Volume 2010, Article ID 360180,15pages doi:10.1155/2010/360180
Research Article
Generalized Order and Best Approximation of
Entire Function in
L
p
-Norm
Mohammed Harfaoui
University Hassan II-Mohammedia, U.F.R Mathematics and Applications, F.S.T, BP 146, Mohammedia, Morocco
Correspondence should be addressed to Mohammed Harfaoui,[email protected]
Received 7 March 2010; Accepted 26 December 2010 Academic Editor: Rodica Costin
Copyrightq2010 Mohammed Harfaoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 compactKwith respect to the setΩr {z∈Cn; expVKz≤r}, whereVKsup{1/dln|Pd|,
Pd polynomial of degree≤d, PdK≤1}is the Siciak extremal function of aL-regular compactK.
1. Introduction
Letfz k∞0ak·zλkbe a no constant entire function in complex planeCand let
Mf, rsupfz, |z|r, r >0. 1.1
It is well known that the function r → lnMf, r is convex and decreasing of lnr. To estimate the growth off, the concept of order, defined by the numberρ0≤ρ ≤∞, such that
ρlim sup
r→∞
ln lnMf, r
lnr 1.2
has been givensee1.
of orderρ0< ρ <∞, is said to be of typeσ0≤σ≤∞if
σlim sup
r→∞
lnMf, r
rρ . 1.3
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 Junejasee2have introduced the concept of index-pair of an entire function. Thus, for p≥q≥1, they have defined the number
ρp, qlim sup
r→∞
logpMf, r
logqr . 1.4
It is easy to show thatb≤ρp, q≤∞whereb0 ifp > qandb1 ifpq.
The functionfis said to be of index-pairp, qifρp−1, q−1is nonzero finite number. The numberρp, qis called thep, q-order off.
Bajpai and Juneja have also defined the concept of the p, q-type σp, q, for b < ρp, q<∞, by
σp, qlim sup
r→∞
logp−1Mf, r
logq−1r ρp,q
. 1.5
In their works, the authors have established the relationship ofp, q-growth offwith respect to the coefficientsakin the Maclaurin series offin complex planeCforp, 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 planeC, of positive logarithmic capacity andf be a complex function defined and bounded onK. Fork∈Nput
Ek
K, ff−TkK, 1.6
where the norm · Kis the maximum onKandTkis thekth Chebytchev polynomial of the
best approximation tofonK. It is knownsee3that
lim
k→∞
k
Ek
K, f0 1.7
if and only iffis the restriction toKof an entire functionginC. This result has been generalized by Reddysee4,5as follows:
lim
k→∞
k
Ek
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 Winiarskisee6has generalized this result for a compactKof the complex planeC, of positive logarithmic capacity notedccapKas follows.
IfKbe a compact subset of the complex planeC, of positive logarithmic capacity then
lim
k→∞k
1/ρk
Ek
K, fceρσ1/ρ 1.9
if and only iffis the restriction toKof an entire function of orderρ0< ρ <∞and typeσ. Recall that cap−1,1 1/2 and capacity of a disk unit is capDO,1 1.
The authors considered the Taylor development offwith respect to the sequenceznn
and the development offwith respect to the sequenceWnndefined by
Wnz jn
j1
z−ηnj
, 1.10
whereanjnis the thnth extremal points system ofK.
The aim of this paper is to establish relationship between the rate at which πkpK, f1/k tends to zero in terms of best approximation inLp-norm, and the generalized growth of entire functions of several complex variables for a compact subsetKofCn, where
Kis a compact well-selected. In this work we give the generalization of these results inCn,
replacing the circle{z ∈ C;|z| r}by the set{z ∈ Cn; expV
Ez < r}, where VE is the
Siciak’s extremal function ofEa compact ofCnwhich 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 2.1see Siciak7. LetKbe a compact set inCnand let ·
Kdenote the maximum
norm onK. The function
VKsup
1
dlog|Pd|, Pd polynomial of degree ≤d, PdK≤1, d∈N
2.1
is called the Siciak’s extremal function of the compactK.
Definition 2.2. A compact K in Cn is said to be L-regular if the extremal function, V K,
associated toKis continuous onCn.
Regularity is equivalent to the following Bernstein-Markov inequalitysee8. For any >0, there exists an openU⊃Ksuch that for any polynomialP
In this case we takeU{z∈Cn; V
Kz< }.
Regularity also arises in polynomial approximation. Forf∈ CK, we let
d
K, finff−PK, P ∈ PkCn
, 2.3
where PkCn is the set of polynomials of degree at most d. Siciak see 7 showed the
following.
IfKisL-regular, then
lim sup
d→∞
εd
K, f1/d 1
r <1 2.4
if and only iffhas an analytic continuation to{z∈Cn; V
Kz<log1/r}.
Letfbe a function defined and bounded onK. Fork∈Nput
πkpK, finff−PLpK,μ, P ∈ PkCn
, 2.5
wherePkCnis the family of all polynomial of degree≤ kandμthe well-selected measure
The equilibrium measureμ ddcV
Knassociated to aL-regular compactE see9and
LpK, μ,p≥1, is the class of all function such that:
fLpK,μ
K
fpdμ 1/p
<∞. 2.6
For an entire functionf ∈Cnwe establish a precise relationship between the general
growth with respect to the set:
Ωr
expVK< r
2.7
and the coefficients of the development of f with respect to the sequence Akk, called
extremal polynomial see 10. Therefore, we use these results to give the relationship between the generalized growth offand the sequenceπkpK, fk.
Recall that the subset ofCn,Ωr{expVK< r}, replaces the unit discDO, r{|z|< r}
in the classical case.
It is known that ifK is an compactL-regular ofCn, there exists a measureμ, called
extremal measure, having interesting propertiessee7,8, in particular, we have:
P1Bernstein-Markov Inequality.
For all >0, there existsCCεis a constant such that
BM:PdKC1εskPdL2K,μ, 2.8
P2Bernstein-WaishB.WInequality.
For every setL-regularKand every realr >1, we have:
f
K≤M·rdegf
K
fp·
dμ 1/p
. 2.9
Note that the regularity is equivalent to the Bernstein-Markov inequality. Letα:N → Nn, k → αk α1k, . . . , α
nkbe a bijection such that
|αk1| ≥ |αk| where|αk|α1k · · ·αnk. 2.10
Z´eriahisee10has constructed according to the Hilbert-Shmidt method a sequence of monic orthogonal polynomial according to a extremal measure see8,Akk, called
extremal polynomial, defined by
Akz zαk k−1
j1
ajzαj 2.11
such that
AkLpK,μ
⎡ ⎢ ⎣inf
⎧ ⎪ ⎨ ⎪ ⎩ zαk
k−1
j1 ajzαj
L2 K,μ
, a1, a2, . . . , an∈Cn
⎫ ⎪ ⎬ ⎪ ⎭ ⎤ ⎥ ⎦
1/αk
. 2.12
We need the following notations which will be used in the sequel:
N1νkνkK AkL2K,μ.
N2akakK AkKmaxz∈K|Akz|andτk ak1/sk, whereskdegAk.
With that notations andB.Winequality, we have
AkΩr ≤ak·rsk, 2.13
whereskdegAk. For more detailssee9.
Let α and β be two positives, strictly increasing to infinity differentiable functions 0,∞to0,∞such that for everyc >0:
lim
x→∞
αcx αx 1,
lim
x→∞
β1xωx βx 1,
2.14
Assume that, for everyc > 0, there exists two constantsaandbsuch that for every x≥a:
dβ−1cαx αlogx
≤b, 2.15
wheredumeans the differential ofu.
Definition 2.3. LetKbe a compactL-regular, we put
Ωr
z∈Cn; expVKz≤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, by
ρα, βlim sup
r→∞
αlogfΩ
r
βlogr ,
σα, βlim sup
r→∞
αfΩ
r
"
βr#ρα,β,
2.17
wherefΩr supΩr|fz|.
Note that in the classical caseαx logxandβx 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, see10.
Lemma 2.4. LetKbe a compactL-regular subset ofCn. Then for everyθ >1, there exists an integer
Nθ≥1 and a constantCθsuch that:
πkpK, f≤Cθ
r1Nθ
r−12N−1 f
Ωrθ
rk . 2.18
Letf k∞0fk·Akbe an entire function. Then for everyθ > 1, there existsNθ ∈ NandCθ >0
such that
fkνk≤Cθ r1Nθ
r−12N−1 f
Ωrθ
rsk , 2.19
for everyk≥1 andr >1.
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
Ωr
expVK< r
3.1
and the coefficient of entire functionf∈Cnof the development with respect to the sequence
of extremal polynomials.
LetAkkbe a basis of extremal polynomial associated to the setKdefined the relation
2.11. We recall thatAkkis a basis ofOCn the set of entire functions onCn. So iffis an
entire function then
f
k≥1
fk·Ak 3.2
and we have the following results.
Theorem 3.1. Iff k≥1fk·Akthen theα, β-orderρα, βoffis given by formula
ρα, βlim sup
k→∞
αsk
β−1/sklnfk·τksk
<∞. 3.3
To prove theorem we need the following lemmas.
Lemma 3.2. LetKbe a compactL-regular subset ofCn. Then
lim
k→∞ $|
Akz|
νk
%1/sk
expVKz, 3.4
for everyz∈Cn\K& the connected component ofCn\K,
lim
k→∞
$ AkK
νk
%1/sk
1. 3.5
Lemma 3.3. For everyr >1 the maximum of the function
x−→ωx, r rxexp
$
−xβ−1
1 μαx
%
3.6
is reached forxxrsolution of the equation
xα−1 '
μβ (
logr−d
β−11/μαx dlogx
)*
Consequence. This relation is equivalent to
x≤α−1βlogr b. 3.8
Indeed, the relations
dβ−1cαx dlogx
≤b,
β−1
αx μ
logr−d
β−1cαx dlogx
3.9
give
logr−b≤logr−d
β−1cαx
dlogx ≤logr b. 3.10
Then
β−1
αx μ
≤logr b,
x≤α−1βlogr b.
3.11
We verify easily with the relation
dβ−1cαx αlogx
≤b, 3.12
that
rnexp
$
−nβ−1
1 μαn
%
≤expbα−1μ·βlogr b 3.13
for everyn > n0andr >0.
Proof ofTheorem 3.1. Putγ lim supk→∞αsk/β−1/skln|fk| ·τkskand show that γ
ρα, β.
1Show thatγ≤ρα, β.
By definition ofγ, we have, for all >0,∃ksuch that for allk > k
αsk
β−1/sk lnfk·νksk
From the second assertion of theLemma 3.2, we have for every θ > 1, there exists Nθ∈NandCθ>0 such that
logfkτksk
≤logCθ log
+
r1Nθ
r−12N−1 ,
−sklogr α−1
"
ρ·βlogrθ#ϕr
3.15
or
−1
sk
logfkτksk
≥ −1
sk
logCθ−
Nθ
sk
logr1−2N−1 sk
logr−1 logr
− 1
skα
−1ρβlogrθϕr, k.
3.16
But limk→∞ϕr, k logrsoϕr, k∼logrforrsufficiently large. Then
− 1
sk log
fkτsk
k
≥1o1logr. 3.17
Letrkbe a real satisfying
rkexp
$ β−1
1 ραsk
%
. 3.18
Then
−1
sk
logfkτksk
≥β−1
1
ρ ·αsk
1o1. 3.19
This is equivalent to
β"−1/sklogfkτksk
#
αsk ≥
1
ρ, 3.20
or
αsk
β"−1/sklogfkτksk
#≤ρ, 3.21
hence
2Show thatγ ≥ ρα, β. According to the definition ofγ, we have for every > 0 there existsksuch that for allk > k
fkτksk ≤exp
$
−skβ−1
αsk
γ %
3.23
fork≥ksufficiently large.
According to the first assertion of Lemma 3.2and BMand BW inequalities, we have
AkΩr ≤ak·rsk 3.24
and for every entire function
fz
k≥0
fk·Akz,
f
Ωr ≤
k
k0 ak·rsk
- ./ 0
1
C kr
kk1
|fk| ·νsk ·1rsk
- ./ 0
2
C
∞
kkr1
|fk| ·τksk·rsk
- ./ 0
3
. 3.25
The term1is a constant denotedC0, and
kr
kk1
fk·τksk·1rsk ≤1rskr
∞
k0
fk·τksk
- ./ 0
4
.
3.26
The series4is convergent. Let, forrsufficiently,
Nr E1α−1γβlog 21r2, 3.27
whereExmeans the integer part ofx. Then
fk·νk·1rsk ≤exp
$
−sk·β−1
αsk
γ %
1εrsk. 3.28
Applying the the relation3.13withμγand if we replacerby1εr, we obtain:
fΩ
r ≤exp
1
And so
fΩ
r ≤C·1εr
sk·exp
$
−sk·β−1
αsk
γ %
, 3.30
or
f
Ωr ≤C·expb·α
−1γ·βlog1εr b. 3.31
Thus
logfΩ
r ≤logC b·α
−1γ·βlog1εr b 3.32
and forrsufficiently large, we have
αb−1logfΩ
r
βlog1εr b ≤γ. 3.33
Butαc·x∼αxnear to the infinity, thus
αb−1logfΩ
r
∼αlogfΩ
r
. 3.34
If we putxlog1εr, thenβlog1r b∼βlog1rand,
αlogfΩ
r
βlogr ≤γ. 3.35
This is true for every >0 henceρα, 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 inLp
-norm1≤p≤ ∞with respect to a measureμonKinCn.
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·Akan element ofLpK, μ, forp≥1, then
lim sup
k→∞
αk
β1−1/klogπkpK, f 2 lim supk→∞
αsk
β"−1/sklogfk·τksk
Proof ofLemma 4.1. The proof is done in two stepsp≥2 and 1< p <2.
Step 1. Iff∈LpK, μwherep≥2, thenf∞
k0fk·Akwith convergence inL2K, μ, hence
fork≥0
fk 1
νk2
K
f·Akdμ 4.2
and therefore
fk
1 ν2
k
K
f−Pk−1·Akdμ 4.3
because degAk sk.
Since the relation,
fk≤
1 ν2
k
K
f−Pk−1·Akμ 4.4
satisfied, is easily verified by using inequalities Bernstein-walsh and Holder that, we have for allε >0
fk·νk≤Cε·1εsk ·πspk−1
K, f. 4.5
for allk≥0.
Step 2. If 1≤p <2, letpsuch that 1/p1/p1, we havep≥2. According to the inequality of H ¨older, we have:
fk·ν2k≤f−Pk−1LpK,μ· AkLpK,μ. 4.6
But,
AkLpK,μ≤C· AkKC·akK. 4.7
This shows, according to inequalityBM, that:
fk·νk2≤C·Cε·1εsk·f−Psk−1LpK,μ. 4.8
Hence the result
fk·ν2
k≤Cε·1εsk·π p sk
In both cases, we have therefore
fk·ν2k≤Aε·1εsk·πspk
K, f, 4.10
whereAεis a constant which depends only onε.
After passing to the upper limit in the relation4.10and Applying the relation3.5
of theLemma 3.2we get
lim sup
k→∞
αk
β1−1/klogπkpK, f 2 ≥lim supk→∞
αsk
β"−1/sklogfk·τksk
#. 4.11
To prove the other inequality we consider the polynomial of degreesk,
Pkz k
sj0
fj·Aj 4.12
then
πsp
k−1
K, f≤
∞
sjsk
fj·Aj
LpK,μ≤C0
∞
sjsk
fj·Aj
K. 4.13
By Bernstein-Walsh inequality, we have
πkpK, f≤C
∞
sjsk
1sjf
j·νj 4.14
fork≥0 andp≥1. If we take as a common factor1εsk·|f
k|·νkthe other factor is convergent
thus, we have
πkpK, f≤C1sk ·f
k·νk 4.15
and by3.5ofLemma 3.2, we have, then
πkpK, f≤C12sk ·f
k·τksk. 4.16
We deduce
lim sup
k→∞
αk
β1−1/klogπkpK, f 2 lim supk→∞
αsk
β"−1/sklogfk·τksk#
This inequality is a direct consequence of the relation 4.10 and the inequality on coefficients|fk|given by
fk·τsk
k ≤exp
1
−skβ−1
αsk1/ρ
2 , fk·ν2 ≤C
ε·1εsk·πspk
K, f.
4.18
Applying this lemma we get the following main result:
Theorem 4.2. Letf ∈LpK, μ, thenf isμ-almost-surely the restriction toKof an entire function
inCnof finite generalized orderρα, βfinite if and only if
ρα, βlim sup
k→∞
αk
β1−1/klogπkpK, f 2 <∞. 4.19
Proof. Suppose thatfisμ-almost-surely the restriction toKof an entire functiongof general orderρ0< ρ <∞and show thatρρα, β.
We haveg ∈LpK, μ,p≥2 andg
k≥0gk·AkinL2K, μSinceg is an element of
L2K, μtheng k∞0gk·Akand according to theTheorem 3.1.
ρg, α, βlim sup
k→∞
αsk
β"−1/skloggk·τksk
# 4.20
and with theLemma 4.1, we have
lim sup
k→∞
αk
β−1/klogπkpK, g
lim sup
k→∞
αsk
β"−1/skloggk·τksk
#. 4.21
ButgfonKhence
ρlim sup
k→∞
αk β−1/klogπt
k
K, f <∞. 4.22
Suppose now thatfis a function ofLpK, μsuch that the relation4.19is verified.
1Letp≥2, thenf k∞0fk·Ak, becausefis an element ofL2K, μ LpK, μp≥1
is decreasing sequence.
Consider inCn the seriesf
k·Ak,k ≥ 0, we verify easily that this series converges
normally on all compact ofCn to an entire function denotedf
1. We havef1 f, obviously, μ-almost surly onK, and byTheorem 3.1, we have
ρf1
lim sup
k→∞
αsk
β"−1/sklogfk·τksk
# lim sup
k→∞
αk
Applying theLemma 4.1, we find
ρf1
lim sup
k→∞
αk
β"−1/klogπktK, f# <∞. 4.24
Consider the functionf1k≥0fk·Ak, we havef1z fzμ-almost surely for every
zinK. Therefore theα, β-order off1is:
ρf1, α, β
lim sup
k→∞
αk β"−1/klogπt
k
K, f#<∞ 4.25
seeTheorem 3.1.
ByLemma 4.1, we checkρf1 ρso the proof is completed.
2Now letp∈1,2andf ∈LpK, μ.
ByBMinequality and H ¨older inequality, we have again the inequality the relation 4.10and by the previous arguments we obtain the result.
References
1 R. P. Boas Jr., Entire Functions, Academic Press, New York, NY, USA, 1954.
2 S. K. Bajpai and O. P. Juneja, “On entire functions of fast growth,” Transactions of the American
Mathematical Society, vol. 203, pp. 275–297, 1975.
3 J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1969.
4 A. R. Reddy, “Approximation of an entire function,” Journal of Approximation Theory, vol. 3, pp. 128– 137, 1970.
5 A. R. Reddy, “Best polynomial approximation to certain entire functions,” Journal of Approximation
Theory, vol. 5, pp. 97–112, 1972.
6 T. Winiarski, “Approximation and interpolation of entire functions,” Annales Polonici Mathematici, vol. 23, pp. 259–273, 1970.
7 J. Siciak, “Extremal plurisubharmonic functions inCn,” in Proceedings of the 1st Finnish-Polish Summer
School in Complex Analysis at Prodlesice, pp. 115–152, University of Lodz, Lodz, Poland.
8 J. Siciak, “On some extremal functions and their applications in the theory of analytic functions of several complex variables,” Transactions of the American Mathematical Society, vol. 105, pp. 322–357, 1962.
9 A. Z´eriahi, “Familles de polyn ˆomes presques partout born´es,” Bulletin des Sciences Math´ematiques, vol. 197, pp. 81–91, 1983.