On the Inclusion Relations of Global Ultradifferentiable Classes Defined by Weight Matrices

We study and characterize the inclusion relations of global classes in the general weight matrix framework in terms of growth relations for the defining weight matrices. We consider the Roumieu and Beurling cases, and as a particular case, we also treat the classical weight function and weight sequence cases. Moreover, we construct a weight sequence which is oscillating around any weight sequence which satisfies some minimal conditions and, in particular, around the critical weight sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p!)^{1/2}$$\end{document}(p!)1/2, related with the non-triviality of the classes. Finally, we also obtain comparison results both on classes defined by weight functions that can be defined by weight sequences and conversely.


Introduction
We deal with global classes of ultradifferentiable functions defined by weight matrices and study and characterize different inclusion relations between these classes.There are basically two ways to introduce the classes of ultradifferentiable functions, the point of view of Komatsu [14], based in previous ideas of Carleman, which pays attention to the growth of the derivatives on compact sets modulated by a sequence (M p ) p of positive numbers, or the point of view of Björck [2], based in ideas of Beurling [1], who used a weight function to estimate the growth of the Fourier transform of compactly supported functions.Braun, Meise and Taylor [7] unified these points of view by introducing weight functions which allow the use of convex analysis techniques.In terms of their topological structure, the classes are of two types, of Beurling type, which are classes whose topology looks like the topology of the space of all smooth functions, and of Roumieu type, whose topology looks like that of the space of real-analytic functions.
More recently, Rainer and Schindl [18] introduced weight matrices to study when the spaces of ultradifferentiable functions are closed under composition treating at the same time the classes in the sense of Komatsu (estimates of the derivatives with a sequence) and in the sense of Braun, Meise and Taylor (estimates of the derivatives via a weight function).They also studied intersections and inclusion relations of the classes in the local sense (i.e. when the estimates are given over the compact sets of a given open set).Since then several papers using weight matrices have been published.We mention, for instance, [4,9,10,13], and the references therein.However, the characterization of inclusion relations in global classes of ultradifferentiable functions, i.e. classes where the estimates on the derivatives are taken in the whole of R d , has not been investigated yet.In this paper (Section 4) we characterize the inclusion relations of global classes defined by weight matrices using the isomorphisms introduced in [4, Section 5].Moreover, given a weight sequence we construct an oscillating weight sequence around the first one to have examples of the opposite situation of the inclusion relations.In particular, in Section 3 we construct an oscillating weight sequence around the sequence (p!) 1/2 , which is very related to the non-triviality of the corresponding ultradifferentiable class (see Remark 3.3).We begin with some notation (Section 2) and continue in Section 4 with the weight function case and the more general weighted matrix case.In Section 5 we compare the classes when defined by weight functions and sequences in the spirit of [6].In Section 6 we give alternative assumptions (non-quasianalyticity), and accordingly different techiques of demonstration, for the inclusion relations.And finally, as an appendix we analyze the extreme case of the weight function t → log t which yields the Schwartz class in our setting.

Notation
2.1.Weight sequences.Let M = (M p ) p∈N 0 be a sequence of positive real numbers where we have set N 0 := N ∪ {0}.A sequence (M p ) p is called normalized if M 1 ≥ M 0 = 1 (without loss of generality).We say that M satisfies the logarithmic convexity condition, i.e. assumption (M1) of [14], if This is equivalent to the fact that the sequence of quotients µ p := Mp M p−1 , p ∈ N, is nondecreasing and we set µ 0 := 1.If M is normalized and log-convex, then (2.2) ∀ p, q ∈ N 0 : M p M q ≤ M p+q ; see e.g.[20,Lemma 2.0.6].Moreover, in this case, M is nondecreasing because µ p ≥ µ 1 ≥ 1 for all p ∈ N.
We say that M satisfies derivation closedness, i.e. condition (M2) ′ of [14], if and M satisfies the stronger condition of moderate growth, i.e. condition (M2) of [14], if For convenience we set where R N >0 denotes the set of strictly positive sequences.
Given two (normalized) sequences M and N we write M N, if If M N and N M, then we write M ≈ N and say that the sequences M and N are equivalent.Moreover we write M ⊳ N if Next we recall [14,Lemmas 3.8,3.10]transferring these growth relations to the associated functions: Given M, N ∈ LC we have = 0.However, the notion of logarithmic convexity is quite delicate in the anisotropic case, and we refer to [5] for more details.
We call M a weight matrix and we say that it is constant if M (λ) ≈ M (κ) for all λ, κ > 0. In the one-dimensional case, we call M standard log-convex if M (λ) ∈ LC for any λ > 0.
We write Denoting by • ∞ the supremum norm, for a normalized weight sequence M we consider the following spaces of weighted rapidly decreasing global ultradifferentiable functions endowed with the inductive limit topology in the Roumieu setting (which may be thought countable if we take h ∈ N) and with the projective limit topology in the Beurling setting (countable for h −1 ∈ N).Next, the matrix type spaces are defined as follows: again endowed with the inductive limit topology in the Roumieu setting (which may be thought countable if we take λ, h ∈ N) and endowed with the projective limit topology in the Beurling setting (countable for λ −1 , h −1 ∈ N).
We denote by E {M} , E (M) , E {M} , E (M) the analogous (local) classes of ultradifferentiable functions replacing x α ∂ β f ∞ by the supremum of ∂ β f on compact sets (and then take the projective limit over compact sets).Moreover D {M} , D (M) , D {M} , D (M) denote the corresponding classes of ultradifferentiable functions with compact support.We refer to [6] and [18] for precise definitions of such classes.
We collect here some conditions, already introduced in [4] and motivated by the assumptions in [15].In the Roumieu case we consider α+β , (2.9) β , (2.11) and in the Beurling case We summarize now some consequences for a given weight matrix M as defined in (2.7): (i) By [ We have an analogous statement for the class S (M) (R d ) when assuming (2.13) and (2.15).When we define the spaces S {M} (R d ) or S (M) (R d ) with the weighted L 2 norms, the similar property holds.From now on we make use of the following conventions: when the spaces are defined in terms of • ∞,M (λ) ,h we will always write "joint growth at infinity"; when they are defined in terms of the seminorms from (2.16), then we will write "separated growth at infinity"; frequently we omit writing the set R d when the context is clear.
We call ω a general weight function, if ω satisfies all listed properties except (β).
It is not restrictive to assume ω| [0,1] ≡ 0 (normalization).As usual, we define the Young conjugate ϕ * ω of ϕ ω by ϕ * ω (s) := sup which is an increasing convex function such that ϕ * * ω = ϕ ω and s → ϕ * ω (s) s is increasing.Condition (γ) guarantees that ϕ * ω is finite (see Appendix A).We introduce the following growth relations between two (general) weight functions arising naturally in the ultradifferentiable framework: We write If ω σ and σ ω are valid, then we write ω ∼ σ and call the weights equivalent.
For any given (general) weight function we set and consider the weight matrix We recall now the following result, which was proved in [4, Lemma 11] for a weight function, but which can be stated for a general weight function, since assumption (β) was not needed in the proof.Lemma 2.2.Let ω be a (general) weight function.Then M ω satisfies the following properties: (i) The spaces of rapidly decreasing ω-ultradifferentiable functions are then defined as follows: In the Roumieu case and in the Beurling case From (iv), (vii) in Lemma 2.2 we have that equivalently the classes can be treated by separated growth at infinity, i.e.
We can also insert h |α+β| at the denominator (for some h > 0 in the Roumieu case and for all h > 0 in the Beurling case) by (v) in Lemma 2.2.In particular, we finally recall from [4, Proposition 5] (where again assumption (β) was not necessary) that, analogously as in the ultradifferentiable setting, we can use the associated weight function in order to have an alternative and useful description of the classes defined by weight matrices: Proposition 2.3.Let ω be a (general) weight function and M ω be the weight matrix defined in (2.17), (2.18).We have and and both equalities are also topological.
We refer to [3] for a more complete characterization of such spaces, and to [7] for the analogous spaces E {ω} /D {ω} and E (ω) /D (ω) of ultradifferentiable functions/with compact support.

Oscillating sequences and a critical example case
The aim of this section is to construct explicitly a weight sequence M which oscillates around a given fixed sequence N ∈ LC.We assume for N some more basic growth properties and show that these requirements can be transferred to M, too.Moreover, these conditions yield the fact that the function ω ≡ ω M is also oscillating around the weight ω N .Since by construction M ∈ LC we focus on the one-dimensional situation (or, equivalently, on the isotropic case, i.e.M α := M |α| ).As a special case we apply this to N ≡ G 1/2 := (p! 1/2 ) p∈N and the corresponding weight function ω(t) = t 2 .This is a crucial case since it is related to the problem of non-triviality of S (ω) and S {ω} (see Remark 3.3).We start the construction as follows.Let from now on (α j ) j≥0 be a sequence of positive real numbers such that Moreover let Q ∈ N, Q ≥ 2, be given, arbitrary but fixed.We introduce a new sequence (β j ) j≥1 by (3.2) Finally M is defined via the quotients (µ j ) j≥1 as follows: We put Using this we have ), then we can estimate as before replacing α n and α n+1 by α 0 and α 1 respectively.The estimate from above in the claim is obtained analogously when taking α max instead of α min .
Claim II: M ∈ LC holds.Since by definition and assumption min which tends to infinity as n → ∞ because α min > 1 by assumption.This proves lim j→∞ µ j = +∞, hence lim j→∞ (M j ) 1/j = +∞ follows (e.g.see [18, p. 104]).Now we start with the definition of M in terms of the aforementioned construction by using the auxiliary sequences (β j ) j resp.(α j ) j .We put so M 1 = M 0 = 1 follows which ensures normalization.The idea is now to define M (via (α j ) j ) piece-wise by considering an increasing sequence (of integers) (k j ) j≥1 .Given a sequence N ∈ LC we consider the sequence of its quotients and, moreover, we assume that N satisfies It is immediate that Q ≥ 2 in the above requirement and (3.5) is crucial to ensure that ω M satisfies (α) (see (III) below) and that M has moderate growth (see (IV ) below).Let now Q be the parameter according to (3.5) and without loss of generality Then we have and for 1 < i < k 1 we have put In the next step we select a number n 1 ∈ N, n 1 ≥ 2, and put Here we choose n 1 sufficiently large in order to ensure ν and get Hence, by (3.3) and (3.4) we get and so one has Hence we get For k 2 < i < k 3 , again according to (3.3) and (3.4), we have put And then we proceed as follows: Case I -from odd to even numbers.Given any k j with j ≥ 3 odd, then we select n j ∈ N, Case II -from even to odd numbers.Given any k j with j ≥ 4 even, then we select n j ∈ N, With these choices, first for all odd j ≥ 3 (starting with the case j = 3 from above) one has and so On the other hand, for all even j ≥ 4 we see and so Moreover, recall that and for all k j < i < k j+1 , according to (3.3) and (3.4), we have set Claim III: (3.5) implies (3.1).First, we treat the upper estimates and note that for Case I we have (2 j (j + 1)) and this is valid because by the second part of (3.5) we have ν 2j ν j ≤ B for some B ≥ 1 and all j ∈ N 0 and so, iterating this estimate cn j -times with c ∈ N such that Q ≤ 2 c , we have Here the first estimate holds by the log-convexity of N.
For Case II with the expression (2 j+1 j) 1 n j we get the bound (2 j+1 j) 1 n j A by the previous comments.And this can be bounded uniformly for all even j ≥ 4 by some A 1 > A when choosing n j large enough.Therefore note that A is not depending on the choice of n j ; it only depends on given (fixed) constants Q and B, both depending only on N via (3.5).Summarizing, the upper estimate in (3.1) is verified for all j ∈ N since the remaining cases are only finitely many indices.Now we treat the lower estimate.By the first part in (3.5) we have that there exists some ǫ > 0 such that (by iteration) provided that k j is chosen sufficiently large.We assume now that we have chosen k 3 large enough (for a fixed ǫ > 0) and so the above estimate holds for all j ≥ 3. Now, concerning Case I for all odd j ≥ 3 we estimate by (2 j (j + 1)) and the last estimate is equivalent to requiring (1 + ǫ) n j > 2 j (j + 1).This can be achieved by choosing n j , j ≥ 3 odd, sufficiently large.
Concerning Case II we observe (2 j+1 j) To guarantee α j > 1 for all j ∈ N, i.e. also for 1 ≤ j ≤ n 1 + n 2 , we recall our choice for n 1 above.Summarizing, we get: (I) M ∈ LC: Normalization is obtained as seen above, log-convexity holds by the fact that α j > 1 for all j ∈ N and so the sequence j → µ j is (strictly) increasing.Since for all odd j ≥ 3 by construction we get µ k j ≥ 2 j+1 ν k j , we see that lim j→∞ µ j = +∞ (since ν k j is nondecreasing by the logarithmic convexity), and so also lim j→∞ (M j )  (λ) : λ > 0} be the matrix associated to ω M .By [18,Lemma 5.9] and (3.6) it follows that M (λ) ≈ M (κ) , i.e.M ω M is constant.In this case we get M ≡ M (1) by definition of M (1) and [14, Prop.3.2] (see also the proof of [21, Thm.6.4]): Note that by normalization we have ω M (t) = 0 for 0 ≤ t ≤ 1 which follows by the known integral representation formula for ω M , see [16, 1.8.III] and also [14, (3.11)], and since t p ≤ 1 for 0 ≤ t ≤ 1, p ∈ N 0 arbitrary.Consequently, M (λ) ≈ M for all λ > 0 and so as topological vector spaces (and the spaces can be defined equivalently by joint or separated growth at infinity).(V I) By construction we have µ k j = 2 j ν k j for all j ≥ 3 odd and µ k j = 1 j ν k j for all j ≥ 4 even.Thus lim inf p→∞ µ p ν p = 0 and lim sup p→∞ In Hence, M and N are not comparable, which means that neither M N nor N M hold (consequently, neither M ⊳ N nor N ⊳ M, too).
3.2.The critical example case.Now we treat a particular case.Namely, when the weight sequence N is the critical sequence G 1/2 := (p! 1/2 ) p∈N ∈ LC, and hence we look for a weight function ω ≡ ω M which oscillates around the critical weight function ω(t) = t 2 , related to the problem of nontriviality of S (ω) and S {ω} .First, we summarize some known facts: (i) The associated weight function which follows from [18, Lemma 5.9, Theorem 5.14] applied to the weight ω ≡ ω G 1/2 .For this note that G 1/2 satisfies (2.4) and so equivalently (see [14,Prop. 3.6]) the function . Moreover, we can define these classes equivalently by using separate or joint growth at infinity which holds by (2.2) (implied by (2.1)) and (2.4).
(iv) However, note that formally [6, Theorem 14] cannot be applied directly to G 1/2 in order to conclude (ii) and/or (iii) since their basic assumption is violated for G 1/2 .Recall that (3.7) for M means that lim inf p→∞ (M p /p!) 1/p > 0, which yields ω M (t) = O(t), see [6, Lemma 12 (4) ⇒ (5)].Note also that (3.7) is slightly stronger than our assumption lim j→∞ (M j ) 1/j = ∞.Translating into the notation of growth relations [4, Lemma 13] (whose proof did not use assumption (β)) we have the following result: Lemma 3.1.Let ω be a (general) weight function.Then Similarly, following the lines in the proof of [4, Lemma 13] we obtain: Let ω be a (general) weight function.Then and Note that these results follow also from [18, Lemma 5.16, Corollary 5.17] and we can replace in all conditions "∀ λ > 0" equivalently by "∃ λ > 0".Now, for M ≡ W (λ) we have that M and G 1/2 are not comparable which means: It follows from Lemmas 3.1 and 3.2 that neither ω M t → t 2 nor t → t 2 ω M is valid (and hence neither ω M ⊳ t → t 2 nor t → t 2 ⊳ ω M , too).
Finally, we mention that M does not satisfy the requirements of [6] because their basic assumption (3.7) is violated: since However, when t 2 = O(ω(t)) as t → ∞, then we prove now that S (ω) = {0}: First, for any f ∈ S (ω) we get ∀ λ > 0 : sup which gives, by the relation Analogously, in the Roumieu case

Characterization of the inclusion relations of global ultradifferentiable classes
In this section we characterize the inclusion relations of spaces of rapidly decreasing ultradifferentiable functions using the isomorphisms with sequence spaces obtained in [4].
Let us distinguish the various cases.
We can then prove the following: Theorem 4.1.Let ω and σ be weight functions.Then the following are equivalent: ) holds for all dimensions d ∈ N with continuous inclusion.More precisely, the spaces in (ii) can be defined by joint or separated growth at infinity; (i) ⇒ (ii) is valid for general weight functions ω and σ and for (ii) ⇒ (i) only the inclusion for d = 1 is required.
, by the iteration of property (α) for ω.The number n of iterations is only depending on given l, more precisely we choose n ∈ N minimal to guarantee l ≤ 2 n .Note that all arising constants C, C 1 ≥ 1 and l ∈ N are not depending on k (hence not on h).
Finally let t ∈ R with for some k ∈ N, then for h := with L ≥ 1 denoting the constant arising in (α) for the weight ω.More precisely, this condition implies ω( All arising constants are only depending on the given weights ω and σ and not on t ∈ R. Thus we have shown lim sup t→∞ σ(t) ω(t) < +∞, i.e. ω σ.Now we treat the mixed case between the Roumieu-and Beurling-type classes.Theorem 4.2.Let ω and σ be weight functions with σ(t) = o(t 2 ) as t → ∞.Then the following are equivalent: ) holds for all dimensions d ∈ N with continuous inclusion.More precisely, the spaces in (ii) can be defined by joint or separated growth at infinity; (i) ⇒ (ii) is valid for general weight functions ω and σ and for (ii) ⇒ (i) only the inclusion for d = 1 is required.
Proof.(i) ⇒ (ii) follows by definition of the classes.
Finally we consider the Beurling case.
Then the following are equivalent: ) holds for all dimensions d ∈ N with continuous inclusion.More precisely, the spaces in (ii) can be defined by joint or separated growth control at infinity; (i) ⇒ (ii) is valid for general weight functions ω and σ and for (ii) ⇒ (i) only the inclusion for d = 1 is required.
Proof.(i) ⇒ (ii) follows again by definition of the classes.
(ii) ⇒ (i) By assumption and the isomorphism (4.2) we get now Λ (ω) ∼ = S (ω) (R d ) ⊆ S (σ) (R d ) ∼ = Λ (σ) with continuous inclusion.We use assumption (ii) for the case d = 1.By the continuity of the inclusion we get For i ∈ N 0 we consider the sequence c i defined by < +∞ for all i ∈ N 0 , j ∈ N. We apply (4.3) to the case j = 1 and the family c i , i ∈ N 0 , and get The iteration of property (α) for ω yields Note that all arising constants are not depending on k.
again by applying (α) similarly as in the proof of Theorem 4.1 before.Since all arising constants are not depending on t we have shown σ(t) = O(ω(t)) as t → ∞, i.e. ω σ.

4.2.
The general weight matrix case.In this case, for a weight matrix M we recall the isomorphisms proved in [4, Theorem 1].In the Roumieu case, if conditions (2.8) and (2.10) are satisfied, then Analogously in the Beurling case, if conditions (2.12) and (2.14) are satisfied, then We can thus prove similar results as in §4.1.We start with the Roumieu case.Then we get the following: (i) ⇒ (ii) is valid for all dimensions d ∈ N and the classes in (ii) can be defined by joint or separated growth at infinity.If (ii) holds for the case d = 1 and both matrices are standard log-convex with (2.8) and (2.10), then (ii) ⇒ (i) is valid, too.
Proof.Again, (i) ⇒ (ii) follows by the definition of the spaces.(ii) ⇒ (i) We use the inclusion for the dimension d = 1 and so the matrices consist only of sequences M (λ) , N (λ) ∈ LC.
By the assumptions on M and N we can apply the isomorphism (4.4) and so (ii) yields We consider the sequence c := (c k ) k∈N 0 ∈ C N defined by c k := e −ω M (j) ( √ k j ) with j ∈ N, j ≥ 2, arbitrary but from now on fixed.So c ∈ Λ {M} follows by choosing l = j in (4.4) and this yields c ∈ Λ {N } as well.Thus Here we have used that k is valid for any k ∈ N when j ≥ 2 and that each ω M (j) is increasing.Finally, if 0 < t < 1, then ω N (l) t l ≤ ω N (l) 1 l .Consequently, by enlarging the constant C if necessary, so far we have shown We use this estimate and the fact that each sequence belonging to the matrices is logconvex and normalized.Hence, by (2.6) we get for all p ∈ N 0 : and so M{ }N is verified.Note that the assumption j ≥ 2 is not restricting the generality in our considerations since we are dealing with Roumieu type spaces.
Next we treat the mixed situation between the Roumieu case and the Beurling case.Then we get the following: (i) ⇒ (ii) is valid, for all dimensions d ∈ N and the classes in (ii) can be defined by joint or separated growth at infinity.If (ii) holds for the case d = 1 and if both matrices are standard log-convex and M does have (2.8) and (2.10), whereas N is required to satisfy (2.12) and (2.14), then (ii) ⇒ (i) is valid, too.
Proof.Again, (i) ⇒ (ii) follows by the definition of the spaces.
(ii) ⇒ (i) We use this inclusion for d = 1.By the assumptions on M and N and the isomorphisms (4.4)-(4.5)we have that (ii) yields As in the previous proof we consider the sequence c := (c k ) k∈N 0 ∈ C N defined by c k := e −ω M (j) ( √ k j ) with j ∈ N, j ≥ 2, arbitrary but from now on fixed.So c ∈ Λ {M} by choosing l = j and now the assumption yields c ∈ Λ (N ) as well.Thus we obtain ) and note that the arising constant C is depending on l and j.
as in the proof of Theorem 4.4.Finally, if 0 Consequently, by enlarging the constant C if necessary, so far we have shown We use this estimate and the fact that each sequence belonging to the matrices is logconvex and normalized, hence by (2.6) we get for all p ∈ N 0 and i ≤ l: This estimate proves M (j) ⊳ N (1/i) for all i, j ∈ N, j ≥ 2: Let i and j ≥ 2 be arbitrary but fixed, then we get by the previous computations that for all l ≥ i and p ∈ N. Assumption j ≥ 2 is not restricting since the matrix M is related to Roumieu-type conditions and small indices can be omitted without changing the corresponding function class.Thus we have verified M ⊳ N .
Finally we treat the general weight matrix case in the Beurling-type setting.
Theorem 4.6.Let M := {M (λ) : λ > 0} and N := {N (λ) : λ > 0} be given and consider the following assertions: ) holds with continuous inclusion.Then we get the following: (i) ⇒ (ii) is valid for all dimensions d ∈ N and the classes in (ii) can be defined by joint or separated growth at infinity.If (ii) holds for the case d = 1 both matrices are standard log-convex with (2.12) and (2.14), then (ii) ⇒ (i) is valid, too.Proof.Again, (i) ⇒ (ii) follows by the definition of the spaces.
(ii) ⇒ (i) We use this inclusion for d = 1.By the assumptions on M and N and the isomorphism (4.5) we have that (ii) yields with continuous inclusion.Now we proceed analogously as in the proof of Theorem 4.3 before.By the continuity of the inclusion we get For i ∈ N 0 we consider again the sequence c i defined by < +∞ for all i ∈ N 0 and j ∈ N. We apply (4.6) to the family c i , i ∈ N 0 , and get Then we estimate by Here we have used that . Consequently, by enlarging the constant C if necessary, so far we have shown Finally, by using this and again (2.6) we get for all p ∈ N 0 : which proves M (1/l) N (1/j) and so M( )N is verified.
) with continuous inclusion.(II) Let M and N satisfy (2.8) and (2.12) respectively, and consider the following assertions: ) with continuous inclusion.(III) Let M and N satisfy (2.12) and consider the following assertions: ) with continuous inclusion.Then all implications (i) ⇒ (ii) are valid for arbitrary sequences M, N ∈ R N >0 , for all dimensions d ∈ N and the spaces can be defined in terms of a joint or separated growth at infinity.If (ii) holds for d = 1, then the implications (ii) ⇒ (i) are valid, too.
Remark 4.8.If we consider the isotropic setting, so we set M (λ) |α| (resp.M α := M |α| ) for any λ > 0 and α ∈ N d 0 , then in all results from Section 4.2 and Section 4.3 we have that (ii) ⇒ (i) is valid if (ii) holds for some/any dimension d ∈ N. Similarly this applies to the results listed in Section 6 as well.For the analogous results in the anisotropic setting we refer to [5].

Comparison of classes defined by weight sequences and weight functions
Gathering the information from the previous section we are now able to prove the following results which are analogous to the statements obtained in [6] and [18] for the spaces E {M} , E (M) , E {ω} , E (ω) (cf.also [4,Remark 4]).
In both cases we can take M ≡ W (λ) for some/each λ > 0 in (ii).
When the above equivalence holds true the space can be defined by separate or joint growth at infinity.The analogous result holds true for the Beurling case as well when M satisfies (2.12) (instead of (2.8)) and (in addition) ω(t) = o(t 2 ) as t → ∞.
In both cases we can take the weight function ω ≡ ω M in (ii).When the above equivalence holds true the space can be defined by separate or joint growth at infinity.
Proof.Again we only treat the Roumieu case.

Alternative assumptions for the characterization of the inclusion relations
The aim of this section is to present alternative proofs for the characterizations of the inclusion relations for Gelfand-Shilov classes.More precisely, we are not using results from [4] in the classes S (with matrix/function weights), but following ideas generally used in the ultradifferentiable setting E (with matrix/function weights).In this case our assumptions are slightly different and we have to involve the property of non-quasianalyticity, which was not required in section 4. We refer to Remark 6.4 to compare the distinct assumptions of sections 4 and 6: stronger conditions on the first weight, but weaker conditions on the second weight.Another difference with respect to Section 4 is that here the Roumieu and the Beurling cases require different techniques.So let us start by the Roumieu case.
6.1.The Roumieu case.Let M ∈ LC be given.We recall (e.g.see [18, Lemma 2.9]): There exists a function θ M belonging to the ultradifferentiable class E {M} (R) and such that |θ (j) M (0)| ≥ M j for all j ∈ N. In fact θ M belongs to the global ultradifferentiable class since the estimates are valid on whole R, more precisely: In [18] such a function has been called a characteristic function.We can assume θ M to be real-or complex-valued (see the proof of [18,Lemma 2.9]).Those functions are in some sense optimal in the ultradifferentiable classes of Roumieu-type.
Please note, that θ M cannot belong to the Beurling-type class E (M) (R).Such functions are useful to characterize the inclusion relations of (global/local) ultradifferentiable function classes in terms of growth relations of weight sequences/functions or even matrices, see [18, Prop.2.12, Prop.4.6, Cor.5.17].
However, for our purposes we need that θ M ∈ S {M} (R), M ∈ LC.To this aim we assume that M is non-quasianalytic, i.e. (6.1) For the last equivalence we refer to [14,Lemma 4.1].
Then we can proceed as follows.First, by the well-known Denjoy-Carleman-theorem we obtain that both the classes D {M} and D (M) are non-trivial (e.g.see [14,Theorem 4.2]).
Hence, in this situation we can define with φ ∈ D {M} having φ (j) (0) = δ j,0 (Kronecker delta).For the existence of such a test function we refer to the proof of [17,Thm. 2.2].Concerning the support of φ we do not make any restriction.
First, ψ M ∈ D {M} ⊆ S {M} : ψ M obviously has compact support K ∋ 0 with supp(ψ M ) ⊆ supp(φ) = K and moreover both θ M and φ admit growth control expressed in terms of the weight sequence M. Hence for all j ∈ N 0 and x ∈ K: Consider the following assertions: ) with continuous inclusion.Then (i) ⇒ (ii) is valid for arbitrary weight sequences (resp.matrices, weight functions) and the spaces in (ii) can be defined by joint or separated growth at infinity.For the implications (ii) ⇒ (i) only the inclusion for d = 1 is required.
6.2.The Beurling case.We call a standard log-convex weight matrix M = {M (λ) : λ > 0} Beurling non-quasianalytic, when for all λ > 0 the sequence M (λ) is non-quasianalytic.This definition is justified by [21, Thm.4.1, Sect.4.6]: A countable intersection of nonquasianalytic ultradifferentiable classes (with totally ordered weight sequences) is still non-quasianalytic.So, if M is standard log-convex and Beurling non-quasianalytic, then is non-trivial.More precisely, by [21, Prop.4.7 (i), Prop.4.4] we know that there exists L ∈ LC such that L is non-quasianalytic and L ⊳ M. (Recall that for huge intersections this statement will fail in general, so an uncountable intersection of nonquasianalytic classes will be quasianalytic in general.)Finally, by the comments given in the previous section we see that for any given weight function ω the associated matrix M ω will be Beurling non-quasianalytic if and only if ω is non-quasianalytic, i.e. in the weight function approach both matrix notions of non-quasianalyticity do coincide.
Let us now consider the following Beurling-type condition: It is immediate to see that M = {M}, M ∈ LC, can never satisfy (6.3) because this would yield sup p≥1 (M p ) 1/p < +∞.
Moreover, if M is standard log-convex and satisfies (2.12), then for all λ > 0 there exist H > 0 and B > 0 such that j j/2 ≤ BH j M (λ) j for all j ∈ N 0 , i.e.G 1/2 M (λ) .Thus it is immediate to see that any standard log-convex matrix having (6.3) and (2.12) is Beurling non-quasianalytic.Condition (2.12) should be considered as a standard assumption when dealing with S (M) , e.g.see [4,Prop. 3].Now we are ready to state the following result which is analogous to Theorem 6.2.Theorem 6.6.Let M and N be arbitrary and consider the following assertions: ) is valid with continuous inclusion.Then (i) ⇒ (ii) and the spaces in (ii) can be defined by joint or separated growth at infinity.If both matrices are standard log-convex and if we assume that M is Beurling non-quasianalytic and satisfies (6.3) and if (ii) holds for the case d = 1, then (ii) ⇒ (i) is valid, too.
Proof.(i) ⇒ (ii) is again clear by the definition of the spaces.
(ii) ⇒ (i) We follow the ideas of the proof given in [18,Prop. 4.6 (2)] which is based on techniques developed in [8, Sect.2] (for the single weight sequence case).
By the continuous inclusion S (M) (R) ⊆ S (N ) (R) we get the following: (6.4) We will apply (6.4) for h = 1 and to the following family of functions.For each a > 0, arbitrary but from now on fixed, we consider a function φ a ∈ D (M) with supp(φ a ) ⊆ [−a, a] and φ (j) a (0) = δ j,0 : The existence of such functions follows again by [17, Thm.2.2], more precisely we take φ a ∈ D {L} ⊆ D (M) with L ⊳ M denoting the non-quasianalytic sequence mentioned before.(Here we use the fact that M is Beurling non-quasianalytic.)Moreover, for t ≥ 0 and x ∈ R we set f t (x) := exp(itx) and finally First, let us prove that g a,t ∈ S (M) .Let t > 0 be fixed (the case t = 0 is trivial), then note that for all h, λ > 0 (small) there exists some C h,λ ≥ 1 (large) such that |f for all j ∈ N 0 because lim j→∞ (M (λ) j ) 1/j = +∞ is valid.This proves that f t ∈ E (M) and the estimates hold globally on the whole R.
We have that supp(g a,t ) ⊆ [−a, a] for all t ≥ 0. We fix t ≥ 0 and a > 0 and estimate for all j ∈ N 0 , λ, h > 0 and x ∈ [−a, a] as follows: because by normalization and log-convexity we have M for all j, k ∈ N 0 with k ≤ j and for each λ.
Thus g a,t ∈ D (M) ⊆ S (M) and we are able to apply (6.4) to this family (with h = 1 in (6.4) and set a := h 1 (≤ 1)).
According to the index κ arising in (6.4), by applying (6.3) we get an index κ 1 and A ≥ 1 such that (M (κ 1 ) p ) 2 ≤ A p M (κ) p for any p ∈ N 0 .
Using this preparation we start now by estimating the right-hand side in (6.4) for all t ≥ 1 (and by using the seminorms with the joint growth control at infinity): = CC h 1 ,κ 1 exp (ω M (κ 1 ) (2At/h 1 )) .
For the estimate of (6.5) we argued as follows: Since φ h 1 ∈ D (M) ⊆ S (M) (R), we get Note that by log-convexity and normalization M , i.e. each sequence is increasing and since we are dealing with the Beurling case we will have 0 < h 1 ≤ 1 (small).Moreover note that we have estimated by 1 for any j, k ∈ N 0 and any index κ, so if we would consider separated growth control at infinity we would estimate at this step by We continue now with the left-hand side in (6.4) and get and the estimate is precisely the same when considering separated growth control at infinity by normalization.Summarizing, we have shown that (6.4) yields the following: ∀ λ > 0 ∃ κ 1 > 0 ∃ C, A, h 1 > 0 ∀ t ≥ 1 : exp (ω N (λ) (t)) ≤ C exp (ω M (κ 1 ) (2At/h 1 )) .
Note that ω is equivalent to ω 1 (t) := max{0, log(t)} and the relation ∼ preserves all the properties listed in Definition 2.1 except the convexity condition (δ) which is clear for ω 1 as well.In addition ω 1 is normalized, more precisely ω 1 (t) = 0 for 0 ≤ t ≤ 1 and ω 1 (t) = log(t) for all t ≥ 1.It is known that the weight ω yields the classical Schwartz class S and the aim of this appendix is to study for this limiting case the associated weight matrix.
More precisely we show that the weight matrix approach is not "well-defined" for the weights a log(1 + t), a > 0, because the matrix associated with such weights does not contain sequences of positive real numbers (as usually required).
Lemma A.1.We get as topological vector spaces the identity S (ω) (R d ) = S (ω 1 ) (R d ) = S(R d ) for any dimension d.However, the analogous result for the Roumieu-type space fails and neither M ω 1 nor M ω is a weight matrix as defined in Section 2.3.
Moreover, from (3.8) again we also get that the sequence M cannot satisfy the conditions in[4, Prop.3].Hence, the spaces S (M) and S {M} do not contain any Hermite function.Still, we do not know if these classes are nontrivial.However, the existence of such an oscillating sequence is important in view of the following: Remark 3.3.Let ω be a given (general) weight function according to Definition 2.1.If ω(t) = o(t 2 ), then[4, Cor.3(b)] yields that S (ω) is nontrivial (at least all Hermite functions are contained in this class).

4. 3 .Theorem 4 . 7 .
The single weight sequence case.It is straight-forward to obtain the analogous results for Theorems 4.4, 4.5 and 4.6 in the single weight sequence setting and we get the following characterization: Let M, N ∈ LC be given such that both satisfy (2.3).(I)Let M and N satisfy (2.8) and consider the following assertions:

=
+∞ for all α ∈ N d 0 with |α| ≥ 1.Thus only for α = 0 we get W (λ) α = 1.By using the matrix M ω 1 we can describe the Beurling-type class S (ω) (R d ) = S(R d ) as a topological vector space, however the Roumieu-type class is not fitting in this framework anymore.