Interpolation by entire functions with growth conditions

0.16 MB·English

by  Myriam Ounaies

INTERPOLATION BY ENTIRE FUNCTIONS WITH GROWTH CONDITIONS
MYRIAMOUNAIES
8
0
0 INTRODUCTION
2

n Letp:C→[0,+∞[beaweight(seeDefinition1.1)andA (C)thevectorspaceofallentirefunctions
a p
J
9 satisfying supz∈C|f(z)| ≤ exp(−Bp(z)) < ∞ for some constant B > 0. For instance, if p(z) = |z|,
1
A (C)isthespaceofallentirefunctionsofexponentialtype.
p
]
V
Following[3],theinterpolationproblemweareconsideringis: letV = {(z ,m )} beamultiplicity
C j j j
.
h variety, that is, {zj}j is a sequence of complex numbers diverging to ∞, |zj| ≤ |zj+1| and {mj}j is
t
a
m a sequence of strictly positive integers. Let {wj,l}j,0≤l<mj be a doubly indexed sequence of complex
[
numbers.

1
v Underwhatconditionsdoesthereexistanentirefunctionf ∈A (C)suchthat
1 p
4
0 f(l)(z )
j
3 =wj,l, ∀j, ∀0≤l<mj?
l!
.
1
0
8 Inotherwords,ifwedenotebyρtherestrictionoperatordefinedonAp(C)by
0
:
v fl(z )
j
Xi ρ(f)={ l! }j,0≤l<mj,
r
a whatistheimageofA (C)byρ?
p
WesaythatV isan”interpolatingvariety”whenρ(A (C))isthespaceofalldoublyindexedsequence
p
W ={w }satisfyingthegrowthcondition
j,l
|w |≤Aexp(Bp(z )) ∀j, ∀0≤l <m ,
j,l j j
forcertainconstantsA,B >0.
Date:February2,2008.
1991MathematicsSubjectClassification. 30E05,42A85.
Keywordsandphrases. discreteinterpolatingvarieties,entirefunctions.
1
2 M.OUNA¨IES
Letusmentiontheimportantfollowingresult:
Theorem0.1. [2,Corollary4.8]
V isaninterpolatingvarietyforA (C)ifandonlyifthefollowingconditionshold:
p
(i) ∀R>0, N(0,R)≤Ap(R)+B
(ii) ∀j ∈N, N(z ,|z |)≤Ap(z )+B,
j j j
forsomeconstantsA,B >0.
Here, N(z,r) denotes the integrated countingfunctionof V in the disc of center z and radius r (see
Definition1.3below).
In [3], Berenstein and Taylor describe the space ρ(A (C)) in the case where there exists a function
p
g ∈ A (C) such that V = g−1(0). They used groupingsof the pointsof V with respect to the connex
p
components of the set {|g(z)| ≤ εexp(−Bp(z))}, for some ε,B > 0 and the divided diffrences with
respecttothisgrouping.
The main aim of this paper is to determine more explicitely the space ρ(A (C)) in the more general
p
casewherecondition(i)issatisfied. ItisclearthatitisthecasewhenV isnotauniquenesssetforA (C),
p
thatis,whenthereexistsf ∈A (C)notidenticallyequaltozerosuchthatV ⊂f(−1)(0).
p
Wereferto[6]and[10]forsimilarresultsinthecasewherep(z)=|z|α.
Asin[3]and[6],thedivideddifferenceswillbeimportanttools. Ourconditionwillinvolvethedivided
differenceswithrespecttotheintersectionsofV withdiscscenteredattheorigin. Tobemoreprecise,the
maintheorem,statedinthecasewhereallthemultiplicitiesareequaltoone,forthesakeofsimplicity,is
thefollowing:
Theorem0.2. AssumethatV verifiescondition(i). ThenW = {w } ∈ ρ(A (C))ifandonlyifforall
j j p
R>0,
| w R/(z −z )|≤AexpBp(R),
k k m
|zXk|<R |zm|<YR,m6=k
whereA,B >0arepositiveconstantsonlydependingonV andW.
INTERPOLATIONBYENTIREFUNCTIONSWITHGROWTHCONDITIONS 3
WewilldenotebyA˜ (V)thespaceofsequencesW = {w } satisfyingtheabovecondition. Wewill
p j j
showthatingeneralρ(A (C))⊂A˜ (V),thus,wecanconsiderρ :A (C)→A˜ (V). Inthiscontext,the
p p p p
theoremstatesthatcondition(i)impliesthesurjectivityofρ.
On the other hand, we will prove that condition (i) is actually equivalent to saying that V is not a
uniquenesssetor,inotherwords,itisequivalenttothenon-injectivityofρ.
As a corollary of the main theorem, we will find the sufficency in the geometric characterization of
interpolatingvarietiesgiveninTheorem0.1.
The difficultpartof the proofof the main theoremis the sufficiency. As in [4, 7, 11], we will follow
a Bombieri-Ho¨rmanderapproach based on L2-estimates on the solution to the ∂¯-equation. The scheme
willbe thefollowing: the conditiononW givesa smoothinterpolatingfunctionF with a goodgrowth,
usinga partitionof the unityandNewtonpolynomials(see Lemma2.5). Thenwe areled to solve the∂¯
equation: ∂¯u=−∂¯F withL2-estimates,usingHo¨rmandertheorem[8]. Todoso,weneedtoconstructa
subharmonicfunctionU withaconvenientgrowthandwithprescribedsingularitiesonthepointsz (see
j
Lemma2.6). FollowingBombieri[5],thefactthate−U isnotsummablenearthepoints{z }forcesuto
j
vanishonthepointsz andwearedonebydefiningtheinterpolatingentirefunctionbyu+F.
j
Afinalremarkaboutthenotations:
A, B andC will denotepositiveconstantsand theiractualvaluemay changefromone occurrenceto
thenext.
A(t) . B(t)meansthatthereexistsaconstantC > 0, notdependingont suchthatA(t) ≤ CB(t).
A≃BmeansthatA.B .A.
The notation D(z,r) will be used for the euclidean disk of center z and radius r. We will denote
∂f ∂f
∂f = ,∂¯f = . Then∆f =4∂∂¯f denotesthelaplacianoff.
∂z ∂z¯
1. PRELIMINARIES AND DEFINITIONS.
Definition1.1. Asubharmonicfunctionp : C −→ R , iscalledaweightif, forsomepositiveconstants
+
C,
(a) ln(1+|z|2)≤Cp(z),
4 M.OUNA¨IES
(b) p(z)=p(|z|),
(c) thereexistsaconstantC >0suchthatp(2z)≤Cp(z).
Property(c)isreferredtoasthe”doublingpropertyoftheweightp”. Itimpliesthatp(z)=O(|z|α)for
someα>0.
LetA(C)bethesetofallentirefunctions,weconsiderthespace
A (C)= f ∈A(C), ∀z ∈C, |f(z)|≤AeBp(z) forsomeA>0,B >0 .
p
n o
Remark1.2. (i) Condition(a)impliesthatA (C)containsallpolynomials.
p
(ii) Condition(c)impliesthatA (C)isstableunderdifferentiation.
p
Examples:
• p(z)=ln(1+|z|2). ThenA (C)isthespaceofallthepolynomials.
p
• p(z)=|z|. ThenA (C)isthespaceofentirefunctionsofexponentialtype.
p
• p(z)=|z|α,α>0. ThenA (C)isthespaceofallentirefunctionsoforder≤αandfinitetype.
p
LetV ={(zj,mj)}j∈N beamultiplicityvariety.
Forafunctionf ∈A(C),wewillwriteV =f−1(0)whenf vanishesexactlyonthepointsz withmul-
j
tiplicitym andV ⊂ f−1(0)whenf vanishesonthepointsz (butpossiblyelsewhere)withmultiplicity
j j
atleastequaltom .
j
We will say that V is a uniquenessset for A (C) if there is no functionf ∈ A (C), exceptthe zero
p p
function,suchthatV ⊂f−1(0).
Weneedtorecallthedefinitionsofthecountingfunctionsandtheintegratedcountingfunctions:
Definition1.3. LetV ={(z ,m )} beamultiplicityvariety. Forz ∈Candr >0,
j j j
n(z,r)= m ,
j
|z−Xzj|≤r
r n(z,t)−n(z,0) r
N(z,r)= dt+n(z,0)lnr = m ln +n(z,0)lnr,
j
t |z−z |
Z0 0<|zX−zj|≤r j
INTERPOLATIONBYENTIREFUNCTIONSWITHGROWTHCONDITIONS 5
An application of Jensen’s formula in the disc D(0,R) shows that, if V is not a uniqueness set for
A (C),thenthefollowingconditionholds:
p
(1) ∃A,B >0, ∀R>0, N(0,R)≤Ap(R)+B
Wewilllatelyshowthattheconversepropertyholds.
ByanalogywiththespacesA(C)andA (C),wedefinethefollowingspaces
p
A(V)={W ={w } ⊂C}
j,l j,0≤l<mj
and
mj−1
A (V)= W ={w } ⊂C, ∀j, |w |≤AeBp(zj) forsomeA>0,B >0 .
p j,l j,0≤l<mj j,l
n Xl=0 o
ThespaceA (C)canbeseenastheunionoftheBanachspaces
p
A (C)={f ∈A(C), kfk :=sup|f(z)|e−Bp(z) <∞}
p,B B
z∈C
and has a structure of an (LF)-space with the topology of the inductive limit. The analog is true about
A (V).
p
Remark1.4. (see[1,Proposition2.2.2])
Letf beafunctioninA (C). Then,forsomeconstantsA>0andB >0,
p
∞ f(k)(z)
∀z ∈C, ≤AeBp(z).
k!
k=0(cid:12) (cid:12)
X(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Asaconsequenceofthisremark,weseethattherestrictionmap:
ρ:A(C)−→A(V)
fl(z )
j
f 7→ { }
l! j,0≤l≤mj−1
maps A (C) into A (V), but in general, the space A (V) is larger than ρ(A (C)). It is clear that ρ is
p p p p
injectiveifandonlyifV isauniquenesssetforA (C).
p
6 M.OUNA¨IES
When ρ(A (C))=A (V), we say that V is an interpolating variety for A (C). As mentioned in the
p p p
introduction,BerensteinandLigaveageometriccharacterizationofthesevarieties:
Theorem1.5. [2,Corollary4.8]
V isaninterpolatingvarietyforA (C)ifandonlyifconditions(1)and
p
(2) ∃A>0, ∃B >0 ∀j ∈N, N(z ,|z |)≤Ap(z )+B
j j j
hold.
In this paper, we are concerned by determining the subpace ρ(A (C)) of A(V) in the case where
p
condition(1)isverified.
To any W = {w } ∈ A(V), we associate the sequenceof divideddifferencesΦ(W) =
j,l j,0≤l≤mj−1
{φ } definedbyinductionasfollows:
j,l j,0≤l≤mj−1
Wewilldenoteby
q
Πq(z)= (z−zk)mk,forallq ≥1.
k=1
Y
φ =w ,forall 0≤l≤m −1,
1,l 1,l 1
w −P (z )
q,0 q−1 q
φ = ,
q,0
Π (z )
q−1 q
w − Pq(−l)1(zq) − l−1 1 Π(l−j)(z )φ
q,l l! j=0 (l−j)! q−1 q q,j
φ = for1≤l≤m −1
q,l q
Π (z )
Pq−1 q
where
q−1 mj−1 j−1
P (z)= φ (z−z )l (z−z )mt .
q−1 j,l j t
!
j=1 l=0 t=1
X X Y
Remark1.6. Actually,P isthepolynomialinterpolatingthevaluesw atthepointsz withmultiplicity
q j,l j
m ,for1≤j ≤q. Itistheuniquepolynomialofdegreem +···+m −1suchthat
j 1 q
P(l)(z )
q j
=w
j,l
l!
forall1≤j ≤qand0≤l ≤m −1.
j
INTERPOLATIONBYENTIREFUNCTIONSWITHGROWTHCONDITIONS 7
Examples.
• LetW ={δ δ } .
0 1,j l,m1−1 j,0≤l<mj
j−1
UsingthefactthatPj(z)mustcoincidewith(z−z1)m1−1 (z−zj)mj andidentifyingthe
k=2
Y
coefficientinfrontofzm1+···+mj−1+l−1,wefind:
φ =φ =···=φ =0, φ =1,
1,1 1,2 1,m1−2 1,m1−1
and,forj ≥2,0≤l ≤m −1,
j
j−1
φl,j =(z1−zj)−(l+1) (z1−zk)−mk.
k=2
Y
• Inthespecialcasewherem =1forallj andW ={w } ,wehaveforallj ≥1,
j j j
j
φ = w (z −z )−1.
j k k l
k=1 1≤l≤j,l6=k
X Y
To compute the coefficients, we may use the fact that P (z) must coincide with the Lagrange
j
j
(z−z )
polynomial w k andidentifythecoefficientinfrontofzj−1.
n
(z −z )
n k
n=1 1≤k≤j,k6=n
X Y
Let us denote by A˜ (V) the subspace of A(V) consisting of the elements W ∈ A(V) such that the
p
followingconditionholds:
(3) foralln≥0, |zj|≤2n and0≤l ≤mj −1, |φj,l|2n(l+m1+...+mj−1) ≤Aexp(Bp(2n)),
whereAandBarepositiveconstantsonlydependingonV andW.
WehavechosentouseacoveringofthecomplexplanebydiscsD(0,2n),butwecanreplace2nbyany
RnwithR>1.
Lemma1.7. Assumez =0. Then,condition(1)holdsifandonlyif
1
W ={δ δ } ∈A˜ (V).
0 1,j l,m1−1 j,0≤l<mj p
8 M.OUNA¨IES
Proof. Supposethat(1)isverified.Letn∈N,0<|z |≤2nand0≤l ≤m −1. Wehavebydefinition,
j j
2n j
N(0,2n)= mkln +m1ln(2n)≥ln 2n(m1+···+mj) |zk|−mk ,
|z |
k !
0<|Xzk|≤2n kY=2
j j
|φj,l|=|zj|mj−l−1 |zk|−mk ≤2n(mj−l−1) |zk|−mk ≤exp(N(0,2n))2−n(m1+···+mj−1+l+1).
k=2 k=2
Y Y
Wereadilyobtaintheestimate(3),usingthatN(0,2n)≤Ap(2n)+B.
Conversely,letnbeaninteger.Usingtheestimate(3)whenj ≥2isthenumberofdistinctpoints{z }
k
inD(0,2n)andl =m −1,wehave
j
j
N(0,2n)=ln 2n(m1+···+mj) |zk|−mk =ln(2n(m1+···+mj)|φj,mj−1|)≤Ap(2n)+B.
!
k=2
Y
Then, we deduce the estimate for N(0,R) using the aboveone with 2n−1 ≤ R < 2n and the doubling
propertyofp. (cid:4)
Wedefinethefollowingnorm:
kWk =supkW(n)k exp(−Bp(2n))
B n
n
where
kW(n)kn = sup sup |φj,l|2−n(l+m1+...+mj−1),
|zj|≤2n0≤l≤mj−1
ThespaceA˜ (V)canalsobeseenasan(LF)-spaceasaninductivelimitoftheBanachspaces
p
A˜ (V)={W ∈A(V), kWk <∞}.
p,B B
Wearenowreadytostatethemainresults.
Proposition1.8. TherestrictionoperatorρmapscontinouslyA (C)intoA˜ (V).
p p
Proposition1.9. Undertheassumptionofcondition(1),A˜ (V)isasubspaceofA (V).
p p
Proposition1.10. Ifconditions(1)and(2)areverified,thenA˜ (V)=A (V).
p p
INTERPOLATIONBYENTIREFUNCTIONSWITHGROWTHCONDITIONS 9
Theorem1.11. Ifcondition(1)holds,then
A˜ (V)=ρ(A (C)).
p p
Inotherwords,condition(1)impliesthatthemapρ:A (C)→A˜ (V)issurjective.
p p
ThecombinationofProposition1.10andTheorem1.11showseasilythesufficiencyinTheorem1.5.
Usingtheresultsgivensofar,wecandeducenexttheorem:
Theorem1.12. Thefollowingassertionsareequivalent:
(i) V isnotauniquenesssetforA (C).
p
(ii) Themapρisnotinjective.
(iii) V verifiescondition(1).
(iv) ThesequenceW ={δ δ } belongstoρ(A (C)).
0 1,j l,m1−1 j,0≤l<mj p
Inparticular,itshowsthatcondition(1)isequivalenttotheexistenceofafunctionf ∈A (C)suchthat
p
V ⊂ f−1(0). Combinedwith Theorem1.11, it shows that, if ρ is notinjective, then it is surjectiveand
that,iftheimagecontainsW ,thenitcontainsthewholeA˜ (V).
0 p
ProofofTheorem1.12. Aswementionedbefore,itisclearthat(i)isequivalentto(ii)andthat(i)implies
(iii).
(iv)implies(i): Wehaveafunctionf ∈A (C)notidenticallyequalto0suchthatf(l)(z )=0forall
p j
j 6=1andforall0≤l<mj. Thefunctiongdefinedbyg(z)=(z−z1)m1f(z)belongstoAp(C),thanks
toproperty(i)oftheweightp,andvanishesoneveryz withmultiplicityatleastm .
j j
(iii)implies(iv):
Up to a translation, we may suppose that z = 0. By Lemma 1.7, we know that W ∈ A˜ (C). By
1 0 p
Theorem1.11,W ∈ρ(A (C)).
0 p
(cid:4)
10 M.OUNA¨IES
2. PROOF OFTHE MAIN RESULTS.
ProofofTheorem1.8. WewillfirstrecallsomedefinitionsaboutthedivideddifferencesanNewtonpoly-
nomials.Wereferthereaderto[1,Chapter6.2]or[9,Chapter6]formoredetails.
Letf ∈A(C)andx ,...,x bedistinctpointsofC. Theqthdivideddifferenceofthefunctionf with
1 q
respecttothepointsx ,...,x isdefinedby
1 q
q
∆q−1f(x ,...,x )= f(z ) (x −x )−1
1 q j j k
j=1 1≤k≤q,k6=j
X Y
andtheNewtonpolynomialoff ofdegreeq−1is
q j−1
P(z)= ∆j−1f(x ,...,x ) (z−x ).
1 j k
j=1 k=0
X Y
Itistheuniquepolynomialofdegreeq−1suchthatP (z)=f(x )forall1≤j ≤q.
q j
Whenx ,1≤j ≤qareeachonerepeatedl times,thedivideddifferencesaredefinedby
j j
∆l1+···+lq−1f(x1,...,x1,...,xq−1,...,xq−1,xq,...,xq)
l1 lq−1 lq
| {z1 } ∂l1|+···+lj{z } | {z }
= ∆q−1f(x ,··· ,x ).
l1!···lq!∂xl11···∂xlqq 1 q
ThecorrespondingNewtonpolynomialistheuniquepolynomialofdegreel +···l −1suchthat,for
1 q
all0≤j ≤qand0≤l ≤l −1,
j
P(l)(x )=f(l)(x ).
j j
Wehavethefollowingestimate
Lemma2.1. [1,Lemma6.2.9.]
Letf ∈A(C),ΩanopensetofC,δ >0andx ,··· ,x inΩ ={z ∈Ω: d(z,Ωc)>δ}. Then
1 k 0
2k−1
|∆k−1f(x ,...,x )|≤ sup|f(z)|.
1 k δk−1
z∈Ω
LetB >0befixedandf ∈A (C).
p,B
Letnbeafixedinteger. Let|z | ≤ 2n and0 ≤ l ≤ m −1. Weconsiderthedivideddifferencesoff
j j
withrespecttothepointsz ,··· ,z ,eachz ,1 ≤ k ≤ j−1repeatedm timesandz repeatedltimes..
1 j k k j