[-(1/k)log⁡(πkp(K,f))]≥limsup⁡k→+∞α(sk)β[-(1/sk)log⁡(|fk|⋅τksk)].
To prove the other inequality we consider the polynomial of degreesk,(4.12)Pk(z)=∑sj=0kfj⋅Aj
then
(4.13)πsk-1p(K,f)≤∑sj=sk+∞|fj|⋅‖Aj‖Lp(K,μ)≤C0∑sj=sk+∞|fj|⋅‖Aj‖K.
By Bernstein-Walsh inequality, we have(4.14)πkp(K,f)≤Cϵ∑sj=sk+∞(1+ϵ)sj|fj|⋅νj
for k≥0 and p≥1. If we take as a common factor (1+ɛ)sk·|fk|·νk the other factor is convergent thus, we have
(4.15)πkp(K,f)≤C(1+ϵ)sk⋅|fk|⋅νk
and by (3.5) of Lemma 3.2, we have, then
(4.16)πkp(K,f)≤C(1+ϵ)2sk⋅|fk|⋅τksk.
We deduce(4.17)limsup⁡k→+∞α(k)β[-(1/k)log⁡(πkp(K,f))]=limsup⁡k→+∞α(sk)β[-(1/sk)log⁡(|fk⋅τksk|)].
This inequality is a direct consequence of the relation (4.10) and the inequality on coefficients |fk| given by
(4.18)|fk|⋅τksk≤exp⁡[-skβ-1(α(sk)1/(ρ+ϵ))],|fk|⋅ν2≤Cɛ⋅(1+ɛ)sk⋅πskp(K,f).Applying this lemma we get the following main result:Theorem 4.2.
Letf∈Lp(K,μ), then f is μ-almost-surely the restriction to K of an entire function in ℂn of finite generalized order ρ(α,β) finite if and only if
(4.19)ρ(α,β)=limsup⁡k→+∞α(k)β[-(1/k)log⁡(πkp(K,f))]<+∞.Proof.
Suppose thatf 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 haveg∈Lp(K,μ), p≥2 and g=∑k≥0gk·Ak in L2(K,μ) Since g is an element of L2(K,μ) then g=∑k=0+∞gk·Ak and according to the Theorem 3.1.(4.20)ρ(g,α,β)=limsup⁡k→+∞α(sk)β[-(1/sk)log⁡(|gk|⋅τksk)]
and with the Lemma 4.1, we have
(4.21)limsup⁡k→+∞⁡α(k)β(-(1/k)log⁡(πkp(K,g)))=limsup⁡k→+∞⁡α(sk)β[-(1/sk)log⁡(|gk|⋅τksk)].
Butg=f on K   hence(4.22)ρ=limsup⁡k→+∞α(k)β(-(1/k)log⁡(πkt(K,f)))<+∞.
Suppose now thatf is a function of Lp(K,μ) such that the relation (4.19) is verified.
(1) Letp≥2, then f=∑k=0+∞fk·Ak, because f is an element of L2(K,μ) ((Lp(K,μ))p≥1 is decreasing sequence).
Consider inℂn the series ∑fk·Ak, k≥0, we verify easily that this series converges normally on all compact of ℂn to an entire function denoted f1. We have f1=f, obviously, μ-almost surly on K, and by Theorem 3.1, we have(4.23)ρ(f1)=limsup⁡k→+∞α(sk)β[-(1/sk)log⁡(|fk|⋅τksk)]=limsup⁡k→+∞α(k)β[-(1/k)log⁡(πkt(K,f))]<+∞.
Applying the Lemma 4.1, we find
(4.24)ρ(f1)=limsup⁡k→+∞α(k)β[-(1/k)log⁡(πkt(K,f))]<+∞.
Consider the functionf1=∑k≥0fk·Ak, we have f1(z)=f(z)μ-almost surely for every z in K. Therefore the (α,β)-order of f1 is:(4.25)ρ(f1,α,β)=limsup⁡k→+∞α(k)β[-(1/k)log⁡(πkt(K,f))]<+∞
(see Theorem 3.1).
By Lemma4.1, we check ρ(f1)=ρ so the proof is completed.(2) Now letp∈[1,2[ and f∈Lp(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.



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