UNIVERSAL APPROXIMATION THEOREM
FOR DIRICHLET SERIES
O. DEMANZE AND A. MOUZE
Received 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 absolutely 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≥1 an n−s be a Dirichlet series and let σa ( f ) be its abscissa of absolute convergence, defined by
σa ( f ) = inf σ ∈ R;
an n−σ converges .
(1.1)
n ≥1
We denote n≥1 an n−s σ = n≥1 |an |n−σ ∈ [0,+∞] for all σ ∈ R. If f is given by a finite
sum of the previous type, then we say that f is a Dirichlet polynomial. Let C+ be the halfplane of complex numbers with strictly positive real part. We denote by Ᏸa (C+ ) the set of
Dirichlet series which are absolutely convergent on C+ · This space Ᏸa (C+ ), endowed with
the topology given by the family of seminorms · σ , is a Fréchet space. In the following,
we fix σ̃ = (σk )k≥0 to be a strictly decreasing sequence of real numbers which converges to
0. Then, the distance associated to the Fréchet space is defined by
dσ̃ ( f ,g) =
1
f − g σn
,
n 1 + f − g
2
σn
n ≥0
f and g are in Ᏸa (C+ ).
(1.2)
The purpose of this work is to obtain, for Dirichlet series in Ᏸa (C+ ), a simultaneous approximation theorem analogous with the one proved by Costakis and Vlachou for
holomorphic functions [5]; see also [6]. Let Ω be a Jordan domain included in C and let
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 37014, Pages 1–11
DOI 10.1155/IJMMS/2006/37014
2
Universal approximation theorem for Dirichlet series
ξ ∈ Ω be a fixed complex number. We denote by ᐃ the set of holomorphic functions f
on Ω ( f ∈ H(Ω)) such that the following holds. For every choice of compact sets K and
L, with L ⊂ Ω and K ⊂ C \ Ω, with connected complement and for every functions φ and
◦
◦
ψ, continuous in K and L, respectively, analytic in the interiors K and L, the following
hold.
There exists a sequence (λn )n≥0 of nonnegative integers such that
λn
f ( j) (ξ)
j
−−−−→ 0,
(z − ξ) − φ(z)
sup n→+∞
j!
z∈K j =0
sup f (λn ) (z) − ψ(z) −−−−→ 0.
(1.3)
n→+∞
z ∈L
Theorem 1.1 [5]. The intersection of ᐃ with the set of univalent functions and constant
ones (denote by Ꮾ) is Gδ -dense in Ꮾ.
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 following. If we have some approximation properties (realized by sequence of natural numbers)
which hold on Gδ and dense subsets of a complete metric space, then by Baire’s theorem,
the intersection of these sets is also Gδ 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 indices.
For the set of Dirichlet series Ᏸa (C+ ), a technical version of Mergelyan’s theorem has
recently been proved by Bayart [3]. This result uses the notion of compact sets K ⊂ C \
C+ admissible for Dirichlet series (see Definition 1.4). For every Dirichlet series f (z) =
+∞
−z
∗
k=1 fk k , we denote by Sn ( f ), n ∈ N , the partial sum
Sn ( f )(z) =
n
f k k −z ,
z ∈ C.
(1.4)
k =1
Definition 1.2. Let ω = (ωn )n∈N∗ be a sequence of strictly positive real numbers. Denote
by Bω the backward weighted shift defined by
Bω (1) = 0,
B ω n −s =
1
ω 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)
D zn = nzn−1 ,
∀n ≥ 2.
(1.6)
O. Demanze and A. Mouze 3
For this paper, we just consider weights ω = (ωn )n∈N∗ satisfying
(H1 )
∀α ≥ 0,
ω1 · · · ωm mα −→ 0,
as m −→ +∞,
(1.7)
(H2 )
∀α > 0,
the sequence
1
jαω j
is bounded.
(1.8)
j ∈N∗
We denote by Mα the supremum of the bounded sequence (1/ j α ω j ) j ∈N∗ . These conditions mean that the infinite product converges to zero but not too quickly. For example,
we can choose
∀k ≥ 2,
ωk = C
log(k)
C ∈ R∗+ , β > 0.
β ,
(1.9)
Simpler examples can be chosen, by taking ωk = 1/λ with λ > 1. Moreover, we can note
that the second condition implies
that the Bω are continuous operators on the Fréchet
space Ᏸa (C+ ). We have for f (s) = n≥1 an n−s ∈ Ᏸa (C+ ),
Bω ( f ) αn
=
−α n
a j+1 j
a j+1 =
j ≥1
≤2
αn+1
ωj
j ≥1
j
j +1
−αn+1
( j + 1)−αn+1
j αn −αn+1 ω j
(1.10)
Mαn −αn+1 f αn+1 .
Therefore, we obtain the following result, as a direct consequence of the main Theorem
4.3, which generalizes Bayart [3, Theorem 6]. This result is Corollary 4.4 and states the
following.
Corollary 1.3. Let f ∈ Ᏸa (C+ ) be a Dirichlet series. Then, for every ε > 0, there exists
h ∈ Ᏸa (C+ ) satisfying the following. For every ϕ ∈ Ᏸa (C+ ), every admissible compact set K
◦
in C− , and every function ψ : K → C continuous on K and holomorphic in K, there exists a
sequence (λn )n≥0 in N∗ such that the following hold:
dσ̃ ( f ,h) < ε,
sup Sλn (h)(z) − ψ(z) −→ 0,
z ∈K
Bωλn (h) −→ ϕ,
as n −→ +∞,
(1.11)
as n −→ +∞, in Ᏸa C+ ,
where Bωλn is the λn th iteration of the weighted backward shift.
The method of the proof is analogous to the one of the main theorem in [5]. We use
principally the version of Mergelyan’s theorem for Dirichlet series given by Bayart [3].
The main tool of approximation by Dirichlet series is a technical lemma from Bagchi [1]
which needs the notion of admissible compact set.
4
Universal approximation theorem for Dirichlet series
Definition 1.4. Let K be a compact set included in C. This set is admissible for Dirichlet
series if C \ K is connected, and if the following representation can be obtained: K =
K1 ∪ · · · ∪ Kd , where the Ki should be contained in disconnected strips Si = {z ∈ C; ai ≤
(z) ≤ bi } breadth strictly less than 1/2, that is, bi − ai < 1/2.
We denote by C− the left half-plane {s ∈ C; (s) < 0}. We can now express the version
of Mergelyan’s theorem for Dirichlet series included in Ᏸa (C+ ).
Theorem 1.5 [3]. Let K ⊂ C− be an admissible compact set for Dirichlet series, let f be a
◦
Dirichlet series in Ᏸa (C+ ), and let g be a continuous function on K which is analytic in K.
For every pair of fixed positive real numbers σ and ε, there exists a Dirichlet polynomial h in
Ᏸa (C+ ) such that
sup h(z) − g(z) < ε,
h − f σ < ε.
z ∈K
(1.12)
Definition 1.6. Denote by ᐃd (ω) the set of all Dirichlet series h ∈ Ᏸa (C+ ) satisfying the
following: for every admissible compact set K in C− , for every f ∈ Ᏸa (C+ ), and for every
◦
function g : K → C continuous on K and holomorphic in K, there exists a sequence of
integers (λn )n≥0 such that the following hold:
sup Sλn (h)(z) − g(z) −−−−→ 0,
n→+∞
z ∈K
Bωλn (h) −−−−→ f
n→+∞
(1.13)
in Ᏸa C+ .
In a first step, we give relations between the set ᐃd (ω) and subsets of Ᏸa (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 Fréchet 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 sets ᏻσk (C+ ,ρ, j,s,n)
Definition 2.1. The family of compact sets is defined as
K ai ;bi ;n = z ∈ C such that ai ≤ (z) ≤ bi et (z) ∈ [−n;n] ,
(2.1)
where ai ∈ Q, bi ∈ Q∩]ai ;ai + 1/2[, n ∈ N∗ , and bi ≤ 0. For all d ∈ N∗ , let K(a1 ;b1 ;... ;ad ;
bd ) be the disjoint union of the sets K(ai ;bi ;ni ) for i ∈ {1,...,d}. The family of compact sets K(a1 ;b1 ;... ;ad ;bd ) is denumerable. This family can be denoted by the sequence
(Kρ )ρ≥0 .
It is obvious that the sets Kρ are well admissible for Dirichlet series and are contained
in C− .
Proposition 2.2. For each admissible (for Dirichlet series) compact set K ⊂ C− , there exists
a nonnegative integer ρ0 such that the following inclusion holds:
K ⊂ Kρ 0 .
(2.2)
O. Demanze and A. Mouze 5
Proof. We just have to prove this property for a compact set K included in only one strip
S = {z ∈ C/a ≤ (z) ≤ b} with b − a < 1/2. Using the notations above, if (a,b) ∈ Q2 , we
just have to choose n ≥ sup{|z|/z ∈ K }. Else, we take a < a and b > b such that (a ,b ) ∈
Q2 and b − a < 1/2. Hence, with the same choice for n, we obtain the desired relation
K ⊂ K(a ;b ;n).
In the following, we denote the set of Dirichlet polynomials with coefficients in Q + iQ
by the sequence ( f j ) j ∈N .
Lemma 2.3. The family of Dirichlet polynomials ( f j ) j ∈N is a dense set for the topology of the
Fréchet space Ᏸa (C+ ).
Proof. Let g =
by
+∞
k=1 αk k
f jn =
−s
n
be a Dirichlet series Ᏸa (C+ ). We define the sequence ( f jn )n≥0
βk,n k−s ,
αk − βk,n ≤
k =1
1
,
2k+n
βk,n ∈ Q + iQ.
(2.3)
Let σ be a fixed strictly positive real number and let ε be a strictly
positive real number too.
Clearly, there exists a positive integer N such that we have +k=∞N+1 |αk |k−σ < ε. Therefore,
we obtain the following inequality:
g − f j n σ =
N
+∞
N
−σ 1
1
αk − βk,n k −σ +
αk k <
+ ε < n + ε.
k+n
k =1
k=N+1
k =1
2
2
(2.4)
The conclusion is obvious.
Remark 2.4. We obviously have the density of the family ( f j ) j ∈N for the topology of the
uniform (on every compact set) convergence.
Definition 2.5. Let σ > 0. According to the preceding definitions, for all positive integers
ρ, j, n, s, the set ᏻσ (C+ ,ρ, j,s,n) ⊂ Ᏸa (C+ ) is defined by
1
1
ᏻσ C+ ,ρ, j,s,n = g ∈ Ᏸa C+ such that sup Sn (g)(z) − f j (z) < , Bωn (g) − f j σ < .
s
s
z ∈K ρ
(2.5)
With these sets, we have a complete representation of ᐃd (ω).
Lemma 2.6. The following equality holds:
ᐃd (ω) =
+∞ +
+
∞ +
∞+
∞
∞
ᏻσk C+ ,ρ, j,s,n .
(2.6)
ρ=1 k=1 j =0 s=1 n=1
Proof. Let g be a Dirichlet series belonging to the right-hand side set. Let K ⊂ C− be an
admissible compact set for Dirichlet series and Φ : K → C be continuous function on K
◦
and analytic in K, and let h be in Ᏸa (C+ ). For all ε > 0 and σ > 0, we just have to find an
6
Universal approximation theorem for Dirichlet series
integer n0 ∈ N such that we have
n
B 0 (g) − h < ε.
ω
σ
sup Sn0 (g)(z) − Φ(z) < ε,
z ∈K
(2.7)
Using Bayart’s Theorem 1.5, there exists a Dirichlet polynomial p satisfying the two inequalities
ε
sup p(z) − Φ(z) < ,
2
z ∈K
p − h σ <
ε
.
2
(2.8)
With no loss of generality, by Lemma 2.3, we may assume that its coefficients are in Q +
iQ. As a consequence, there exists an integer j0 ∈ N such that p = f j0 . We let s0 and ρ0 be
nonnegative integers satisfying 2 ≤ s0 ε and K ⊂ Kρ0 . Moreover there exists n0 such that
sup Sn0 (g)(z) − p(z) <
z ∈K
n
B 0 (g) − p < 1 .
ω
σ
1
,
s0
(2.9)
s0
The last four inequalities allow us to conclude that g belongs to ᐃd (ω). The inverse in
clusion is obvious.
3. The sets ᏻσk (C+ ,ρ, j,s,n) are open
In this section, we prove that each set n≥1 ᏻσ (C+ ,ρ, j,s,n) is open according to the
Fréchet topology defined by the seminorms ( · β )β on Ᏸa (C+ ). We divide the proof
into two parts.
Proposition 3.1. For all j, s, n, and ρ positive integers, the subsets Γ(ρ, j,s,n) of Ᏸa (C+ )
are open in the Fréchet space Ᏸa (C+ ), where
Γ(ρ, j,s,n) = g ∈ Ᏸa C+ such that sup Sn (g)(z) − f j (z) <
z ∈K ρ
1
.
s
(3.1)
Proof. We denote by M the minimum of the real parts of the complex numbers included
in Kρ . Let f be a Dirichlet series in Γ(ρ, j,s,n), which means that we have
1
sup Sn ( f )(z) − f j (z) < .
s
z ∈K ρ
(3.2)
Let ε1 be the following strictly positive real number:
ε1 =
1/s − supz∈Kρ Sn ( f )(z) − f j (z)
n
r =1 r
σ0 r −M
> 0.
(3.3)
Define a = ε1 /(1 + ε1 ) and let g be a Dirichlet series satisfying dσ̃ (g, f ) < a. In particular,
we obtain
f − g σ0
< a which means f − g σ0 < ε1 .
1 + f − g σ0
(3.4)
O. Demanze and A. Mouze 7
Now, we can overestimate |Sn (g)(z) − f j (z)|. One has
Sn (g)(z) − f j (z) ≤ Sn (g − f )(z) + Sn ( f )(z) − f j (z).
Afterwards, if we denote Sn (g − f )(z) =
inequality:
n
r =1 br r
−z
(3.5)
, using (3.4), we have the following
n
−σ
br ≤ r σ0
br r 0 < r σ0 ε1 .
(3.6)
r =1
Consequently, one has
n
sup Sn (g − f )(z) <
z ∈K ρ
ε1 r σ0 −M .
(3.7)
r =1
Hence, we obtain the strict inequality
sup Sn (g)(z) − f j (z) < ε1
z ∈K ρ
n
1
r σ0 −M + sup Sn ( f )(z) − f j (z) = .
s
z ∈K ρ
r =1
Therefore, the set Γ(ρ, j,s,n) is open in the Fréchet topology.
(3.8)
Proposition 3.2. For all positive integers j, s, n, and ρ, the subsets Δσk (ρ, j,s,n) are open
in the Fréchet space Ᏸa (C+ ), where
Δσk (ρ, j,s,n) = g ∈ Ᏸa C+ such that Bωn (g) − f j σk <
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
n
B ( f ) − f j < 1 .
ω
σk
(3.10)
s
Take ε2 to be a strictly positive real number satisfying
1/s − Bωn ( f ) − f j σk
,
ε2 ≤
An,k
with An,k = sup
j ≥1
j
j +n
−σk+1
sup
j
j σk+1 −σk
.
ω j · · · ω j+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, define a = ε2 /2k+1 (1 + ε2 ) and let g be a Dirichlet series satisfying dσ̃ (g, f ) < a. In particular, we obtain f − g σk+1 /(1 + f − g σk+1 ) < 2k+1 a so f −
g σk+1 < ε2 . Now we can estimate Bωn (g) − f j σk . It follows that
n
B (g) − f j ≤ B n (g − f ) + B n ( f ) − f j .
ω
ω
ω
σk
σk
σk
Using the notation (g − f )(z) =
+∞
+∞
n
B (g − f ) =
ω
σk
r =1 br r
−z
(3.12)
, we also have
b j+n ω · · · ω j+n−1
j =1 j
j −σk ≤ An,k g − f σk+1 .
(3.13)
8
Universal approximation theorem for Dirichlet series
As a consequence, we obtain the following inequality:
n
B (g) − f j ≤ B n (g − f )σ + B n ( f ) − f j ω
ω
ω
k
σk
σk
n
≤ An,k g − f σk+1 + Bω ( f ) − f j σk
1
< An,k ε2 + Bωn ( f ) − f j σk ≤ .
(3.14)
s
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 integers j, s, n, ρ and k, the set ᏻσk (C+ ,ρ, j,s,n) is open in
the Fréchet space Ᏸa (C+ ).
Proof. We just have to remark that the set ᏻσk (C+ ,ρ, j,s,n) is the intersection of Δσk (ρ, j,s,
n) with Γ(ρ, j,s,n) which are open sets due to Propositions 3.1 and 3.2.
4. Main results
Proposition 4.1. Let g be a Dirichlet polynomial denoted by g(z) = Ak=2 gk k−z . Then,
there exists a sequence of Dirichlet polynomials, (pm )m≥0 , such that pm σ → 0, as m → +∞,
for all σ ∈ R, and
∀z ∈ C, ∀m ∈ N,
Bωm pm (z) = g(z).
(4.1)
Proof. Using the notations of the proposition g(z) = Ak=1 gk k−z , we consider the Dirich
let polynomials pm (z) = Ak=1 gk ωk ωk+1 · · · ωk+m−1 (k + m)−z . With an easy computation,
we obtain the desired equalities. Moreover, we have for all σ ∈ R,
case σ ≥ 0 : pm σ ≤
A
gk ωk ωk+1 · · · ωk+m−1 k −σ ,
k =1
case σ < 0 : pm σ ≤
A
gk ωk ωk+1 · · · ωk+m−1 (k + m)−σ .
(4.2)
k =1
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=0 ᏻσk (C+ ,ρ,
j,s,n) are dense subsets of the Fréchet space Ᏸa (C+ ).
−z
Proof. Let d ∈ Ᏸa (C+ ) be written d(z) = ∞
j =1 a j j . We want to obtain a sequence of
∞
Dirichlet polynomials in n=0 ᏻσk (C+ ,ρ, j,s,n)
which converges to d. Fix ε ∈]0;1/2s[ and
β ∈ σ̃. We denote by dr (z) the partial sum rj =1 a j j −z and we choose r such that d −
dr β <ε. Now, we approximate the Dirichlet polynomial dr with an element in ∞
n=0 ᏻσk(C+,
ρ, j,s,n). According to Theorem 1.5, for all m > 0, there exists a Dirichlet polynomial pm
O. Demanze and A. Mouze 9
satisfying the following inequalities:
sup pm (z) − f j (z) <
z ∈K ρ
1
,
m
pm − dr < 1 .
β
m
(4.3)
Due to Proposition 4.1, there exists a sequence of Dirichlet polynomials, (qm )m≥0 , such
that qm σ → 0, as m → +∞, for all σ ∈ R and Bωm (qm )(z) = f j (z). With the notation
N j
f j (z) = l=1 αl, j l−z , we know that qm can be described by
qm (z) =
Nj
αl, j ωl ωl+1 · · · ωl+m−1 (l + m)−z .
(4.4)
l=1
With this representation, we obtain Sm (qm )(z) ≡ 0. From (4.3), there exists an integer m0
Nm
such that we have supz∈Kρ | pm0 (z) − f j (z)| < 1/s. We denote pm0 (z) = l=10 bl,m0 l−z and
take m1 > max(m0 ;Nm0 ) such that
N
j qm = αl, j ωl ωl+1 · · · ωl+m −1 l + m1 −β < 1 .
1 β
1
2s
l=1
(4.5)
As m1 > Nm0 , we have the equality Sm1 (pm0 ) = pm0 . Let P be defined by
P(z) = pm0 (z) + qm1 (z).
(4.6)
We deduce from above that P satisfies
m
B 1 (P) − f j = B m1 pm = 0 < 1 ,
ω
ω
0
σk
σk
2s
(4.7)
because m1 > Nm0 . Using Sm1 (qm1 ) ≡ 0, supz∈Kρ |Sm1 (P)(z) − f j (z)| = supz∈Kρ | pm0 (z) −
f j (z)| < 1/s. Using formula (4.3), we also obtain pm − dr β < ε. Up to choosing a bigger
integer m0 , we may assume that pm0 − dr β < ε. As we have
P − dr ≤ pm − dr + qm < 2ε,
0
1 β
β
β
(4.8)
we conclude thatfor a fixed strictly positive real β, for all ε < 1/(2s), there exists a Dirichlet
polynomial P ∈ n ᏻσk (C+ ,ρ, j,s,n) such that P − dr β < 2ε, which implies the inequality P − dβ < 3ε. As it is true for every strictly positive real β, we obtain the density of
+∞
n=0 ᏻσk (C+ ,ρ, j,s,n) in Ᏸa (C+ ).
Theorem 4.3. The set ᐃd (ω) is a Gδ and dense set included in Ᏸa (C+ ).
Proof. Lemma 2.6 implies that ᐃd (ω) is a countable intersection of dense open sets of
Ᏸa (C+ ). Hence, the result is a direct consequence of Baire’s theorem.
As a straightforward consequence of Theorem 4.3, we obtain the following explicit
result of approximation.
10
Universal approximation theorem for Dirichlet series
Corollary 4.4. Let f ∈ Ᏸa (C+ ) be a Dirichlet series. Then, for every ε > 0, there exists
h ∈ Ᏸa (C+ ) satisfying the following. For every ϕ ∈ Ᏸa (C+ ), every admissible compact set K
◦
in C− and every function ψ : K → C continuous on K and holomorphic in K, there exists a
sequence (λn )n≥0 in N∗ such that the following hold:
dσ̃ ( f ,h) < ε,
sup Sλn (h)(z) − ψ(z) −→ 0,
z ∈K
Bωλn (h) −→ ϕ,
as n −→ +∞,
as n −→ +∞, in Ᏸa C+ ,
(4.9)
where Bωλn is the λn th 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 4.3 is a universal approximation result in the sense of [7]. Note
that here the intersection between compact sets and C+ 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 and g2 in
ᐃd (ω) such that f = g1 + g2 .
Proof. We use a translation homeomorphism T f : Ᏸa (C+ ) → Ᏸa (C+ ) defined by T f (h) =
f − h. As ᐃd (ω) is a Gδ and dense set, its image by T f is also a Gδ and dense set. Using
Baire’s theorem, ᐃd (ω) ∩ [ f − ᐃd (ω)] is no void. Let g1 be an element of this intersec
tion. It follows that g1 ∈ ᐃd (ω) and g1 = f − g2 for an element g2 ∈ ᐃd (ω).
As for the classical case, there exists a relation between the set ᐃd (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. Let S = n≥1 an n−s ∈ ᐃd (ω) and let g and h be two measurable functions from R into [−∞;+∞]. Then there exists a subsequence Skm of the partial sums of S
such that
Skm (it) −→ g(t),
Skm (it) −→ h(t),
as m −→ +∞, almost everywhere.
(4.10)
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O. Demanze and A. Mouze 11
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O. Demanze: Laboratoire de Mathématiques, UMR 8524, Université des Sciences et
Technologies de Lille 1 (USTL), Cité Scientifique, 59650 Villeneuve d’Ascq, France
E-mail address: olivier.demanze@math.univ-lille1.fr
A. Mouze: Laboratoire de Mathématiques, UMR 8524, Université des Sciences et
Technologies de Lille 1 (USTL), Cité Scientifique, 59650 Villeneuve d’Ascq, France
E-mail address: augustin.mouze@math.univ-lille1.fr
Current address: École Centrale de Lille, Cite Scientifique, BP 48, 59651 Villeneuve
d’Ascq Cedex, France