On the construction of large Algebras not contained in the image of the Borel map

The Borel map $j^{\infty}$ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. It is well known that the restriction of $j^{\infty}$ to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In a recent paper the authors have studied the size of the image of $j^{\infty}$ by using different approaches and worked in the general setting of quasianalytic ultradifferentiable classes defined by weight matrices. The aim of this paper is to show that the image of $j^{\infty}$ is also small with respect to the notion of algebrability and we treat both the Cauchy product (convolution) and the pointwise product. In particular, a deep study of the stability of the considered spaces under the pointwise product is developed.


Introduction
Classes of ultradifferentiable functions on an open subset U ⊆ R are classically defined by imposing growth restrictions on their derivatives. In the case these restrictions are controlled by a weight sequence M = (M j ) j∈N , given a sequence a = (a j ) j∈N of complex numbers, many authors have investigated under which conditions on M and a there exists a function f in the class associated to M satisfying f (j) (0) = a j for every j ∈ N, see [12,22,30]. This coincides with the study of the surjectivity of the Borel map f → f (j) (0) j∈N in the corresponding spaces. Following the work of [10], it is also very classical to consider growth restrictions defined by using weight functions ω. In this situation, the study of the surjectivity of the Borel map has been proposed in [8,6]. More recently, new classes of ultradifferentiable functions have been introduced in order to obtain a general framework that covers both previous situtations, but also different ones, see [23] and [28]. These classes are based on weight matrices M and the study of the surjectivity of the Borel map in this context has been carried out in [24]. In any situation, it appears that if the considered class is quasianalytic, which means that on this class the Borel map is injective, and if it contains strictly the analytic functions, then the Borel map is never surjective onto the corresponding weighted sequence space. In this context, the authors have studied in the recent paper [13] the question of knowing how far is the Borel map from being surjective. More precisely, they obtained that the image of the Borel map is "small" in the corresponding sequence space, where the notion of smallness is defined using different approaches: the notion of residual sets based on Baire categories, the notion of prevalence, and the notion of lineability. This paper aims at obtaining the corresponging result in the algebraic sense, using the notion of algebrability. While the concept of lineability consists in space, i.e. a countable inductive limit of Banach spaces with compact connecting mappings, see [20,Proposition 2.2].
Note that the special case M = (j!) j∈N yields E {M} (U ) = C ω (U ) the space of real analytic functions on U , whereas E (M) (U ) consists of the restrictions of all entire functions provided that U is connected.
Definition 2.2. The spaces of germs at 0 ∈ R of the M -ultradifferentiable functions of Roumieu and Beurling types are defined respectively by Again, if one considers the sequence M = (j!) j∈N in the Roumieu case, we obtain the space of germs of real analytic functions at 0 ∈ R; it is denoted by O 0 .
Let M ∈ R N >0 be arbitrary and define the sets of weighted formal power series by We endow these spaces with their natural topology: , the sequence space has been introduced in [13, Def. 2. 1.4], by identifying the coefficients (F j ) j with a sequence (of complex numbers). So all results from [13] (and from [24]) are also valid for the sets F [M] instead of Λ 1 [M] . Note that in [13] we have preferred to work with classes Λ 1 [M] , but in this present work it seems to be more natural to consider instead classes of weighted formal power series as defined above since the Cauchy product * seems to be more natural when considered on F [M] . Note however that we will also obtain results using the pointwise product.
We introduce the Borel map j ∞ (at 0) by setting We consider the following definition, according to [ So for any M ∈ LC, assumption (III) above is not necessarily required.
Let us also introduce some classical conditions on a sequence M ∈ R N >0 : • M has moderate growth, denoted by (mg), if If M is log-convex, then using Carleman's inequality one can show (for a proof see e.g. [ for every x > 0 small enough.
Keeping the notations of this Theorem, we directly obtained in [13, Corollary 3.1.2] the following important result. It will be the key for the proofs of algebrability.
Corollary 2.7. Let M be a quasianalytic weight sequence. If F = +∞ j=0 F j x j is a formal power series for which there exists a sequence of positive real numbers (a n ) n∈N decreasing to 0 such that We call the weight sequences M and N equivalent, denoted by M ≈ N , if M N and N M.
In the relations above one can replace M and N simultaneously by m and n because the factorial term is cancelling out.
Those relations between weight sequences imply inclusions between ultradifferentiable classes, see e.g. [ In particular, C ω E (N ) if and only if lim j→+∞ (n j ) 1/j = +∞.
Let us close this section by gathering some comments from [13].
• In the following sections we will study the Borel map j ∞ defined on quasianalytic ultradifferentiable classes such that C ω E [M] holds true. The general assumptions (I) − (III) on M are not restricting the generality of our considerations: For any M ∈ R N >0 with C ω ⊆ E [M] we have lim inf j→+∞ (m j ) 1/j > 0 in the Roumieu and lim j→+∞ (m j ) 1/j = +∞ in the Beurling case and we can replace M by its log-convex minorant M lc (see [21, Chapitre I] and [20, (3.2)]) without changing the associated ultradifferentiable class whereas only F [M lc ] ⊆ F [M] follows (and the weight matrix/function setting is reduced to the sequence case situation).
• In this paper all the spaces and results are considered in R, but everything goes similarly in R r by using a simple reduction argument. • Finally by translation all results below also hold true if 0 ∈ R is replaced by any other point a ∈ R.

Algebrability with respect to the Cauchy product
The classical product that can be considered on the space F [M] is the Cauchy product (or convolution). It is defined by The aim of this section is to obtain results of algebrability in F [M] endowed with the Cauchy product. Then we extend them to the weight matrix and weight function settings. By the Leibnitz formula, we have that pointwise multiplication of functions is transferred to the Cauchy product for their formal power series, i.e. one has .
which is the case if M is a normalized log-convex sequence (see [ Proof. Indeed, if 3.1. The weight sequence setting. We start with the single weight sequence case and prove the following result.
Theorem 3.2. Let M and N be two quasianalytic weight sequences. Then Proof. By assumption, we can consider an increasing sequence (k p ) p∈N of natural numbers satisfying: (i) k 0 = 1 and k p > pk p−1 for every p ∈ N >0 , ω M j,kp − 1 n j ≤ 1 for every p ∈ N >0 , where the numbers (ω M j,k ) j,k∈N are those arising in Theorem 2.6. Let (A, B) be an open interval with 0 < A < B < 1. Let us also consider a Hamel basis H of R (i.e. a basis of R seen as a Q vector space). We can assume that the elements of H are in (A, B). Indeed, if h ∈ H is not in (A, B), it suffices to consider q h ∈ Q such that q h h ∈ (A, B), and we keep a basis. For an arbitrary given value b ∈ H, we define the formal power series . Let us note that if F := F b and if we define the formal power series F (i) := F * · · · * F i times , then one has where A i (j) := (k p1 , . . . , k pi ) ∈ A i : k p1 + · · · + k pi = j . In particular, if j = ik p for some p ≥ i and if k p1 + · · ·+ k pi = ik p , then one has k p1 = · · · = k pi = k p since the sequence (k q ) q∈N is strictly increasing and since k p+1 > (p + 1)k p > ik p . Consequently, one has A i (ik p ) = {(k p , . . . , k p )} and Now, let us consider the algebra G generated by F b : b ∈ H and let us show that G has the desired property. Any element of this algebra can be written as G = L l=1 α l F b1 * · · · * F b1 i l,1 times * · · · * F bJ * · · · * F bJ i l,J times , where α 1 , . . . , α L = 0, b 1 , . . . , b J ∈ H are pairwise distinct, for every l ∈ {1, . . . , L} there is at least one m ∈ {1, . . . , J} such that i l,m = 0 and for every l, l ′ ∈ {1, . . . , L}, l = l ′ , there is at least one m ∈ {1, . . . , J} such that i l,m = i l ′ ,m . For every l ∈ {1, . . . , L}, let us set P l := i l,1 + · · · + i l,J .
In order to show that the formal power series G does not belong to the image j ∞ (E 0 {M} ) of the Borel map, by Corollary 2.7 it suffices to show that ω M j,kp G j a j = +∞.
If p ≥ P , then by (3.4), one has The first term of the sum is a power series, so its convergence or divergence properties are easy to study. So, let us start with this expression. We have lim sup Note that the exponents i l,1 b 1 + · · · + i l,J b J are pairwise distinct. Indeed since the i l,j are natural numbers and since if l = l ′ there is at least one number j such that i l,j = i l ′ ,j , it is impossible to have because this would contradict the linear independence of the values b 1 , . . . , b J ∈ H. Hence, the desired behavior will be given by the largest one (since n kp → +∞ as p → +∞) and we can write by recalling i l,1 + · · · + i l,J = P and assumption (ii) from above. Hence this first term of the sum in (3.5) cannot be bounded.
Let us now study the second term of the sum in (3.5). Since F {N } is an algebra for the Cauchy product, we know that G ∈ F {N } . So there exist h, C > 0 such that Using assumption (iii), we obtain if p ≥ P and a < 1 h . The conclusion follows.
We wish to mention that each algebra contained in F [N ] \ j ∞ (E 0 {M} ), hence in particular the algebra G constructed in the previous result, does not contain the identity E = 1 for the convolution * anymore. Here E j = δ j,0 and clearly E = j ∞ (1) with 1 : x → 1 for all x ∈ R. Similarly this will be the case for the weight matrix and weight function case below as well.
3.2. The general weight matrix case. The aim of this subsection is to establish an equivalent of Theorem 3.2 in the more general setting supplied by weight matrices. First we recall the definitions given in [13,Section 4.1], see also the literature citations therein.
We introduce classes of ultradifferentiable function of Roumieu type E {M} and of Beurling type E (M) as follows (only the pointwise order in Def. 3.3 is required), see [28,Section 7] and [23,Section 4.2]. and For a compact set K ⊆ R, one has the representations and so for U ⊆ R non-empty open Similarly we get for the Beurling case Consequently, since the sequences of M are pointwise ordered, Finally, as done in the case of weight sequences, we introduce the corresponding spaces of weighted power series sequences, and we endow them with their classical topology: Using notations similar as before, the Borel map j ∞ is defined in the weight matrix case by

Given a quasianalytic weight matrix M, both classes E {M} and E (M) and all classes E {M
Let us now prove the generalization of Theorem 3.2 for the matrix setting. The idea of the proof is based on the following lemma, which allows to reduce the general case of two weight matrices N and M to the case of a weight matrix N and a single weight sequence M (analogously as done in [13, Section 4.2]). The general result can be stated as follows.
Proof. Using Lemma 3.7, we can consider a quasianalytic weight sequence L such that The Roumieu case is a consequence of Theorem 3.2: indeed, it suffices to fix a weight sequence N (λ0) ∈ N and use the obvious inclusion For the Beurling case, we will follow the proof of Theorem 3.2. First, by induction we can construct an increasing sequence (k p ) p∈N of natural numbers satisfying: (i) k 0 = 1 and k p > pk p−1 for every p ∈ N >0 , It is straightforward to check that F b ∈ F (N ) for any b ∈ H. We follow then the lines of the proof of Theorem 3.2 where (3.3) turns into as soon as p ≥ P l . We consider again the splitting (3.5) and proceed for the first term as in Theorem 3.2. Concerning the estimation of the second term of the sum in (3.5), since G ∈ F {N } there exist an index λ 0 > 0 and h, C > 0 such that It follows that if p ≥ max{P, λ 0 } and a < 1 h , and using assumption (iii). This concludes the proof.
3.3. The weight function case. In this section we will study classes of ultradifferentiable functions defined using weight functions ω in the sense of Braun, Meise and Taylor, see [10]. As done in [24] and [13], we will see that this case can be reduced to the weight matrix situation by using the matrix associated with ω. First, let us start by recalling the basic definitions.
Classical additional conditions can be imposed on the considered weight functions. More precisely, let us define the following conditions: ( For convenience, we define the set Note that (ω 2 ) is sometimes also considered as a general assumption on ω (e.g. see [24,Sect. 4.1]) and note also that (ω 5 ) implies (ω 2 ).
For ω ∈ W, we define the Legendre-Fenchel-Young-conjugate of ϕ ω by , and the ω-ultradifferentiable Beurling type class by .
Analogously as in the sections above, we also consider the spaces of germs at 0, denoted by E 0 {ω} and E 0 (ω) , and the associated spaces of weighted power series F {ω} and F (ω) . Again, we endow these spaces with their natural topology: F {ω} is an (LB)-space and F (ω) a Fréchet space. In this setting, the Borel map is given by As pointed out in [24,Section 4.2], that to ensure C ω E {ω} resp. C ω E (ω) , one has to assume that which follows from the characterizations given in [23,Lemma 5.16,Cor. 5.17] and the fact that the weight ω(t) = t (up to equivalence) defines the class C ω . Moreover, in the present setting, the definition of quasianalyticity takes the following form.
Definition 3.11. A weight function is called quasianalytic if it satisfies In [28] and [23, Section 5], a matrix Ω := W (λ) = (W (λ) j ) j∈N : λ > 0 has been associated with each ω ∈ W: This matrix is defined by and holds as locally convex vector spaces. Moreover, the following results have been obtained (for which (ω 1 ) is not needed necessarily): (i) Each W (λ) satisfies the basic assumptions (I) and (II) and lim j→+∞ (W ∈ Ω is a weight sequence according to the requirements from Definition 2.4, provided ω ∈ W has (ω 2 ). Moreover, by [23,Corollary 5.8] and [29,Corollary 4.8], one has that the following assertions are equivalent (again (ω 1 ) is not needed but then Proposition 2] (and in the same spirit as in [23, Section 5]), for any ω ∈ W one gets F [ω] = F [Ω] as locally convex spaces, too. Since each W (λ) satisfies (3.1) and the sequences W (λ) are pointwise ordered, as already commented in the general weight matrix case above, by following the proof of Lemma 3.1 it is immediate to see that for any ω ∈ W the sets F {ω} and F (ω) are always rings w.r.t. the convolution product.
The weight function approach is again reduced to the more general weight matrix setting by using the weight matrices N = Ω and M = Σ associated with ω and σ and Theorem 3.8 turns into the following form.
Theorem 3.12. Let σ, ω ∈ W be two quasianalytic weight functions. Assume that ω satisfies (ω 2 ) and lim inf t→+∞ ω(t) t = 0 in the Roumieu resp. (ω 5 ) in the Beurling case. Then and F [ω] , one can also treat the pointwise product, in the literature also known under Hadamard product: On the one hand, the study of the problem of algebrability with respect to this product might be a quite natural question. Moreover this product has become important very recently by the development of a convenient theory of multisummability of formal power series, see [15,Chapter 4] and [16]. Concerning these recent insights, in a private communication Prof. J. Sanz has told the authors the following explanations.
Remark 4.1. The natural procedure for assigning a sum to a summable series (in a one step procedure) precisely starts by termwise dividing the coefficients of the series by a moment sequence (equivalent to the weight sequence defining the level) to make the new series (the formal Borel transform) convergent. Correspondingly, the formal Laplace transform multiplies coefficients by the weight sequence. Moreover, sometimes series are not summable but multisummable, i.e. a sum is assigned to them after a finite number of summability procedures, each associated to a different (that is, associated to nonequivalent weight sequences) level, and then one needs to move from one level to another one, which means one has to termwise multiply or divide the coefficients of a given series by a sequence which measures the "jump" between two different levels.
Consequently, when working within the framework of weight matrices, one can control these movements/jumps in the sense that one can stay within a given matrix M by multiplying pointwise one sequence M 1 ∈ M by another one M 2 ∈ M; and for this behavior closedness under ⊙ of F [M] becomes interesting and crucial.
But the study of ⊙ has also been motivated by the following approach (cf. [24], [13] It is not difficult to see that such a function θ M does not belong to the Beurling type class associated to M . On the opposite direction, [24,Thm. 2] and its proof show that if the derivatives of a smooth function f at 0 have "large size" and all have the same sign, then f cannot belong to any quasianalytic germ class E 0 {M} . More precisely we have: These two results lead to the following observation: If M is a quasianalytic weight sequence such Conclusion: Multiplying a given F ∈ j ∞ (E 0 {M} ) pointwise by a formal power series S given in terms of a sequence of suitable complex numbers on the unit circle, and so . Connected to this observation is the notion of solid sub-and superspaces for spaces of (complex) sequences, e.g. see [1]. Let A be a vector spaces of sequences, then A is said to be solid if (a j ) j ∈ A does imply (b j ) j ∈ A for all sequences satisfying |b j | ≤ |a j |, ∀j ∈ N.
In [1, Lemma 2] it has been shown that for any given sequence space A there does exist s(A), the largest solid subspace (or solid core) of A, and there does exist S(A), the smallest solid superspace (or solid hull), of A. We have e.g. see [9, p. 594]. In our context, the two following results will show that this notion of solidness is not helping answering the question which F does belong to the image of j ∞ or not (again by identifying a formal weighted power series F = +∞ j=0 F j x j by its sequence of coefficients (F j ) j ). In particular, we see that the image j ∞ (E 0 [M] ) of the Borel map is solid if and only if the Borel map is surjective. Hence if F ∈ F (M) , then F ∈ F {L} for some L ∈ LC with L⊳M . The Roumieu part shows F ∈ S(j ∞ (E 0 {L} )) and so, by L⊳M , also F ∈ S(j ∞ (E 0 (M) )) follows because A ⊆ B implies S(A) ⊆ S(B). The conclusion follows.
Concerning the solid core, we have the following result.
{M} )) and the Roumieu case allows to conclude.
Let us mention that using unions and intersections, the two previous results easily generalize to the case of weight matrices (and so to weight functions by using the associated weight matrix).

4.2.
Characterization of the closedness under the pointwise product. The aim is now to characterize, as a first step, the closedness of F [M] and F [M] under ⊙ defined in (4.1). For the weight function case F [ω] we need some more preparation and we will study this situation in Section 5 below in detail.
First we observe that, if M ∈ R N >0 , then one clearly has that F [M] is a ring under ⊙ provided that M has which is also equivalent to sup j∈N>0 (m j ) 1/j < +∞ (i.e. M (j!) j∈N ).
In the general weight matrix setting we consider the following generalizations of (4.4): In the Roumieu case we require j , and in the Beurling case

It is immediate to see that (4.5) is preserved under {≈} and (4.6) under (≈).
In this situation we can estimate as follows for all j ∈ N: j , by taking λ 3 := max{λ 1 , λ 2 }. This shows the Roumieu case, the Beurling case holds true analogously. So these conditions are sufficient to have closedness under the pointwise product. We will show now that under mild additional assumptions on M, (4.5) and (4.6) are also necessary for the particular case (and thus in the single weight sequence case (4.4)).
The proof of the stability of F {M} under the pointwise product will use the following classical result, see [21, Chapitre I] and [20,Proposition 3.2]. Note that it allows also to construct the log-convex minorant of a sequence. , ∀ j ∈ N.
We can now state and prove the result of stability under the pointwise product. Beurling case. We follow the ideas from [11, Section 2] and [23, Proposition 4.6 (1)]. We set Note that both F 2 (M) and F (M) are Fréchet space spaces under the canonical projective topology over all h = h −1 1 and λ = λ −1 1 , h 1 , λ 1 ∈ N >0 . By assumption F (M) is closed under the pointwise product which amounts to F (M) ⊆ F 2 (M) . The closed graph theorem implies that this last inclusion is continuous. Consequently, for each λ > 0 and h > 0, there exist κ > 0 and C, h 1 > 0 such that .
For every s ≥ 0, let us consider the function f s (t) := sin(st) + cos(st), t ∈ R, and let us show that . Indeed, if s > 0 (the case s = 0 is obvious), note that |f ) for all s ≥ 0, where the associated function is defined in Proposition 4.6. Using (4.7) we get for all j ∈ N: for all j ∈ N follows. Using the log-convexity of M (κ) , one knows that the sequence (M (κ) 2j for all j ∈ N. This finally yields and so (4.6) follows. Instead of (4.5) resp. (4.6), it would have been natural to assume on M also the following assumptions: Note that (4.9) ⇒ (4.5) resp. (4.10) ⇒ (4.6) whereas the equivalences will fail in general, see also the example in Section 4.3 below.

4.3.
Example of a quasianalytic weight matrix. In contrast to the single weight sequence case we will construct now an example which shows that (4.5) and/or (4.6) can even hold true for quasianalytic weight matrices M satisfying C ω E [M] , i.e. for M having (3.6). So this weight matrix satisfies the requirements of Theorem 3.8 and hence it illustrates that in the general matrix setting an equivalent of Theorem 3.8 using the pointwise product makes sense, see Theorem 4.9 below.
Note that M violates both (4.9) and (4.10). Indeed, for all j > j 0 we have But this cannot hold true for all j ∈ N for any given numbers C and h large, since, by Stirling's formula, the left-hand side is increasing like j → j e j √ 2πj, whereas the right-hand side is bounded by above by j 0 !Ch j log(log(j)) j(κ−2λ) .
It shall be noted that, by the characterization shown in Proposition 4.7, we have stability under ⊙ for both F {M} and F (M) . However, even in this situation it is still impossible to obtain closedness under ⊙ for j ∞ (E 0 {M} ): Take θ M (λ 0 ) for some λ 0 > 0 and put F := j ∞ (θ M (λ 0 ) ). Then clearly for any quasianalytic weight sequence L (see Proposition 4.3) and so in particular this holds true for the sequence L coming from Lemma 3.7.
We close this section with the following observation: Not for all (quasianalytic) weight matrices the characterizing conditions (4.5) and (4.6) are satisfied simultaneously.
For this we consider N := {(j!) j , M (λ0) } with M (λ0) denoting one of the sequences belonging to the matrix M constructed above. So N is a weight matrix consisting only of two non-equivalent (quasianalytic) weight sequences and so F (N ) = F ((j!)j ) , F {N } = F {M (λ 0 ) } . Then (4.6), which amounts to (4.4) for (j!) j holds true, whereas (4.5) for N , i.e. (4.4) for M (λ0) , fails. Note that j! ≤ M (λ0) j only holds true for all j ∈ N large, but M (λ0) can be replaced by an equivalent sequence satisfying this pointwise estimate for all j ∈ N (as required in Definition 3.3) and defining the same matrix.

4.4.
Algebrability for the general matrix setting. As seen by the example constructed in Section 4.3, in the general weight matrix setting it makes also sense to consider on F [M] the pointwise product. We show the following result analogous to Theorem 3.8 for the convolution product but the proof will simplify at several steps due to the fact that multiplying two lacunary series w.r.t. ⊙ does not change and mix the indices j ∈ N with F j = 0.
Proof. As in the proof of Theorem 3.8, one can use Lemma 3.7 to reduce the proof to the case of a quasianalytic weight sequence L instead of M. By assumption, one can construct an increasing sequence (k p ) p∈N of natural numbers satisfying (i) k 0 = 1 and k p > k p−1 for every p ∈ N >0 , (ii) lim p→+∞ n (1/(p+1)) kp 1 kp = +∞, j ≤ 1 for every p ∈ N >0 . We proceed then exactly as in the proof of Theorem 3.8 to construct formal power series F b , b ∈ H, and we remark that if To conclude, one follows the same ideas as in the proofs of Theorem 3.2 and Theorem 3.8.
The identity for ⊙ is given by E ⊙ = +∞ j=0 1x j and so E = j ∞ (f ) with f (x) := +∞ j=0 x j representing a real analytic germ at 0. Consequently also in this setting each algebra contained in F [N ] \j ∞ (E 0 {M} ) does not contain the identity E ⊙ anymore.

On the stability under the pointwise product of F [ω]
The goal of this Section is to show that, similarly as commented in Remark 4.8 for the single weight sequence situation, the problem of algebrability with respect to ⊙ cannot be considered for F [ω] within the quasianalytic setting. More precisely we will show that all required assumptions on ω can never be satisfied simultaneously. While in the weight function case we can have the situation that F [Ω] = F [ω] is closed under the pointwise product ⊙ and E [Ω] = E [ω] is strictly containing the real analytic functions, we will see below that this situation forces already non-quasianalyticity for ω. Consequently the matrix constructed in Section 4.3 above cannot be associated with a weight function ω.
In order to do so first recall that, as shown in Lemma 4.7 above, (4.5) resp. (4.6) are characterizing the closednees under the pointwise product for F {Ω} = F {ω} resp. F (Ω) = F (ω) . Hence we have to show which condition on ω guarantees that Ω satisfies (4.5) resp. (4.6) and for this we have to introduce some notation and recall several results.
Proof. First, let us assume that Ω satisfies (4.5) and/or (4.6) with indices λ and κ. We will prove here the Roumieu case, the Beurling case can be treated in a similar way. If we put W (λ) : j . Hence for all t ≥ 0 and j ∈ N we , and applying logarithm to this inequality yields From [19,Lemma 3.4 (ii), (3.6)] applied to Q = M = W (κ) (recall that W (κ) ∈ Ω), we know that The second inequality of (5.3) yields By using the first inequality of (5.1) we see for all t ≥ 0 that 2ω W (λ) ( √ het) ≤ 2 λ ω( √ het) and the second inequality of (5 . Thus, combining everything, we have shown for all t (large enough) that Conversely, assume now that (5.2) holds true with constants C > 0 and H > 0. First, let in the following computations λ, κ > 0 be arbitrary but fixed. The second inequality of (5.1) yields , and altogether Now take κ = Cλ and with this choice, by using Proposition 4.6, we can estimate as follows for all for all j ∈ N. This proves both (4.5) and (4.6) since Cλ = κ and C is only depending on given ω. The characterizing property (5.2) is looking similar to the following growth property on ω, see [28], [14], [23,Theorem 5.14 (4)] (called (ω 8 ) in there) and [17, Appendix A] (denoted by (ω 7 ) there): In [17,Lemma A.1] it has been shown that for any ω ∈ W with (5.4) the associated matrix Ω does have both (4.9) and (4.10) (by having a precise relation between the indices λ and κ). Following the proof of [17, Lemma A.1 (ii) ⇒ (i)] and replacing Al by l 1 there it is straightforward to see that (4.9) and/or (4.10) are implying (5.4), see [28,Lemma 5.4.1] and also the first half of the proof of Theorem 5.1 (in fact for this implication one only needs that the inequalities in (4.9) or (4.10) are valid for some pair of indices λ and κ).
However, (5.4) implies quite strong, and in our situation undesired, properties for the associated weight matrix Ω. More precisely, by the results shown in [17, Appendix A] we have that for any ω satisfying (ω 0 ), (ω 3 ) and (ω 4 ) property (5.4) does imply the strong non-quasianalyticity condition for weight functions and so in particular (3.8) has to fail. By the results shown in [5] (see also [8]) it follows that for given ω ∈ W condition (5.5) is char- , i.e. the surjectivity of the Borel mapping. Note that in [5] and [8] non-quasianalyticity for ω was a basic assumption but which is superfluous provided (characterized by (3.7)): On the one hand it is clear that (5.5) forces non-quasianalyticity for ω.
We are gathering now some more observations.
Note that (5.6) is clearly stable under relation ≈.
Proof. Let ω M satisfy (5.2) and w.l.o.g. we can assume C ∈ N ≥1 . We follow the ideas in the proof of [19, Lemma 3.4 (i)] (for M instead of m). First, for all j ∈ N, we get .
The supremum in the last expression yields By studying for every j ∈ N and s > 0 fixed the function f j,s (t) := j log(t) − 1 C ω M (1/s) − st C , t > 0, one gets that its supremum is given by log (jC) j (es) j − 1 C ω M (1/s) (if j = 0 we use the convention 0 0 := 1). Using this we can continue the above estimation for all j ∈ N as follows: Summarizing everything we have shown so far that there exist some C 1 , h 1 > 0 such that for all j ∈ N we get (M j ) 2 ≤ M 2j ≤ C 1 h j 1 j!(M Cj ) 1/C (using for the first estimate that the log-convexity for M implies that (M Since by Stirling's formula (Cj)! is growing like j! C up to a factor with exponential growth, we obtain (m j ) 2C ≤ C 3 h j 3 m Cj for all j ∈ N and for some constants C 3 , h 3 not depending on j, thus (5.6) is verified.
Conversely, assume that (5.6) is valid. By going back in the equivalences above, we get (M j ) 2 ≤ D 1 h j 1 j!(M Cj ) 1/C for all j ∈ N.
If Ω := {W (λ) : λ > 0} denotes the matrix associated with ω M , then it is known and straightforward to verify that M ≡ W (1) (e.g. see the proof of [29,Thm. 6.4]) and moreover W for all j ∈ N. Then follow the first part in the proof of Theorem 5.1 with λ = 1 and κ = C in order to conclude. By combining now Proposition 4.7, (5.1), Theorem 5.1, (iii) in the previous observations and Proposition 5.2 we get the following result. On the other hand, starting with a weight sequence satisfying an additional assumption, we have the following characterization. The equivalence (i) ⇔ (ii) can also be seen directly as follows: On the one hand, (i) ⇒ (ii) holds by having (m j ) 2 ≤ Ab j m j and so take C = 1 in (5.6). Conversely, by assumption M has (mg), i.e. M j+k ≤ A j+k 0 M j M k for all j, k ∈ N and some constant A 0 . Consequently, m j+k ≤ A j+k 1 m j m k for all j, k ∈ N and some constant A 1 . By (5.6), we have (m j ) 2C ≤ Dh j m Cj and by iteration of m j+k ≤ A j+k 1 m j m k , we get Dh j m Cj ≤ Bb C 2 j (m j ) C and so (m j ) 2 ≤ B 1/C b Cj m j for some constants b, B > 0 which is precisely (4.4).
The next result establishes a connection between (5.6) and the non-quasianalyticity of a sequence M .
By Stirling's formula j! 2/j (Cj)! 1/(Cj) is asymptotically growing like j → D 1 j and so M has (5.6) if and only if Note that the assumption sup j∈N>0 (m j ) 1/j = +∞ implies that in (5.7) we have C ≥ 2: indeed, the case C = 1 would yield (4.4) and so sup j∈N>0 (m j ) 1/j < +∞, hence a contradiction. Since we have sup j∈N>0 (m j ) 1/j = +∞, for all A ≥ 1 there does exist a number q A ∈ N ≥1 (which can be chosen minimal) such that we get (m qA ) 1/qA ≥ A, or equivalently (M qA ) 1/qA ≥ A(q A !) 1/qA . Thus, by a consequence of Stirling's formula, we obtain (M qA ) 1/qA ≥ AqA e and so also eCC1 A ≥ CC1qA (Mq A ) 1/q A follows with C and C 1 denoting the constants arising in (5.7) (which are not depending on given q A ). Let now A ≥ 1 be chosen sufficiently large in order to have eCC1 A < 1 and set q := q A . By the above we see that CC1q (Mq) 1/q < 1 holds true. Since M ∈ LC we have that j → (M j ) 1/j is increasing. As we will see this property is sufficient to conclude and for convenience we put now L j := (M j ) 1/j . For the sum under consideration we estimate by
Using the above Lemma we can prove now the final statement of this section showing that the problem of algebrability cannot be considered within the quasianalytic weight function setting.
Proof. Let Ω = {W (λ) : λ > 0} be the matrix associated with ω. We apply Lemma 5.5 to some/each sequence W (λ) which can be done by the assumptions on ω and the equivalences obtained in Corollary 5.3 above. Then W (λ) has (nq) and so ω does not enjoy (3.8) (recall that this last step holds by [20, Lemma 4.1] and (5.1)).
Note that this result deals with a property of the associated matrix Ω and (ω 1 ) is not required necessarily. If ω has in addition (ω 1 ), then we have F [ω] = F [Ω] in Theorem 5.6.
Acknowledgements. The authors wish to thank the referee for his comments which have improved the presentation and the structure of this work. The authors also wish to thank Javier Jiménez-Garrido and Javier Sanz from the Universidad de Valladolid for their helping discussions concerning the results and proofs of Section 5.