FOR DIRICHLET SERIES
O. DEMANZE AND A. MOUZEReceived 14 September 2005; Revised 11 May 2006; Accepted 30 May 2006
The paper deals with an extension theorem by Costakis and Vlachou on simultaneous approximation for holomorphic function to the setting of Dirichlet series, which are ab-solutely convergent in the right half of the complex plane. The derivation operator used in the analytic case is substituted by a weighted backward shift operator in the Dirichlet case. We show the similarities and extensions in comparing both results. Several density results are proved that finally lead to the main theorem on simultaneous approximation.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Let f(s)=n≥1ann−sbe a Dirichlet series and letσa(f) be its abscissa of absolute con-vergence, defined by
σa(f)=inf
σ∈R;
n≥1
ann−σ converges. (1.1)
We denoten≥1ann−sσ=n≥1|an|n−σ∈[0, +∞] for allσ∈R. If f is given by a finite sum of the previous type, then we say that f is a Dirichlet polynomial. LetC+be the
half-plane of complex numbers with strictly positive real part. We denote byᏰa(C+) the set of
Dirichlet series which are absolutely convergent onC+·This spaceᏰa(C+), endowed with
the topology given by the family of seminorms · σ, is a Fr´echet space. In the following, we fix ˜σ=(σk)k≥0to be a strictly decreasing sequence of real numbers which converges to
0. Then, the distance associated to the Fr´echet space is defined by
dσ˜(f,g)=
n≥0
1 2n
f −gσn
1 +f−gσn, f andgare inᏰa(C+). (1.2)
The purpose of this work is to obtain, for Dirichlet series inᏰa(C+), a
simultane-ous approximation theorem analogsimultane-ous with the one proved by Costakis and Vlachou for holomorphic functions [5]; see also [6]. LetΩbe a Jordan domain included inCand let
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 37014, Pages1–11
ξ∈Ωbe a fixed complex number. We denote byᐃthe set of holomorphic functions f
onΩ(f ∈H(Ω)) such that the following holds. For every choice of compact setsKand
L, withL⊂ΩandK⊂C\Ω, with connected complement and for every functionsφand
ψ, continuous inK andL, respectively, analytic in the interiorsK◦ andL◦, the following hold.
There exists a sequence (λn)n≥0of nonnegative integers such that
sup z∈K
λn
j=0 f(j)(ξ)
j! (z−ξ)
j−φ(z)
−−−−→n→+∞ 0,
sup z∈L
f(λn)(z)−ψ(z)−−−−→
n→+∞ 0.
(1.3)
Theorem 1.1 [5]. The intersection ofᐃwith the set of univalent functions and constant ones (denote byᏮ) isGδ-dense inᏮ.
This universal theorem can be proved using category type of arguments and applying an approximation theorem due to Mergelyan for analytic functions. The arguments are now well known. We can mention [4,6,7] for similar proofs. The idea of [6] is the follow-ing. If we have some approximation properties (realized by sequence of natural numbers) which hold onGδand dense subsets of a complete metric space, then by Baire’s theorem, the intersection of these sets is alsoGδ and dense. Therefore, there is an object realizing all those approximations (but with different sequences of indices for each one). Now if we repeat simultaneously the proofs of these generic approximation properties in some cases, a miracle happens. We realize all of them generically with the same choice of in-dices.
For the set of Dirichlet seriesᏰa(C+), a technical version of Mergelyan’s theorem has
recently been proved by Bayart [3]. This result uses the notion of compact setsK⊂C\ C+ admissible for Dirichlet series (seeDefinition 1.4). For every Dirichlet series f(z)=
+∞
k=1fkk−z, we denote bySn(f),n∈N∗, the partial sum
Sn(f)(z)= n
k=1
fkk−z, z∈C. (1.4)
Definition 1.2. Letω=(ωn)n∈N∗ be a sequence of strictly positive real numbers. Denote byBωthe backward weighted shift defined by
Bω(1)=0, Bω
n−s=ω1
n−1(n−1)
−s, ∀n≥2. (1.5)
The purpose of this paper is to give an analogy for Dirichlet series to the results of [5]. The study of derivative for analytic function can be seen as the study of a weighted backward shift (with the canonical basis)
For this paper, we just consider weightsω=(ωn)n∈N∗ satisfying (H1)
∀α≥0, ω1···ωmmα−→0, asm−→+∞, (1.7)
(H2)
∀α >0, the sequence 1
jαωj
j∈N∗
is bounded. (1.8)
We denote byMαthe supremum of the bounded sequence (1/ jαωj)j∈N∗. These condi-tions mean that the infinite product converges to zero but not too quickly. For example, we can choose
∀k≥2, ωk= C
log(k)β, C∈R
∗
+,β >0. (1.9)
Simpler examples can be chosen, by takingωk=1/λwithλ >1. Moreover, we can note that the second condition implies that theBω are continuous operators on the Fr´echet spaceᏰa(C+).We have for f(s)=n≥1ann−s∈Ᏸa(C+),
Bω(f) αn=
j≥1
aj+1j−αn
ωj =
j≥1
aj+1 j j+ 1
−αn+1
(j+ 1)−αn+1
jαn−αn+1ωj
≤2αn+1M
αn−αn+1fαn+1.
(1.10)
Therefore, we obtain the following result, as a direct consequence of the mainTheorem 4.3, which generalizes Bayart [3, Theorem 6]. This result isCorollary 4.4and states the following.
Corollary 1.3. Let f ∈Ᏸa(C+) be a Dirichlet series. Then, for everyε >0, there exists h∈Ᏸa(C+) satisfying the following. For everyϕ∈Ᏸa(C+), every admissible compact setK
inC−, and every functionψ:K→Ccontinuous onKand holomorphic inK◦, there exists a
sequence (λn)n≥0inN∗such that the following hold:
d˜σ(f,h)< ε,
sup z∈K
Sλn(h)(z)−ψ(z)−→0, asn−→+∞,
Bλnω(h)−→ϕ, asn−→+∞, inᏰaC+,
(1.11)
whereBλnω is theλnth iteration of the weighted backward shift.
Definition 1.4. LetK be a compact set included inC.This set is admissible for Dirichlet series ifC\K is connected, and if the following representation can be obtained:K=
K1∪ ··· ∪Kd, where theKishould be contained in disconnected stripsSi= {z∈C;ai≤ (z)≤bi}breadth strictly less than 1/2, that is,bi−ai<1/2.
We denote byC−the left half-plane{s∈C;(s)<0}. We can now express the version
of Mergelyan’s theorem for Dirichlet series included inᏰa(C+).
Theorem 1.5 [3]. LetK⊂C−be an admissible compact set for Dirichlet series, let f be a
Dirichlet series inᏰa(C+), and letgbe a continuous function onKwhich is analytic inK ◦
. For every pair of fixed positive real numbersσandε, there exists a Dirichlet polynomialhin Ᏸa(C+) such that
sup z∈K
h(z)−g(z)< ε, h−fσ< ε. (1.12)
Definition 1.6. Denote byᐃd(ω) the set of all Dirichlet seriesh∈Ᏸa(C+) satisfying the
following: for every admissible compact setKinC−, for every f ∈Ᏸa(C+), and for every
functiong:K→Ccontinuous onK and holomorphic inK◦, there exists a sequence of integers (λn)n≥0such that the following hold:
sup z∈K
Sλn(h)(z)−g(z)−−−−→ n→+∞ 0, Bωλn(h)−−−−→n→+∞ f inᏰa
C+
.
(1.13)
In a first step, we give relations between the setᐃd(ω) and subsets ofᏰa(C+) realizing
analogous estimations with Dirichlet polynomials. On the other hand, we prove that these subsets are open and that their union is dense in the Fr´echet space studied. We conclude using category-type arguments (see also [4,5,7]). Other recent developments related to universal Dirichlet series have been obtained in the same way in [8].
2. The setsᏻσk(C+,ρ,j,s,n)
Definition 2.1. The family of compact sets is defined as
Kai;bi;n
=z∈Csuch thatai≤ (z)≤biet(z)∈[−n;n]
, (2.1)
whereai∈Q,bi∈Q∩]ai;ai+ 1/2[,n∈N∗, andbi≤0.For alld∈N∗, letK(a1;b1;. . .;ad;
bd) be the disjoint union of the sets K(ai;bi;ni) for i∈ {1,. . .,d}. The family of com-pact setsK(a1;b1;. . .;ad;bd) is denumerable. This family can be denoted by the sequence (Kρ)ρ≥0.
It is obvious that the setsKρare well admissible for Dirichlet series and are contained inC−.
Proposition 2.2. For each admissible (for Dirichlet series) compact setK⊂C−, there exists
a nonnegative integerρ0such that the following inclusion holds:
Proof. We just have to prove this property for a compact setKincluded in only one strip
S= {z∈C/a≤ (z)≤b}withb−a <1/2. Using the notations above, if (a,b)∈Q2, we
just have to choosen≥sup{|z|/z∈K}. Else, we takea< aandb> bsuch that (a,b)∈ Q2andb−a<1/2. Hence, with the same choice forn, we obtain the desired relation
K⊂K(a;b;n).
In the following, we denote the set of Dirichlet polynomials with coefficients inQ+iQ by the sequence (fj)j∈N.
Lemma 2.3. The family of Dirichlet polynomials (fj)j∈Nis a dense set for the topology of the Fr´echet spaceᏰa(C+).
Proof. Letg=+∞
k=1αkk−sbe a Dirichlet seriesᏰa(C+). We define the sequence (fjn)n≥0
by
fjn= n
k=1
βk,nk−s, αk−βk,n≤ 1
2k+n, βk,n∈Q+iQ. (2.3)
Letσbe a fixed strictly positive real number and letεbe a strictly positive real number too. Clearly, there exists a positive integerNsuch that we have+k=∞N+1|αk|k−σ< ε. Therefore, we obtain the following inequality:
g−fjnσ= N
k=1
αk−βk,nk−σ+
+∞
k=N+1
αkk−σ< N
k=1
1 2k+n+ε <
1
2n+ε. (2.4)
The conclusion is obvious.
Remark 2.4. We obviously have the density of the family (fj)j∈Nfor the topology of the
uniform (on every compact set) convergence.
Definition 2.5. Letσ >0. According to the preceding definitions, for all positive integers
ρ,j,n,s, the setᏻσ(C+,ρ,j,s,n)⊂Ᏸa(C+) is defined by
ᏻσ
C+,ρ,j,s,n
=
g∈Ᏸa
C+
such that sup z∈Kρ
Sn(g)(z)−fj(z)<1
s,B
n
ω(g)−fjσ<1s
.
(2.5)
With these sets, we have a complete representation ofᐃd(ω). Lemma 2.6. The following equality holds:
ᐃd(ω)=
+∞
ρ=1 +∞
k=1 +∞
j=0 +∞
s=1 +∞
n=1
ᏻσkC+,ρ,j,s,n
. (2.6)
Proof. Letg be a Dirichlet series belonging to the right-hand side set. LetK⊂C−be an
admissible compact set for Dirichlet series andΦ:K→Cbe continuous function onK
integern0∈Nsuch that we have
sup z∈K
Sn0(g)(z)−Φ(z)< ε, Bn0
ω(g)−hσ< ε. (2.7)
Using Bayart’sTheorem 1.5, there exists a Dirichlet polynomialpsatisfying the two in-equalities
sup z∈K
p(z)−Φ(z)< ε
2, p−hσ<
ε
2. (2.8)
With no loss of generality, byLemma 2.3, we may assume that its coefficients are inQ+
iQ. As a consequence, there exists an integerj0∈Nsuch thatp=fj0. We lets0andρ0be
nonnegative integers satisfying 2≤s0εandK⊂Kρ0. Moreover there existsn0such that
sup z∈K
Sn0(g)(z)−p(z)< 1
s0,
Bn0
ω(g)−pσ< 1
s0. (2.9)
The last four inequalities allow us to conclude thatgbelongs toᐃd(ω). The inverse
in-clusion is obvious.
3. The setsᏻσk(C+,ρ,j,s,n) are open
In this section, we prove that each set n≥1ᏻσ(C+,ρ,j,s,n) is open according to the
Fr´echet topology defined by the seminorms ( · β)β onᏰa(C+). We divide the proof
into two parts.
Proposition 3.1. For all j,s,n, andρpositive integers, the subsetsΓ(ρ,j,s,n) ofᏰa(C+)
are open in the Fr´echet spaceᏰa(C+), where
Γ(ρ,j,s,n)=
g∈ᏰaC+such that sup
z∈Kρ
Sn(g)(z)−fj(z)<1
s
. (3.1)
Proof. We denote byMthe minimum of the real parts of the complex numbers included inKρ. Let f be a Dirichlet series inΓ(ρ,j,s,n), which means that we have
sup z∈Kρ
Sn(f)(z)−fj(z)<1
s. (3.2)
Letε1be the following strictly positive real number:
ε1=
1/s−supz∈KρSn(f)(z)−fj(z) n
r=1rσ0r−M
>0. (3.3)
Definea=ε1/(1 +ε1) and letgbe a Dirichlet series satisfyingdσ˜(g,f)< a. In particular,
we obtain
f −gσ0 1 +f−gσ0
Now, we can overestimate|Sn(g)(z)−fj(z)|. One has
Sn(g)(z)−fj(z)≤Sn(g−f)(z)+Sn(f)(z)−fj(z). (3.5)
Afterwards, if we denoteSn(g−f)(z)=nr=1brr−z, using (3.4), we have the following inequality:
br≤rσ0 n
r=1
brr−σ0< rσ0ε
1. (3.6)
Consequently, one has
sup z∈Kρ
Sn(g−f)(z)<n r=1
ε1rσ0−M. (3.7)
Hence, we obtain the strict inequality
sup z∈Kρ
Sn(g)(z)−fj(z)< ε1n r=1
rσ0−M+ sup z∈Kρ
Sn(f)(z)−fj(z)=1
s. (3.8)
Therefore, the setΓ(ρ,j,s,n) is open in the Fr´echet topology. Proposition 3.2. For all positive integers j,s,n, andρ, the subsetsΔσk(ρ,j,s,n) are open in the Fr´echet spaceᏰa(C+), where
Δσk(ρ,j,s,n)=
g∈Ᏸa
C+
such thatBωn(g)−fjσk< 1
s
. (3.9)
Proof. We begin this proof as the preceding one. Let f be a Dirichlet series included in Δσk(ρ,j,s,n). Hence
Bn
ω(f)−fjσk< 1
s. (3.10)
Takeε2to be a strictly positive real number satisfying
ε2≤
1/s−Bn
ω(f)−fjσk
An,k , withAn,k=supj≥1
j
j+n
−σk+1 sup
j
jσk+1−σk
ωj···ωj+n−1. (3.11)
Note that An,k is well defined because j/(j+n)→1 and the second supremum is well defined by (H2). Now, definea=ε2/2k+1(1 +ε2) and letg be a Dirichlet series
satisfy-ing d˜σ(g,f)< a. In particular, we obtain f−gσk+1/(1 +f−gσk+1)<2k
+1asof − gσk+1< ε2. Now we can estimateBωn(g)−fjσk. It follows that
Bn
ω(g)−fjσk≤Bnω(g−f)σk+Bnω(f)−fjσk. (3.12) Using the notation (g−f)(z)=+∞
r=1brr−z, we also have Bn
ω(g−f)σk=
+∞
j=1
bj+n
ωj···ωj+n−1j −σk≤A
As a consequence, we obtain the following inequality: Bn
ω(g)−fjσk≤ Bωn(g−f)σk+Bωn(f)−fjσk ≤An,kg−fσk+1+Bωn(f)−fjσk
< An,kε2+Bωn(f)−fjσk≤1
s.
(3.14)
Therefore, there exists an open ball, with f as central point, which is included into the set
Δσk(ρ,j,s,n).
Corollary 3.3. For all positive integersj,s,n,ρandk, the setᏻσk(C+,ρ,j,s,n) is open in
the Fr´echet spaceᏰa(C+).
Proof. We just have to remark that the setᏻσk(C+,ρ,j,s,n) is the intersection ofΔσk(ρ,j,s,
n) withΓ(ρ,j,s,n) which are open sets due to Propositions3.1and3.2.
4. Main results
Proposition 4.1. Letg be a Dirichlet polynomial denoted by g(z)=A
k=2gkk−z. Then, there exists a sequence of Dirichlet polynomials, (pm)m≥0, such thatpmσ→0, asm→+∞, for allσ∈R, and
∀z∈C,∀m∈N, Bmω
pm
(z)=g(z). (4.1)
Proof. Using the notations of the propositiong(z)=A
k=1gkk−z, we consider the Dirich-let polynomialspm(z)=
A
k=1gkωkωk+1···ωk+m−1(k+m)−z.With an easy computation,
we obtain the desired equalities. Moreover, we have for allσ∈R,
caseσ≥0 :pmσ≤ A
k=1
gkωkωk+1···ωk+m−1k−σ,
caseσ <0 :pmσ≤ A
k=1
gkωkωk+1···ωk+m−1(k+m)−σ.
(4.2)
Using (H1), we obtain the result.
Note that the sequence (pm)m≥0 constructed in the previous proposition converges
also uniformly (on each compact set) to 0.
We possess all the arguments to state and to prove the main theorem of this work. Theorem 4.2. The sets∞n=0ᏻσk(C+,ρ,j,s,n) are dense subsets of the Fr´echet spaceᏰa(C+).
Proof. Letd∈Ᏸa(C+) be writtend(z)=∞j=1ajj−z. We want to obtain a sequence of Dirichlet polynomials in∞n=0ᏻσk(C+,ρ,j,s,n) which converges tod. Fixε∈]0; 1/2s[ and β∈σ˜. We denote bydr(z) the partial sumrj=1ajj−z and we choosersuch thatd−
satisfying the following inequalities:
sup z∈Kρ
pm(z)−fj(z)< 1
m, pm−drβ< 1
m. (4.3)
Due toProposition 4.1, there exists a sequence of Dirichlet polynomials, (qm)m≥0, such
that qmσ →0, asm→+∞, for allσ ∈R andBωm(qm)(z)= fj(z). With the notation
fj(z)=Njl=1αl,jl−z, we know thatqmcan be described by
qm(z)= Nj
l=1
αl,jωlωl+1···ωl+m−1(l+m)−z. (4.4)
With this representation, we obtainSm(qm)(z)≡0. From (4.3), there exists an integerm0
such that we have supz∈Kρ|pm0(z)−fj(z)|<1/s. We denote pm0(z)= Nm0
l=1bl,m0l−z and takem1>max(m0;Nm0) such that
qm 1β=
Nj
l=1
αl,jωlωl+1···ωl+m 1−1
l+m1
−β
< 1
2s. (4.5)
Asm1> Nm0, we have the equalitySm1(pm0)=pm0. LetPbe defined by
P(z)=pm0(z) +qm1(z). (4.6)
We deduce from above thatPsatisfies Bm1
ω (P)−fjσk=Bmω1
pm0σk=0< 1
2s, (4.7)
becausem1> Nm0. Using Sm1(qm1)≡0, supz∈Kρ|Sm1(P)(z)−fj(z)| =supz∈Kρ|pm0(z)−
fj(z)|<1/s. Using formula (4.3), we also obtainpm−drβ< ε. Up to choosing a bigger integerm0, we may assume thatpm0−drβ< ε. As we have
P−dr
β≤pm0−drβ+qm1β<2ε, (4.8)
we conclude that for a fixed strictly positive realβ, for allε <1/(2s), there exists a Dirichlet polynomialP∈nᏻσk(C+,ρ,j,s,n) such thatP−drβ<2ε, which implies the inequal-ityP−dβ<3ε. As it is true for every strictly positive realβ, we obtain the density of +∞
n=0ᏻσk(C+,ρ,j,s,n) inᏰa(C+).
Theorem 4.3. The setᐃd(ω) is aGδand dense set included inᏰa(C+).
Proof. Lemma 2.6implies thatᐃd(ω) is a countable intersection of dense open sets of Ᏸa(C+). Hence, the result is a direct consequence of Baire’s theorem.
Corollary 4.4. Let f ∈Ᏸa(C+) be a Dirichlet series. Then, for everyε >0, there exists h∈Ᏸa(C+) satisfying the following. For everyϕ∈Ᏸa(C+), every admissible compact setK
inC−and every functionψ:K→Ccontinuous onK and holomorphic inK ◦
, there exists a sequence (λn)n≥0inN∗such that the following hold:
d˜σ(f,h)< ε, sup
z∈K
Sλn(h)(z)−ψ(z)−→0, asn−→+∞,
Bλnω(h)−→ϕ, asn−→+∞, inᏰa
C+
,
(4.9)
whereBλnω is theλnth iteration of the weighted backward shift.
Remark 4.5. In the article [5], the authors obtain an approximation result using only constant and univalent functions. Here, it is not realistic to obtain such a result because it is well known that a Dirichlet series cannot be univalent on its convergent domain (see, e.g., [2]). The present paper can also be seen as an extension of [6] to universal Dirichlet series. Moreover,Theorem 4.3is a universal approximation result in the sense of [7]. Note that here the intersection between compact sets andC+is not necessary empty.
Remark 4.6. According to a suggestion of Kahane (see [7, Proposition 3.2]), we have the following proposition.
Proposition 4.7. Let f be a Dirichlet series in Ᏸa(C+). Then, there exist g1 andg2 in ᐃd(ω) such that f =g1+g2.
Proof. We use a translation homeomorphismTf :Ᏸa(C+)→Ᏸa(C+) defined byTf(h)=
f −h. Asᐃd(ω) is aGδ and dense set, its image byTf is also aGδand dense set. Using Baire’s theorem,ᐃd(ω)∩[f −ᐃd(ω)] is no void. Letg1be an element of this
intersec-tion. It follows thatg1∈ᐃd(ω) andg1=f −g2for an elementg2∈ᐃd(ω). As for the classical case, there exists a relation between the setᐃd(ω) and the notion of universality in the sense of Menchoff. We do not repeat the proof, it suffices to follow [7, Proposition 3.1].
Proposition 4.8. LetS=n≥1ann−s∈ᐃd(ω) and letg andhbe two measurable func-tions fromRinto [−∞; +∞]. Then there exists a subsequenceSkm of the partial sums ofS such that
Skm(it)
−→g(t), Skm(it)
−→h(t), asm−→+∞, almost everywhere. (4.10)
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O. Demanze: Laboratoire de Math´ematiques, UMR 8524, Universit´e des Sciences et Technologies de Lille 1 (USTL), Cit´e Scientifique, 59650 Villeneuve d’Ascq, France
E-mail address:[email protected]
A. Mouze: Laboratoire de Math´ematiques, UMR 8524, Universit´e des Sciences et Technologies de Lille 1 (USTL), Cit´e Scientifique, 59650 Villeneuve d’Ascq, France
E-mail address:[email protected]
Current address: ´Ecole Centrale de Lille, Cite Scientifique, BP 48, 59651 Villeneuve
