UNIVERSAL APPROXIMATION THEOREM FOR DIRICHLET SERIES

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.


Introduction
Let f (s) = n≥1 a n n −s be a Dirichlet series and let σ a ( f ) be its abscissa of absolute convergence, defined by σ a ( f ) = inf σ ∈ R; n≥1 a n n −σ converges . (1.1) We denote n≥1 a n n −s σ = n≥1 |a n |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 Ᏸ a (C + ) the set of Dirichlet series which are absolutely convergent on C + • This space Ᏸ a (C + ), endowed with the topology given by the family of seminorms • σ , is a Fréchet space.In the following, we fix σ = (σ k ) 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 , f and g are in Ᏸ a (C + ). (1. 2) The purpose of this work is to obtain, for Dirichlet series in Ᏸ a (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 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 (λ n ) n≥0 of nonnegative integers such that sup (1.3) Theorem 1.1 [5].The intersection of ᐃ with the set of univalent functions and constant ones (denote by Ꮾ) is G δ -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 G δ and dense subsets of a complete metric space, then by Baire's theorem, the intersection of these sets is also G δ 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 Ᏸ a (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 (1.4) Definition 1.2.Let ω = (ω n ) n∈N * be a sequence of strictly positive real numbers.Denote by B ω the backward weighted shift defined by 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) ( O. Demanze and A. Mouze 3 For this paper, we just consider weights ω = (ω n ) n∈N * satisfying (H 1 ) ∀α > 0, the sequence 1 j α ω j j∈N * is bounded. (1.8) We denote by M α 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 Simpler examples can be chosen, by taking ω k = 1/λ with λ > 1.Moreover, we can note that the second condition implies that the B ω are continuous operators on the Fréchet space Ᏸ a (C + ).We have for f (s (1.10) 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 ∈ Ᏸ a (C + ) be a Dirichlet series.Then, for every ε > 0, there exists h ∈ Ᏸ a (C + ) satisfying the following.For every ϕ ∈ Ᏸ a (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 (λ n ) n≥0 in N * such that the following hold: where B λn ω is the λ n 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.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: 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 Ᏸ a (C + ).
Theorem 1.5 [3].Let K ⊂ C − be an admissible compact set for Dirichlet series, let f be a Dirichlet series in Ᏸ a (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 (1.12) Definition 1.6.Denote by ᐃ d (ω) the set of all Dirichlet series h ∈ Ᏸ a (C + ) satisfying the following: for every admissible compact set K in C − , for every f ∈ Ᏸ a (C + ), and for every function g : K → C continuous on K and holomorphic in K • , there exists a sequence of integers (λ n ) n≥0 such that the following hold: ( In a first step, we give relations between the set ᐃ d (ω) and subsets of Ᏸ a (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 ᏻ σk (C + ,ρ, j,s,n) Definition 2.1.The family of compact sets is defined as where It is obvious that the sets K ρ 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 ρ 0 such that the following inclusion holds: ( O. Demanze and A. Mouze 5 Proof.We just have to prove this property for a compact set K included in only one strip 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 Ᏸ a (C + ).
Proof.Let g = +∞ k=1 α k k −s be a Dirichlet series Ᏸ a (C + ).We define the sequence ( f jn ) n≥0 by 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 |α k |k −σ < ε.Therefore, we obtain the following inequality: ( 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 ᏻ σ (C + ,ρ, j,s,n) ⊂ Ᏸ a (C + ) is defined by With these sets, we have a complete representation of ᐃ d (ω).
Lemma 2.6.The following equality holds: 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 Ᏸ a (C + ).For all ε > 0 and σ > 0, we just have to find an 6 Universal approximation theorem for Dirichlet integer n 0 ∈ N such that we have Using Bayart's Theorem 1.5, there exists a Dirichlet polynomial p satisfying the two inequalities 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 j 0 ∈ N such that p = f j0 .We let s 0 and ρ 0 be nonnegative integers satisfying 2 ≤ s 0 ε and K ⊂ K ρ0 .Moreover there exists n 0 such that (2.9) The last four inequalities allow us to conclude that g belongs to ᐃ d (ω).The inverse inclusion is obvious.

The sets ᏻ σk (C + ,ρ, j,s,n) are open
In this section, we prove that each set n≥1 ᏻ σ (C + ,ρ, j,s,n) is open according to the Fréchet topology defined by the seminorms ( • β ) β on Ᏸ a (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 Ᏸ a (C + ) are open in the Fréchet space Ᏸ a (C + ), where Proof.We denote by M the minimum of the real parts of the complex numbers included in K ρ .Let f be a Dirichlet series in Γ(ρ, j,s,n), which means that we have Let ε 1 be the following strictly positive real number: Define a = ε 1 /(1 + ε 1 ) and let g be a Dirichlet series satisfying d σ (g, f ) < a.In particular, we obtain Demanze and A. Mouze 7 Now, we can overestimate |S n (g)(z) − f j (z)|.One has Afterwards, if we denote S n (g − f )(z) = n r=1 b r r −z , using (3.4), we have the following inequality: Consequently, one has Hence, we obtain the strict inequality Proof.We begin this proof as the preceding one.Let f be a Dirichlet series included in Δ σk (ρ, j,s,n).Hence Take ε 2 to be a strictly positive real number satisfying

11)
Note that A n,k is well defined because j/( j + n) → 1 and the second supremum is well defined by (H 2 ).Now, define a = ε 2 /2 k+1 (1 + ε 2 ) and let g be a Dirichlet series satisfying d σ (g, f ) < a.In particular, we obtain Using the notation (g − f )(z) = +∞ r=1 b r r −z , we also have 8 Universal approximation theorem for Dirichlet As a consequence, we obtain the following inequality: Therefore, there exists an open ball, with f as central point, which is included into the set Δ σk (ρ, j,s,n).
Corollary 3.3.For all positive integers j, s, n, ρ and k, the set Proof.We just have to remark that the set ᏻ σk (C + ,ρ, j,s,n) is the intersection of Δ σk (ρ, j,s, n) with Γ(ρ, j,s,n) which are open sets due to Propositions 3.1 and 3.2.

Main results
Proposition 4.1.Let g be a Dirichlet polynomial denoted by g(z) = A k=2 g k k −z .Then, there exists a sequence of Dirichlet polynomials, (p m ) m≥0 , such that p m σ → 0, as m → +∞, for all σ ∈ R, and Proof.Using the notations of the proposition g(z) = A k=1 g k k −z , we consider the Dirichlet polynomials p m (z With an easy computation, we obtain the desired equalities.Moreover, we have for all σ ∈ R, case σ ≥ 0 : Using (H 1 ), we obtain the result.
Note that the sequence (p m ) 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 ᏻ σk (C + ,ρ, j,s,n) are dense subsets of the Fréchet space Ᏸ a (C + ).Proof.Let d ∈ Ᏸ a (C + ) be written d(z) = ∞ j=1 a j j −z .We want to obtain a sequence of Dirichlet polynomials in ∞ n=0 ᏻ σk (C + ,ρ, j,s,n) which converges to d. Fix ε ∈]0; 1/2s[ and β ∈ σ.We denote by d r (z) the partial sum r j=1 a j j −z and we choose r such that d − d r β <ε.Now, we approximate the Dirichlet polynomial d r with an element in ∞ n=0 ᏻ σk (C + , ρ, j,s,n).According to Theorem 1.5, for all m > 0, there exists a Dirichlet polynomial p m satisfying the following inequalities: Due to Proposition 4.1, there exists a sequence of Dirichlet polynomials, (q m ) m≥0 , such that q m σ → 0, as m → +∞, for all σ ∈ R and B m ω (q m )(z) = f j (z).With the notation f j (z) = Nj l=1 α l, j l −z , we know that q m can be described by With this representation, we obtain S m (q m )(z) ≡ 0. From (4.3), there exists an integer m 0 such that we have sup z∈Kρ |p m0 (z) − f j (z)| < 1/s.We denote p m0 (z) = Nm 0 l=1 b l,m0 l −z and take m 1 > max(m 0 ;N m0 ) such that As m 1 > N m0 , we have the equality S m1 (p m0 ) = p m0 .Let P be defined by P(z) = p m0 (z) + q m1 (z).(4.6) We deduce from above that P satisfies because m 1 > N m0 .Using S m1 (q m1 ) ≡ 0, sup z∈Kρ |S m1 (P)(z 3), we also obtain p m − d r β < ε.Up to choosing a bigger integer m 0 , we may assume that p m0 − d r β < ε.As we have we conclude that for a fixed strictly positive real β, for all ε < 1/(2s

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The main objective of this Special Issue is to publish topics that are under study in one of those lines.The idea is to get the most recent researches and published them in a very short time, so we can give a step in order to help scientists and engineers that work in this field to be aware of actual research.All the published papers have to be peer reviewed, but in a fast and accurate way so that the topics are not outdated by the large speed that the information flows nowadays.
Before submission authors should carefully read over the journal's Author Guidelines, which are located at http://www .hindawi.com/journals/mpe/guidelines.html.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http://mts.hindawi.com/according to the following timetable: Using the notations above, if (a,b) ∈ Q 2 , we just have to choose n ≥ sup{|z|/z ∈ K}.Else, we take a < a and b > b such that (a ,b ) ∈ Q 2 and b − a < 1/2.Hence, with the same choice for n, we obtain the desired relation ), there exists a Dirichlet polynomial P ∈ n ᏻ σk (C + ,ρ, j,s,n) such that P − d r β < 2ε, which implies the inequality P − d β < 3ε.As it is true for every strictly positive real β, we obtain the density of +∞ n=0 ᏻ σk (C + ,ρ, j,s,n) in Ᏸ a (C + ).Theorem 4.3.The set ᐃ d (ω) is a G δ and dense set included in Ᏸ a (C + )..Lemma 2.6 implies that ᐃ d (ω) is a countable intersection of dense open sets of Ᏸ a (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.