# Best Polynomial Approximation inLp-Norm and (p,q)-Growth of Entire Functions

Authors: Mohamed El Kadiri; Mohammed Harfaoui
Journal: Abstract and Applied Analysis
 (2013)

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## 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 ℂn by means of the best polynomial approximation in terms of Lp-norm, with respect to the set Ωr={z∈Cn;expVK(z)≤r}, where VK=sup{(1/d)log|Pd|,Pdpolynomialofdegree≤d,∥Pd∥K≤1} is the Siciak's extremal function on an L-regular nonpluripolar compact K is not pluripolar.


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## Body

## 1. Introduction

Letf(z)=∑k=0+∞‍akzλk 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≤+∞):
(1)ρ=limsupr→+∞loglog(M(f,r))log(r).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 orderρ, 0<ρ<+∞, is said to be of type σ, 0≤σ≤+∞, if
(2)σ=limsupr→+∞log(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 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
(3)ρ(p,q)=limsupr→+∞log[p](M(f,r))log[q](r),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 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 et al. have also defined the concept of the(p,q)-type σ(p,q), for b<ρ(p,q)<+∞, by
(4)σ(p,q)=limsupr→+∞log[p-1]((M(f,r)))(log[q-1](r))ρ(p,q).In their works, the authors established the relationship of(p,q)-growth of f with respect to the coefficients ak in the Maclaurin series of f.We have also many results in terms of polynomial approximation in classical case. LetK 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
(5)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.Bernstein showed (see [3, page 14]), for K=[-1,1], that there exists a constant ρ>0 such that
(6)limk→+∞k1/ρEk(K,f)k
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:
(7)limk→+∞Ek(K,f)k=(ρeσ)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]) generalized this result to a compact K of the complex plane ℂ of positive logarithmic capacity, denoted c=cap(K) as follows.IfK is a compact subset of the complex plane ℂ, of positive logarithmic capacity, then
(8)limk→+∞k(Ek(K,f))ρ/k=cρeρσ
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 off with respect to the sequence (zn)n and the development of f with respect to the sequence (Wn)n defined by
(9)Wn(z)=∏j=1j=n‍(z-ηnj),n=1,2,…,
where η(n)=(ηn0,ηn1,…,ηnn) 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(Ek(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 Lp-norm for a compact of ℂn.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 Lp-norm for a compact of ℂn 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
(10)α(x)=logp-1(x),β(x)=logq-1(x)for(p,q)∈ℕ2.So we establish relationship between the rate at which(πkp(K,f))1/k, for k∈ℕ, 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 and
(11)πkp(K,f)=inf{∥f-P∥Lp(K,μ);P∈𝒫k(ℂn)},
where 𝒫k(ℂn) is the family of all polynomials of degree ≤k and μ is the well selected measure (the equilibrium measure μ=(ddcVK)n associated to a L-regular compact K) (see [8]) and Lp(K,μ), p≥1, is the class of all functions such that
(12)∥f∥Lp(K,μ)=(∫K‍|f|pdμ)1/p<∞.In this work we give the generalization of these results inCn, replacing the circle {z∈ℂ;|z|=r} by the set {z∈ℂn;exp(VK(z))<r}, where VK is the Siciak’s extremal function of K, a compact of ℂn satisfying some properties (see [9, 10]), and using the development of f with respect to the sequence (Ak)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, letK be a compact of ℂn and let ∥·∥K denote the maximum norm on K.Multivariate polynomial inequalities are closely related to the Siciak extremal function associated with a compact subsetK of ℂn,
(13)VK=sup{1dlog|Pd|,Pdpolynomialinℂnof degree≤d,∥Pd∥K≤1{1d}}.Siciak’s function establishes an important link between polynomial approximation in several variables and pluripotential theory.It is known (see [10]) that
(14)VK(z)=sup{u(z):u∈ℒ(ℂn);u≤0onK},
where
(15)ℒ(ℂn)={u∈PSH(ℂn):u(z)-log(|z|)≤O(1)as|z|→∞(ℂn)}
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 VK*(z)=limsupw→zVK(w) is the unique function in the class ℒ(ℂn) which vanishes on K except perhaps for a pluripolar subset and satisfies the complex Monge-Ampère equation (see [12]):
(16)(ddcVK)n=0onℂn∖K.Ifn=1, the Monge-Ampère equation reduces to the classical Laplace equation.For this reason, the functionVK* 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(17)VK=sup{1dlog|Pd|,Pdpolynomialofdegree≤d,∥Pd∥K≤1{1d}}
is called the Siciak’s extremal function of the compact K.Definition 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 [9]).For anyϵ>0, there exists an open U⊃K such that for any polynomial P(18)∥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
(19)ϵk(K,f)=inf{∥f-P∥K,P∈𝒫k(ℂn)},
where 𝒫k(ℂn) is the set of polynomials of degree at most d. Siciak showed that (see [10]).IfK is L-regular, then
(20)limsupk→+∞(εk(K,f))1/k=1R<1
if and only if f has an analytic continuation to
(21){z∈Cn;VK(z)<log(1R)}.It is known that ifK is a compact L-regular of ℂn, 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=Cε    such that
(22)(BM):∥Pd∥K=C(1+ε)sk∥Pd∥L2(K,μ),for every polynomial ofn 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
(23)∥f∥K≤Mrdeg(f)(∫K|f|pdμ)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
(24)|α(k+1)|≥|α(k)|,where|α(k)|=α1(k)+⋯+αn(k).Zeriahi (see [11]) has constructed according to the Hilbert-Schmidt method a sequence of monic orthogonal polynomials according to an extremal measure (see [9]), (Ak)k, called extremal polynomial, defined by
(25)Ak(z)=zα(k)+∑j=1k-1‍ajzα(j)
such that
(26)∥Ak∥Lp(K,μ)=[inf{∥zα(k)+∑j=1k-1‍ajzα(j)∥L(K,μ)p,(a1,a2,…,an)∈ℂn∥zα(k)+∑j=1k-1‍ajzα(j)∥}]1/αk.
We need the following notations and lemma which will be used in the sequel (see [2]): (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).Forp∈ℕ, put, for p≥1 and x>0,
(27)log[p](x)=log(log[p-1](x)),exp[p](x)=exp(exp[p-1](x)),Λ[p]=∏k=1p‍log[k](x),E[p](x)=∏k=0p‍expk(x),log[0](x)=x,exp[0](x)=x.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)
(28)E[p]-1(x)={x,ifp=0,log[p-1]{log(x)-log[2](x)+o(log[3](x))},ifp=1,2,…,(RR5)
(29)limx→+∞exp(E[p-2](x))={e,ifp=2,1,ifp≥3,(RR6)
(30)limx→+∞[exp[p-1](E[p-2]-1(x))]1/x={e,ifp=2,1,ifp≥3.For more details of these results, see [2].Definition 4.
LetK be a compact L-regular and put
(31)Ωr={z∈ℂn;expVK(z)≤r}.
An entire functionf is said to be of (K,p,q)-order ρK(p,q) if it is of index-pair (p,q) such that
(32)ρ=ρK(p,q)=limsupr→+∞log[p](∥f∥Ω¯r)log[q]r.
Ifρ∈[β,+∞[, the (K,p,q)-type is defined by
(33)σ=σK(p,q)=limsupr→+∞log[p-1](∥f∥Ω¯r)(log[q-1]r)ρ,
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
(34)Ωr={exp(VK)<r}
and the coefficients of entire function f on ℂn 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(Ak)k be the basis of extremal polynomials associated to the set K defined by (25). Recall that (Ak)k is a basis of the vector space of entire functions, hence if f is an entire function, then
(35)f=∑k≥1‍fkAk.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.
LetK be a compact L-regular subset of  ℂn and let f be an entire function such that f=∑k=0+∞‍fkAk. Then for every θ>1, there exists an integer Nθ≥1 and a constant Cθ such that
(36)πkp(K,f)≤Cθ(r+1)Nθ(r-1)2N-1∥f∥Ω¯rθrk,(37)|fk|νk≤Cθ(r+1)Nθ(r-1)2N-1∥f∥Ω¯rθrsk,
where Nθ∈ℕ and Cθ>0 are constant not depending on (r,k,f).Lemma 6.
IfK is an L-regular, then the sequence of extremal polynomials (Ak)k satisfies
(38)limk→+∞(|Ak(z)|νk)1/sk=exp(VK(z))
for every z∈ℂn, and
(39)limk→+∞(∥Ak∥Kskνk)1/sk=1.Recall that the second assertion (37) of Lemma 5 replaces the Cauchy inequality for complex function defined on the complex plane ℂ.Theorem 7.
Letf=∑k≥1‍fkAk be an entire function. Then f is said of a finite (K,p,q)-order ρK(p,q) if and only if
(40)L(p,q)=limsupk→+∞log[p-1](sk)log[q-1][(-1/sk)log(|fk|·τksk)]<+∞
and ρK(p,q)=P1(L(p,q)), where
(41)P1(L(p,q))={L(p,q),ifp>q,1+L(2,2),ifp=q=2,max(1,L(p,q)),if3≤p=q<+∞,+∞,ifp=q=+∞,
for (p,q)∈ℕ2 with p≥q.Proof.
Putρ=ρK(p,q). Let us prove that ϱ≥P1(L(p,q)). If f is of finite (p,q)-order ρ, then we have
(42)ρ=limsupr→+∞log[p]∥f∥Ω¯rlog[q]r=limsupr→+∞log[p]∥f∥Ω¯rθlog[q](rθ).
Thus for every ε>0 there exists r(ε) such that for every r>r(ε)(43)log(∥f∥Ω¯rθ)≤exp[p-2](log[q-1](rθ))ρ+ε.
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)(44)log(|fk|τksk)≤sklog(1+ε)+log(Cθ)+Nθlog(1+r)-(2N-1)log(r-1)-sklog(r)+log(∥f∥Ω¯rθ)
for r>r(ε) and k>k(ε). But for r>r(ε) and k>k(ε) we have
(45)-1sklog(|fk|τksk)≥-1sk·log(Cθ(1+ε)·log(1+r)Nθ(r-1)(2N-1))+exp[p-2](log[q-1](rθ))ρ+ε.
Then, by proceeding to limits ask→∞, we get for r sufficiently large
(46)log(|fk|τksk)≤(1+o(1))log(r).
(i) For(p,q)≠(2,2) with p>q, let
(47)rk=1θexp[q-1](log[p-2](skρ+ε)1/ρ+ε)>r(ε).
Then if we replace in the equality (46) r by rk, we get easily that for k sufficiently large
(48)-1sklog(|fk|τksk)≥exp[q-2](log[p-2](skρ+ε)1/(ρ+ε))[1+o(1)].
After passing to the upper limit, we get for p>q(49)limsupk→+∞log[p-1](sk)log[q-1]((-1/sk)log(|fk|·τksk))≤ρ.
(ii) For3≤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 rk=(1/θ)exp(sk/(ρ+ε))1/(ρ-1+ε)>r(ε) and in the same way we show that
(50)-1sklog(fk∣τksk)≥(skρ+ε)1/(ρ-1+ε)·ρ-1+ερ+ε·(1+o(1))
for k sufficiently large, thus
(51)limsupk→+∞log(sk)log((-1/sk)log(|fk|·τksk))≤ρ-1.
By combining (i), (ii), and (iii) we haveρ≥P1(L(p,q)). This result holds obviously if ρ=+∞.
We prove now reverse inequalityρ≤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)(52)|fk|τksk≤exp(-skexp[q-2](log[p-2](sk))1/(L+ε)),
where L=L(p,q), for simplification.
Letk(r) be a positive integer such that, for k≥k(r),
(53)sk(r)≤exp[p-2](log[q-1](2(1+ε)r)L+ε)<sk(r)+1
and k(r)>k(ε), by (39), (37), (BM) and (BW) inequalities, there exists k0∈ℕ, such that
(54)∥f∥Ω¯r≤∑k=0k=k(ε)‍|fk||Ak(z)|+Cε∑k=k(ε)+1k=k(r)‍|fk|τkskrsk+Cε′·∑k=k(r)+1+∞‍|fk|τksk((1+ε)r)sk.
Indeed,(55)∥f∥Ω¯r≤∑k=0k=+∞‍|fk|·|Ak(z)|
(because z satisfies exp(VK(z))≤r).
By (38) and (39), for k sufficiently large we have
(56)|Ak|≤(1+ε)sk[exp(VK(z))]sk,τksk≤(1+ε)skνk.
Therefore, forr sufficiently large we have
(57)∥f∥Ω¯r≤Ak(ε)+∑k=kε+1N(r)‍fkτksk︸1+∑k=N(r)+∞‍2-sk︸2,
where Ak(ε) is a polynomial of degree not exceeding kε. By using (46) we get
(58)∥f∥Ω¯r≤Ak(ε)+cε(2(1+ε)r)sk(r)×∑k=0+∞‍exp{-skexp[q-2](ln[p-2](sk)L+ε)}︸1+∑k=0+∞‍2-sk︸2.
By (52) the series (1) is convergent, and (2) is obviously convergent, hence we have for r sufficiently large
(59)∥f∥Ω¯r≤Ak(ε)+A1(2(1+ε)r)sk(r)+1,
where A1 is a constant. Thus, for r sufficiently large we obtain
(60)log(∥f∥Ω¯r)≤sklog((1+ε)r)(1+o(1)).
Therefore, for r sufficiently large
(61)log[2](∥f∥Ω¯r)≤log(sk)+log[2]((1+ε)r)+o(1).
Forr sufficiently large let
(62)sk=E[exp[p-2]((log[q-1](2r))L+ϵ)],
where E(x) is the integer part of x. Replacing sk in the inequality (61) we get
(*)log[2](∥f∥Ω¯r)≤exp[p-3](log[q-1](2r)L+ε)+log[2]((1+ε)r)+o(1).
To prove the result we proceed in three steps.
Step1.  For (p,q)=(2,2), we have
(63)log[2](∥f∥Ω¯r)≤(L+ε)log[2](2r)+log[2]((1+ε)r)+o(1).
Then
(64)log[2](∥f∥Ω¯r)log[2](2r)≤(L+ε)+1+o(1).
Proceeding to the upper limit we get ρ≤1+L(2,2)=P1(L(2,2)).
Step2.  For 3≤p=q, since ρ≥1, we get ρ≤max(1(L(p,p))=P1(L(p,q)).
Step3.  For p<q, the relation (*) is equivalent to
(65)log[2](∥f∥Ω¯r)≤exp[p-3](log[q-1](2r)L+ε)(1++o(1)).
Then, for r sufficiently large
(66)log[p](∥f∥Ω¯r)≤log(log[q-1](2r)L+ε)(1+o(1))=(L+ε)(log[q](2r))(1+o(1)).
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 forp≥q≥1, ρ≤P1(L(p,q)). The inequality is obviously true for L(p,q)=+∞.Theorem 8.
Letf be an entire function of (K,p,q)-order ρK(p,q)∈]β,∞[. Then f is of finite  (K,p,q)-type σK(p,q) if and only if
(67)γ(p,q)=limsupk→+∞log[p-2](sk)[log[q-2]((-1/sk)log|fk|τksk)]ρ-C<+∞
and σK(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), sk=deg(Ak), and
(68)M(p,q)={1,ifp≥3,1e·ρ(2,1),if(p,q)=(2,1),(ρ(2,2)-1)ρ(2,2)-1(ρ(2,2))ρ(2,2),(p,q)=(2,2).Proof.
Let us first prove thatσK(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),
(69)|fk|τksk≤exp(-skexp[q-2](log[p-2](sk)γ+ε)1/(ρ-C)).
Letk(r) be a positive integer such that
(70)k(r)≤exp[p-2]((γ+ε)(log[q-1](2(1+ε)r))ρ-C)<k(r)+1
and k≥k(r). By the estimate (39) from Lemma 6 and the (BM) and (BW) inequalities, there exists k0∈ℕ such that
(71)∥f∥Ω¯r≤k(r)+maxk≥0{fk·τksk((1+ε)r)sk}+∑k=0‍2-sk.
Put
(72)H(r)=maxx≥0{ω(r,x)},
where
(73)ω(r,x)=-xexp[q-2]([log[p-2](x)γ+ε]1/ρ-C)+xlog((1+ε)r).
By repeating the argument used in the proof of Theorem7 one may easily check that
(74)σK(p,1)≤M(p,1)γ(p,1).
For example we will show thatσK(p,1)≤M(p,1)γ(p,1). By the relation (69) we have
(75)|fk|τksk≤(γ+εlog[p-2](sk))sk/ρ.
The maximum of the  functionx↦ω(x,r) is reached for x=xr, where xr is the solution of the equation
(76)-log(log[p-2](x)γ+ε)1/ρ+log((1+ε)r)=E[1-p](x)ρ.
Forp=2 the relation (75) becomes
(77)-log(xγ+ε)1/ρ+log((1+ε)r)=1ρ.
Thus
(78)xr=1e(γ+ε)((1+ε)r)ρ.
Therefore H(r)=ω(r,xr)=xr/ρ and ω(r,xr)=(1/eρ)(γ+ε)((1+ε)r)ρ and by (69) we get
(79)∥f∥Ω¯r≤C0+(γ+ε)((1+ε)r)ρexp(H(r))
which gives for r sufficiently large
(80)log(∥f∥Ω¯r)≤H(r)+ρlog((1+ε)r)(1+o(1)).
Passing to the upper limit when r→+∞ we get
(81)σK(2,1)≤M(2,1)γ(2,1).
Forp≥3 we have
(82)ω(r,xr)=xrρK[1-p](x)=xr2ρ⋀[p-2](xr).
Therefore
(83)log(ω(r,xr))=2log(xr)-log(ρ)-log(⋀[p-2](xr))=2log(xr)-∑k=1p-1‍logk(xr).
Hence
(84)log(ω(r,xr))log(xr)=1-∑k=1p-1‍logk(xr)log(xr).
Then, for r sufficiently large,
(85)log(ω(r,xr))~log(xr)~exp[p-2][(γ+ε)((1+ε)r)ρ],∥f∥Ω¯r≤exp[p-2][(γ+ε)((1+ε)r)ρ].
We obtain the result after passing to the upper limit.Remark 9.
Ifn=1 and (p,q)=(2,1), we know that
(86)limk→+∞(τk)1/sk=τ(K)=cap(K).
Then by using Theorems 7 and 8 we get
(87)ρK(2,1)=limk→+∞-sklog(sk)log(|fk|)=ρ(2,1),σK(2,1)=limsupr→+∞sk·([|fk|·τk]1/sk)ρK(2,1)=σ(2,1)(cap(K))ρK(2,1)
which gives the result of Winiarski.Remark 10.
The notion of the type associated to a compact inℂ 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 ℂn, (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 ofLp-Norm

Letf be a bounded function defined on a L-regular compact K of ℂn.The object of this section is to study the relationship between the rate of the best polynomial approximation off in Lp-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 inLp-norm (1≤p≤∞) with respect to a measure μ on K in ℂn. 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
(88)πkp(K,f)=inf{∥f-P∥Lp(K,μ),P∈𝒫k(ℂn)}
which extend the classical results of Reddy and Winiarsk in ℂn. We need the following lemmas.Lemma 11.
IfK is compact L-regular in ℂn, then every function f∈L𝒫2(K,μ) can be written in the form
(89)f=∑k=0+∞‍fk·Ak,
where L𝒫2(K,μ) is the closed subspace of L2(K,μ) generated by the restrictions to E of polynomials ℂn and (Ak) is the sequence defined by (25).Lemma 12.
Let(Ak) be the sequence defined by (25) and f=∑k≥0‍fkAk an element of Lp(K,μ), for p≥1, then
(90)limsupk→+∞log[p-1](k)log[q-1]((-1/k)log(πkp(K,f)))=limsupk→+∞log[p-1](sk)log[q-1]((-1/sk)log(|fk|·τksk)),(91)limsupk→+∞log[p-2](k)(log[q-2]((-1/k)log(πkp(K,f))))ρ-C=limsupk→+∞log[p-2](sk)(log[q-2]((-1/sk)log(|fk|·τksk)))ρ-C.Proof of Lemma12.
The proof is done in two steps (p≥2 and 1<p<2). Let f=∑k≥0‍fk·Ak be an element of Lp(K,μ).
Step1. If f∈Lp(K,μ) with p≥2, then f=∑k=0+∞‍fk·Ak with convergence in L2(K,μ), where fk=1/νk2∫K‍fA¯kdμ, k≥0 and therefore fk=(1/νk2)∫K‍(f-Psk-1)·A¯kdμ (because deg(Ak)=sk). Since |fk|≤(1/νk2)∫K‍|f-Psk-1|·|A¯k|dμ, we obtain easily, using Bernstein-Walsh inequality and de Hölder inequality, that we have for any ε>0(92)|fk|·νk≤Cε·(1+ε)skπsk-1p(K,f)
for every k≥0.
Step2.  If 1≤p<2, let p′ such 1/p+1/p′=1, then p′≥2. By the Hölder inequality we have
(93)|fk|νk2≤∥f-Pk-1∥Lp(K,μ)∥Ak∥Lp′(K,μ).
But ∥Ak∥Lp′(E,μ)≤C∥Ak∥K=Cak(K), therefore, by the (BM) inequality, we have
(94)|fk|νk2≤CCε(1+ε)sk∥f-Psk-1∥Lp(K,μ).
Hence
(95)|fk|νk2≤Cε′(1+ε)skπskp(K,f).
Thus in both cases we have
(96)|fk|νk≤Aε(1+ε)skπskp(K,f),
where Aε is a constant which depends only on ε. After passing to the upper limit (96) gives
(97)limsupk→+∞log[p-1](k)log[q-1]((-1/k)log(πkp(K,f)))≥limsupk→+∞log[p-1](sk)log[q-1]((-1/sk)log(|fk|·τksk)).
To prove the other inequality we consider the polynomial of degree sk(98)Psk(z)=∑j=0j=k‍fjAj,
then
(99)πsk-1p(K,f)≤∑sj=sk+∞‍|fj|∥Aj∥Lp(K,μ)≤C0∑sj=sk+1+∞‍|fj|∥Aj∥K.
By the Bernstein-Walsh inequality we have
(100)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+ε)skνk, the other factor is convergent, thus we have
(101)πkp(K,f)≤C(1+ε)sk·νk
and by (39) of Lemma 6 we have then
(102)πkp(K,f)≤C(1+ε)sk·|fk|·τksk.
We then deduce that
(103)limsupk→+∞log[p-1](k)log[q-1]((-1/k)log(πkp(K,f)))≤limsupk→+∞log[p-1](sk)log[q-1]((-1/sk)log(|fk|·τksk)).
This inequality is a direct consequence of (102) and the inequality on coefficients |fk| given by
(104)|fk|·τksk≤exp(-skexp[q-2](ln[p-2](sk)1/(ρ+ε))).Applying the above lemma we get the following main result.Theorem 13.
Letf be an element of Lp(K,μ), then(1)
f is μ-a·s the restriction to K of an entire function in ℂn of finite (K,p,q)-order ρ if  and only if
(105)ρ1(p,q)=limsupk→+∞log[p-1](k)log[q-1]((-1/k)log(πkp(K,f)))<+∞
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
(106)σ1(p,q)=limsupk→+∞log[p-2](k)(log[q-2]((-1/k)log(πkp(K,f))))ρ-C<+∞
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 thatf 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∈Lp(K,μ), p≥2 and g = ∑k≥0‍gk·Ak in L2(K,μ), where gk=(1/νk)∫K‍fA¯kdμ, k≥0. From (40) of Theorem 7 we get ρ=P1(L(p,q)), where
(107)L(p,q)=limsupk→+∞log[p-1](sk)log[q-1]((-1/sk)log|gk|·τksk(E)).
Butg=f on K, thus by Lemma 11 we have ρ=L(ρ1(p,q)).
Conversely, suppose now thatf is a function of Lp(K,μ) such that the relation (105) holds.
(1) Letp≥2, then we have f=∑k≥0‍fkAk because f∈L2(K,μ), (Lp(K,μ)⊂L2(K,μ)) and (Ak)k is a basis of L2(K,μ) as in Section 3. Consider in ℂn the series ∑‍fkAk. By (90) of Lemma 12 one may easily check that this series converges normally on every compact subset of ℂn to an entire function denoted f1 (this result is a direct consequence of the inequality (BM) and the inequality on coefficients |fk|). We have obviously f1=fμ-a.s on K, and by Theorem 8, the (K,p,q)-order of f1 is
(108)ρ(f1,p,q)=limsupk→+∞log[p-1](sk)log[q-1]((-1/sk)log|fk|·τksk).
By Lemma 12 we check that ρ(f1,p,q)=ρ so the proof is completed for p≥2.
(2) Now letp∈[1,2[ and  f∈Lp(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 Theorem8 and the arguments discussed above, hence we omit the details.Remark 14.
(1) Ifn=1 and (p,q)=(2,1), using the results of Theorem 13 we obtain the result of Winiarski (see [6]):
(109)limsupk→+∞k(πkp(K,f))ρ/k=(eρσ)(cap(K))ρ.
(2) Ifn=1, p>2, and q=1, using the results of Theorem 13 we obtain the result of Nguyen (see [13]):
(110)limsupk→+∞log[p-2](k)(πkp(K,f))ρ/k=σ(cap(K))ρ.Remark 15.
The above result holds for0<p<1 (see [14]).
Let0<p<1; of course, for 0<p<1, the Lp-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),
(111)∥f∥Lp(E,μ)≤M∥f∥Lr(E,μ).
Using the inequality(112)∫E|f|pdμ≤∥f∥Ep-r∫E|f|rdμ
we get
(113)∥f∥Lp(E,μ)≤∥f∥E1-(r/p)∥f∥Lr(E,μ)r/p.
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
(114)∥P∥E≤C(1+ϵ)deg(P)∥P∥Lp(E,μ).
Thus if(E,μ) satisfies the Bernstein-Markov inequality for one p>0, then (92) and (95) are satisfied for all p>0.



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*Source: 845146-2013-02-28.xml*