1.6 Fonction de Green avec pˆole logarithmique `a l’infini
1.6.3 Ensembles plurir´eguliers de C n
On dit que K v´erifie la condition polynomiale de Leja en a ¡L0
¢
si ; Pour toute famille F de polynˆome de Cn tele que
sup©f (z) : P ∈ Fª< ∞, ∀ z ∈ K,
et tout ² > 0, il existe un voisinage U de a et une constante M tels que
¯
¯P (z)¯¯ ≤ Me².d0P
, ∀P ∈ F, ∀z ∈ U.
D´efinition 1.6.4
K est dit L-r´egulier au point a si V∗
K(a) = 0, avec
VK∗ = Regsup h
sup©u ∈ L; u ≤ 0, surKªi,
o`u L est l’ensemble des fonctions plurisouharmoniques u sur Cn v´erifiant
u(z) ≤ Cu+ Log(1+ | z |), ∀z ∈ Cn.
La deuxi`eme d´efinition est l’extension naturelle de la notion de non effilement dans C.
Lorsque K est compact les deux d´efinitions sont ´equivalentes. Proposition 1.6.2 Si la fonction V∗
K est continue sur K alors elle est continue sur
Cn et V∗
K = VK. Dans ce cas on dit que K est pluri-r´egulier.
Proposition 1.6.3 lLe compact K est pluri-r´egulier si et seulement si K ⊂ {V∗ K <
²} pour tout ² > 0.
En cons´equence
Proposition 1.6.4 lLe compact K est pluri-r´egulier si et seulement si il v´erifie l’in´egualit´e de Bernstein- Markov :
¡
B.M¢ : ∀² > 0 ∃ un onvert Utel que k P kU≤ e².d
◦P
k P kK, pour tout polynˆome de Cn
On prend U = {V∗
Ensembles pluripolaires M.Harfaoui D´efinition 1.6.5
L’enveloppe convexe d’un compact est donn´e par :
b
K = {z ∈ Cn :k P (z) k≤k P kK, pour tout polynome P }.
Un compact K est dit polynomialement convexe si bK = K.
Exemple 1.6.3.1 Si K1 = {(z1, z2) :| z1 |≤ 1, | z2 |≤ r < 1} et K2 = {(z1, z2), |
z2 |≤ 1, | z1 |≤ r < 1}, on peut montrer que
K = {(z1, z2) :| z1 |≤ 1, | z2 |≤ 1, | z1.z2 |≤ r}.
On pend P (z1, z2) = z1.z2.
Proposition 1.6.5 Si bK est l’enveloppe convexe d’un compact K alors :
1. K = {z ∈ Cb n: V
K(z) = 0},
2. VK = VKb.
tout compact K on a :
Le compact K est pluri-r´egulier si et seulement si K ⊂ {V∗
K < ²} pour tout ² > 0.
Par la d´efinition de VK on obtient l’in´egalit´e de type Bernstein-Walsh :
¯ ¯P (z)¯¯ ≤°°P°° K ¡ exp(VK(z)) ¢degP (1.6.6) pour tout polynˆome P ∈ Cn[z].
Par cons´equent, le point crucial pour les applications est d’´etablir la continuit´e de VK dans Cn, ce qui est ´equivalent, par Zakharyuta 1976 et Siciak 1983, `a la
propri´et´e suivante :
Pour tout b > 1, il existe un voisinage U de K et une constante M > 0 tel que °
°P°°
U ≤ M.b
degP°°P°°
K (1.6.7)
pour tout polynˆome P ∈ Cn[z].
Dans un tel cas, l’ensemble K est dit pluriregulier ou bien L-r´egulier. D’apr`es la formule int´egrale de Cauchy , on a l’in´egalit´e de Markov : Pour tout polynˆome P ∈ Cn[z],
¯
¯Gradient¡P (z)¢≤ M.(degP )r°°P°°K (1.6.8) pour z ∈ K, M et r des constantes ind´ependantes de P .
L’in´egalit´e de Markov, et l’in´egalit´e de Bernstein-Walsh, est l’un des outils fon- damentaux de la th´eorie des fonctions constructives.
Dans le plan on le r´esultat suivant :
Th´eor`eme 1.6.1 Soit K ⊂ C un compact avec C \ K connexe. Alors pour toute fonction f holomorphe sur un voisinage de K, il existe une suite Pn de polynˆomes
holomorphes qui converge uniformements vers f sur K.
M.Harfaoui Ensembles pluripolaires Dans Cn ce r´esultat devient :
Th´eor`eme 1.6.2 Soit K ⊂ Cn un compact, avec bK = K. Alors toute fonction f
holomorphe sur un voisinage de K, il existe une suite Pn de polynˆomes holomorphes
Chapitre 2
Best polynomial approximation in
L
p
-norm and (p, q)-growth of
entire functions
2.1
Introduction.
Let f (z) = +∞ X k=0akzλk be a nonconstant entire function and M(f, r) = max |z|=r
¯ ¯f(z)¯¯. It is well known that the function r 7→ log(M(f, r)) is indefinitely increasing convex function of log(r). To estimate the growth of f precisely, R. P. Boas, (see [4]), has introduced the concept of order, defined by the number ρ (0 ≤ ρ ≤ +∞) :
ρ = lim sup
r→+∞
log log(M(f, r))
log(r) . (2.1.1)
The concept of type has been introduced to determine the relative growth of two functions of same nonzero finite order. An entire function, of order ρ, 0 < ρ < +∞, is said to be of type σ, 0 ≤ σ ≤ +∞, if
σ = lim sup
r→+∞
log(M(f, r))
rρ . (2.1.2)
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 [1]) 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))
log[q](r) , (2.1.3)
b ≤ ρ(p, q) ≤ +∞, where b = 0 if p > q and b = 1 if p = q, where log[0](x) = x, and
M.Harfaoui Introduction 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 σ(p, q) = lim sup r→+∞ log[p−1]((M(f, r))) ³ log[q−1](r) ´ρ(p,q) . (2.1.4)
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. 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
Ek(K, f ) = ° °f − Tk ° ° K (2.1.5)
where the norm °°.°°K is the maximum on K and Tk is the k-th Chebytchev
polynomial of the best approximation to f on K.
S. N. Bernstein showed (see [3], p.14), for K = [−1, 1], that there exists a constant
ρ > 0 such that
lim
k→+∞k
1/ρpk E
k(K, f ) (2.1.6)
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 A. R. Reddy (see [28] and [29]) as follows : lim
k→+∞
k
p
Ek(K, f ) = (ρeσ)2−ρ (2.1.7)
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 T. Winiarski (see [33]) 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→+∞k ³ Ek(K, f ) ´ρ/k = cρeρσ (2.1.8)
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.
Introduction M.Harfaoui The authors considered respectively, the Taylor development of f with respect to the sequence (zn)n and the development of f with respect to the sequence (Wn)n
defined by Wn(z) = j=n Y j=1 (z − ηnj), n = 1, 2, ... (2.1.9) where η(n) = (η
n0, ηn1, ..., ηnn) is the n-th extremal points system of K (see [33], p.
260).
We remark that the above results suggest that rate at which the sequence ³
k
p
Ek(K, f )
´
k tends to zero depends on the growth of the entire function (order
and type).
M. Harfaoui (see [12]) obtained a result of generalized order in term of approxi- mation 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 A.R. Reddy (see [28] and [29]) and T.Winiarski (see [33]), in term 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 M. Harfaoui (see [12]) in term of (p, q)-order and (p, q)-type for the functions
α(x) = logp−1(x), β(x) = logq−1(x) for (p, q) ∈ N2. (2.1.10)
So we establish relationship between the rate at which³πkp(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 πkp(K, f ) = inf n°° °f − P ° ° ° Lp(K,µ); P ∈ Pk(C n)o, (2.1.11)
where Pk(Cn) is the family of all polynomial of degree ≤ k and µ the well-selected
measure (the equilibrium measure µ = (ddcV
K)n associated to a L-regular compact
K) (see [34]) and Lp(K, µ), p ≥ 1, is the class of all function such that :
° ° °f ° ° ° Lp(K,µ)= ³ Z K | f |p dµ´1/p < ∞. (2.1.12)
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(V
K(z)) < r}, where VK is the
Siciak’s extremal function of K a compact of Cnsatisfying some properties (see [30]
and [31]), and using the development of f with respect to the sequence ³
Ak
´
k∈N
M.Harfaoui Definitions and notations Recall that in the paper of T. Winiarski (see [33]) the author used the Cauchy inequality. In our work we replace this inequality by an inequality given by A. Zeriahi (see [35])