Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting

We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending the previous work by Langenbruch. As a consequence, we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results for spaces defined by weight functions.


Introduction
The systematic study of nuclear locally convex spaces began in 1951 with the fundamental dissertation of Grothendieck [20] to classify those infinite dimensional locally convex spaces which are not normed, suitable for mathematical analysis. Among the properties of a nuclear space, the existence of a Schwartz kernel for a continuous linear operator on the space is of crucial importance for the theory of linear partial differential operators. In our setting of ultradifferentiable functions, this fact helps, for instance, to study the behaviour (propagation of singularities or wave front sets) of a differential or pseudodifferential operator when acting on a distribution. See, for example, [1,7,16,17,33,38] and the references therein.
Since the middle of the last century, several authors have studied the topological structure of global spaces of ultradifferentiable functions and, in particular, when the spaces are nuclear. See [31], or the book [19]. More recently, the first three authors in [9] used the isomorphism established by Langenbruch [28] between global spaces of ultradifferentiable functions in the sense of Gel'fand and Shilov [18] and some sequence spaces to see that under the condition that appears in [11,Corollary 16(3)] on the weight function (as in [12]) the space S ( ) (ℝ d ) of rapidly decreasing ultradifferentiable functions of Beurling type in the sense of Björck [3] is nuclear. However, there was the restriction that the powers of the logarithm were not allowed as admissible weight functions. Later, the authors of the present work proved in [10] that S ( ) (ℝ d ) is nuclear for any weight function satisfying log(t) = O ( (t)) and (t) = o(t) as t tends to infinity. The techniques used in [10] come especially from the field of time-frequency analysis and a mixture of ideas from [7,21,22,38]. In both [9] and [10], we use (different) isomorphisms between that space S ( ) (ℝ d ) and some sequence space and prove that S ( ) (ℝ d ) is nuclear by an application of the Grothendieck-Pietsch criterion [32,Theorem 28.15]. Very recently, Debrouwere, Neyt and Vindas [14,15] (cf. [27] for related results about local spaces), using different techniques have extended our previous results in a very general framework. In [14], they characterize when mixed spaces of Björck [3] of Beurling type or of Roumieu type are nuclear under very mild conditions on the weight functions. In [15], using weight matrices in the sense of [37], the same authors characterize the nuclearity of generalized Gel'fand-Shilov classes which extend their previous work [14] and treat also many other mixed classes defined by sequences.
The aim of the present paper is twofold. On the one hand, we extend the work of Langenbruch [28] to the matrix weighted setting in the sense of [37,40]. In particular, we prove that the Hermite functions are a Schauder basis of many global weighted spaces of ultradifferentiable functions. Moreover, we determine the coefficient spaces corresponding to this Hermite expansion (Theorem 1). These results are applied to spaces defined by weight functions S [ ] (ℝ d ) , being [ ] = ( ) (Beurling setting) or [ ] = { } (Roumieu setting). Hence, we extend part of the previous work of Aubry [2] to the several variables case. As a consequence we are able to generalize our previous study [9,10] about the nuclearity of the space S ( ) (ℝ d ) to global spaces of ultradifferentiable functions defined by weight matrices (Corollary 2). An application to particular matrices gives that S ( ) (ℝ d ) is nuclear when (t) = o(t 2 ) as t tends to infinity. Similarly, we also prove the analogous result for the Roumieu setting, namely that S { } (ℝ d ) is nuclear when (t) = O(t 2 ) as t tends to infinity (see Theorem 6 for both results). For weights of the form (t) = log (1 + t) with > 1 , our results hold and, hence, we generalize the results of [28] to spaces that could not be treated there since, as is easily deduced from [11,Example 20], S [(M p (ℝ) for any sequence of positive numbers (M p ) p∈ℕ in the sense of [26] (see Remark 4). We do not treat here the classical case (t) = log (1 + t) , for which S ( ) (ℝ) = S(ℝ) , the Schwartz class, because in this case infinitely many entries of our weight matrices are not well defined. However, the results presented here are already well known for the Schwartz class.
The classes of functions treated in [15] are in general different from ours. In fact, here we consider spaces of functions f that are bounded in the following sense: for some (or any) h > 0 , there is C > 0 such that for all x ∈ ℝ d and every multi-indices and , we have And we pass to the matrix setting for the multi-sequence (M ) , i.e. we make M depend also on a parameter > 0 (see the precise definition in the next section). In [15], the authors consider spaces of functions f bounded in the following sense: there is C > 0 such that for all x ∈ ℝ d and every multi-index they have where w is a positive continuous function. They pass to the matrix setting by making M and w depend on the same parameter > 0 . Hence, taking unions (Roumieu setting) or intersections (Beurling setting) in in the situation (A) gives different classes of functions than in the situation (B) in general. On the other hand, it is a very difficult problem to determine when the classes treated in this work are nontrivial, a question not considered in [14,15]. We characterize in a very general way (Propositions 2 and 3) when the Hermite functions are contained in our classes and this fact is closely related to classes being non-trivial. Indeed, we can deduce from our results that, in the Beurling setting, the space S ( ) (ℝ d ) contains the Hermite functions if and only if (t) = o(t 2 ) as t tends to infinity (Corollary 3). However, it is not difficult to see from the uncertainty principle [23,Theorem] ( (t)) as t tends to infinity. In the same way, in the Roumieu case, the space S { } (ℝ d ) contains the Hermite functions if and only if (t) = O(t 2 ) as t tends to infinity (Corollary 3), but again from [23,Theorem] we can deduce ( (t)) as t tends to infinity. For more information on the uncertainty principle for S [ ] (ℝ d ) where, as stated above, [ ] = ( ) or { } , see the nice introduction to the paper of Aubry [2] and the references therein. Moreover, our classes are well adapted for Fourier transform (Corollary 1). We should also mention that throughout this paper we assume, on the multi-sequence (M ) , that (M ) 1∕| | tends to infinity when | | tends to infinity, which is stronger than the [26,Def. 3.1] (for the one-dimensional case). The reason is that it is not clear how the results read when the associated function is infinite (see Remark 1).
The paper is organized as follows: in the next section, we give some necessary definitions, in Sect. 3 we introduce the classes under study in the matrix weighted setting and establish the analogous conditions to [28] to determine in Sect. 4 when the Hermite functions belong to our classes. In Sect. 5, we introduce the suitable matrix sequence spaces and prove that they are isomorphic to our classes, which is the fundamental tool to see that our spaces are nuclear. We finally apply these results to the particular case of spaces defined by weight functions in Sect. 6.

the associated function is denoted by
We say that (M p ) p satisfies the logarithmic convexity condition (M1) of [26] if The following lemma is well known (see Lemmas 2.0.6 and 2.0.4 of [39] for a proof). Lemma 1 Let (M p ) p∈ℕ 0 be a normalized sequence satisfying (2.2). Then From Lemma 1(c) and [26,Prop. 3.2], we have that a normalized sequence = (M p ) p satisfies (2.2)

if and only if
We say that (M p ) p satisfies the stability under differential operators condition (M2) � of [26] if and (M p ) p satisfies the stronger moderate growth condition (M2) of [26] if The following lemma extends [26,Proposition 3.4] for two sequences. We give the proof for the convenience of the reader.

turns into
Now, for t ∈ ℝ d , we denote (At) .
The associated weight function of a normalized = (M ) ∈ℕ d 0 is given by where by convention 0 0 ∶= 1 . Note that for a normalized sequence we have (0) = 0.

Remark 1
As it has already been pointed out in the geometric construction in [30,Chap. I] for the one dimensional weight function (see (2.1)), we have that 0 and now let t min → +∞. Conversely, let lim | |→∞ (M ) 1∕| | = ∞ and so for any A > 0 large, we can find some C > 0 large enough such that A | | ≤ CM . Since |t | ≤ |t| | | for all t ∈ ℝ d and ∈ ℕ d 0 , we see that for any given t ∈ ℝ d we get |t | M ≤ |t| | | M ≤ C for some C > 0 and all ∈ ℕ d 0 .
Then for t ∈ ℝ d ⧵ ℝ d , we have t = 0 , and so it is enough to prove that We have observe that ∈ ℕ d 0,t and so, choosing = , we get (t∕h) |t | = inf which proves (2.11), and then the proof is complete. Note that if (t∕h) = +∞ , then (2.10) is clear and so we could restrict in the estimates above to all t ∈ ℝ d such that (t∕h) is finite. ◻ In the following, we use two normalized sequences as above This clearly implies In [28], Langenbruch uses his condition (1.2) to prove that the Hermite functions belong to the spaces considered there. In the present paper we need, for the same reason, a mixed condition that involves two sequences: We call M a weight matrix and consider matrix weighted global ultradifferentiable functions of Roumieu type defined as follows (from now on ‖ ⋅ ‖ ∞ denotes the supremum norm): first, for a given normalized sequence , we set 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, we define the matrix type spaces as follows: again endowed with the inductive limit topology in the Roumieu setting (which may be thought countable if we take , h ∈ ℕ ) and endowed with the projective limit topology in the Beurling setting (countable for −1 , h −1 ∈ ℕ). Now we consider different conditions on the weight matrices that we use following the lines of [28]. The next basic condition extends (1.2) of [28] in the Roumieu case and is needed to show that the Hermite functions belong to S {M} (see Proposition 3): The analogous condition to (3.2) in the Beurling case, which is needed to show that the Hermite functions belong to S (M) is the following (see Proposition 3):

Remark 3
Similarly, as commented in Remark 2 for (2.12), property (3.2) (property (3.3)) yields that lim | |→∞ (M ( ) ) 1∕| | = +∞ for some > 0 , and hence for all We also need to extend condition (3.7) of [28] to the matrix weighted setting. First, we state it in the Roumieu case: and in the Beurling case: ( ) ,h ≤ C}, The extensions of condition (2.7) (mixed derivation closedness properties) for a weight matrix M in the Roumieu and Beurling cases read as follows: The following conditions generalize (2.8) to the weight matrix setting: It is immediate that for any given matrix M satisfying (3.8) and (3.4) we can replace in the definition of S {M} the seminorm ‖ ⋅ ‖ ∞, ( ) ,h by We have an analogous statement for the class S (M) under (3.9) and (3.5). When we define the spaces S {M} or S (M) with the weighted L 2 norms treated below in (3.17), the similar property holds.

Lemma 4
Let M be a weight matrix as defined in (3.1).

.7) holds, then
Proof First, we consider the Roumieu case.
Therefore, by the definition of the associated weight function, choosing ≥ > 0 as in (3.12), we have, assuming |t| ∞ = t j for some 1 ≤ j ≤ d and |t| ∞ ≥ 1: On the other hand, if t ∈ ℝ d with |t| ∞ ≤ 1 , then �t� ≤ √ d and hence, for as in (3.12), with C depending on since depends on .
We have thus proved (3.10) with In the Beurling case, by 2(d + 1) iterated applications of (3.7), we find and ∶= 2d+2 . Then we proceed as in the Roumieu case and prove that (3.12)

Lemma 5 Let M be a weight matrix that satisfies (3.7). Then
Let M be a weight matrix that satisfies (3.6). Then , then by the definition of the associated weight function, for where ℕ d 0,t is defined by (2.9). This estimate is valid for any given index > 0. In the Beurling case, by N iterated applications of (3.7) we find ∶= N , we have, proceeding as in (3.13), ( ) . Therefore, and we conclude that (3.14) is satisfied for B ∶= max{ N 2 log d, 1}. In the Roumieu case, we make N iterated applications of (3.6) and we find indices ( ) and hence from (3.16): . ◻ Now, we consider the different system of seminorms Under suitable conditions on the weight matrix M , it turns out to be equivalent to the previous one given by sup norms, as we prove in the following: Proposition 1 Let M be a weight matrix as defined in (3.1) that satisfies (3.3) and (3.7) ((3.2) and (3.6)). Then the system of seminorms ‖ ⋅ ‖ ∞, ( ) More precisely, in the Beurling case we have the following two conditions for every f ∈ C ∞ (ℝ d ): in the Roumieu case we have the following two conditions, for every f ∈ C ∞ (ℝ d ), On the other hand, if |x| ∞ ≥ 1 then Therefore, for any fixed , h > 0, ( ) ,h , ( ) ,h , and hence Now, we consider separately the Beurling and Roumieu cases. In the Beurling case, for every , h > 0 , we first estimate ‖x +(d+1)e j f ‖ 2, ( ) ,h to use (3.22).
such that, proceeding as in (3.13), we obtain M ( ) Hence, we deduce Therefore, from (3.22) and the fact that ( ) we obtain This shows (3.18).
by Leibniz's rule and [28, formula (2.3)] that, for some C 2 > 0, On the other hand, by | | iterated applications of (3.7), there exist Observe that ̃ may depend on . From (3.1) we can consider in the previous estimates, instead of ̃ , the minimum of all these ̃ for | | ∞ ≤ 2d + 2 , so that we can For taking into account that we finally have that for all , h > 0 there exist ̃ , C ,h > 0 and h > 0 , such that (3.24) Since neither H nor A are depending on h, we have h → 0 as h → 0 . This shows (3.19) and concludes the proof in the Beurling case.

Hermite functions: properties in the matrix setting
We recall the definition of the Hermite functions H for ∈ ℕ d 0 : where h are the Hermite polynomials As in [28] we consider, for f ∈ C ∞ (ℝ d ) , the operators with A 0 ±,i ∶= id. By [32, Example 29.5 (2) It follows that, for , ∈ ℕ d 0 , We also recall the following two lemmas from [28]: for some coefficients C , ( ) satisfying We can generalize Lemma 3.1(b) of [28] in the following way: Applying now (2.12) and ∑ As a corollary, we immediately have the following:

Lemma 9
Let M be a weight matrix satisfying (3.2) and assume that f ∈ C ∞ (ℝ d ) satisfies, for some , C 1 > 0 for the constant C of (3.2). Then with , B, H, C as in (3.2).

If M satisfies (3.3) and if, for some
for the constant ≤ of (3.3) and for some C, C 1 > 0 , then where H = H( ) and B = B(C, ) are given by (3.3).

Proposition 3 Let M be a weight matrix that satisfies
If M satisfies (3.7), (3.5), then the following are equivalent:

Matrix sequence spaces
Let us consider, for = (M ) ∈ℕ d 0 , the following sequence spaces in the Roumieu and the Beurling cases: Since h ↦ ( 1∕2 ∕h) is decreasing we can also write Now, for a weight matrix M as in (3.1), we denote It follows that we can write Λ {M} ( Λ (M) ) as inductive (projective) limit: Note that by Remark 1, it seems natural to require that lim | |→∞ (M ) 1∕| | = +∞ for the definition of Λ { } and Λ ( ) . In fact, otherwise and we get Λ ( ) = {0} and Λ { } consisting of sequences having only finitely many values ≠ 0. However, in our next main result, by Remark 3 and assumption (3.2) ((3.3), respectively), we have the warranty of the finiteness of all associated weight functions under consideration. Theorem 1 Let M be a weight matrix satisfying (3.2) and (3.6). Proof By Proposition 1 we can assume that S {M} and S (M) are defined by L 2 norms. First, we consider the Roumieu case. If f ∈ S {M} , there exist , C, C 1 > 0 such that By (4.1) and Lemma 9, there exist ≥ , B, C, H > 0 such that for all , ∈ ℕ d 0 , since ‖H ‖ 2 = 1 for all ∈ ℕ d 0 , we have Therefore, by definition of the associated weight function, and using the notation of (2.9), since Hence, ( (f By Lemma 4, there exist ≥ and B 1 , B 2 ≥ 1 such that Then, by (3.2), there exist ′ ≥ and B, C, H > 0 with C ≥ B 2 C * , such that, by Lemma 10, Since here | 1∕2 | denotes the Euclidean norm of the multi-index 1∕2 , we have Hence, Hence, This shows that T −1 is continuous and, moreover, that (H ) is an absolute Schauder basis in S {M} .
Let now f ∈ S (M) and , C > 0 be given. We consider 0 < ≤ , H, B > 0 as in (3.3) (with and H depending only on ) and we set By Lemma 8, we have Hence, proceeding as in the Roumieu case, we deduce that, for all , C > 0 , there exist 0 < ≤ and B, H > 0 such that This shows that T is continuous in the Beurling case. (M) , then by (3.3) and Lemma 10, for all , C > 0 there exist 0 < ≤ , and H, B > 0 (with and H depending only on ) such that By Lemma 4, there exist 0 < ′ ≤ and B 1 , B 2 ≥ 1 such that Since ∈ Λ (M) , we have Therefore, arguing as in the Roumieu case, For all , h > 0 , there exist then ′ ≤ and h = h∕(2HB 2 ) > 0 such that This shows that T −1 is continuous on S (M) and that (H ) is an absolute Schauder basis in S (M) , which finishes the proof. ◻ As in [28, Corollary 3.6], we also have that the Fourier transform is well adapted to our spaces and it is an isomorphism: Corollary 1 Let M be a weight matrix satisfying (3.2) and (3.6) ((3 .3) and (3.7)). Then the Fourier transform is an isomorphism in S {M} ( S (M) ). Now, we prove that the spaces of sequences are nuclear.
be a weight matrix satisfying (3.7). Then Λ (M) is nuclear.
Proof By (5.2) and [32,Prop. 28.16] (see also [10, Theorem 3.1] for a self-contained proof in the case of countable lattices), the sequence space Λ (M) is nuclear if and only if Moreover, by Lemma 5, condition (3.14) is satisfied. We can thus proceed as in the proof of Theorem 1 of [9] to prove that (3.14) implies that the series in (5.9) converges, and hence Λ (M) is nuclear. To this aim, we fix an index > 0 and N ∈ ℕ with N > 2d and remark that if the inequality (3.14) holds for = 1∕j and ≤ , then it holds also if, instead of , we put � = 1∕h with h ∈ ℕ , h > [ 1 ] + 1 , since ( � ) ≤ ( ) for ′ ≤ and hence ( ) ≤ ( � ) . Then for ≥ Ah (so that ≥ Aj and ≥ h > j and note that the constant A is also depending on the chosen N): by our choice of N > 2d . ◻ Concerning the Roumieu case, we have the following result.
be a weight matrix satisfying (3.6). Then Λ {M} is nuclear.
Proof For we consider the matrices We observe that A is a Köthe matrix since its entries are strictly positive and a ,j ≤ a ,j+1 for every j ∈ ℕ . We consider now the space and v ,j > 0 for every and j, we have that the matrix V satisfies the condition (D) of [5] (see also [4]). From [4, Proof If condition (b) is satisfied, then (3.7) is satisfied and hence also condition (a), as we already saw in the proof of Theorem 2.
Let us now assume condition (a) and prove (b). To this aim let us first remark that is decreasing. Indeed, is convex (see the proof of Theorem 1 in [9] for the implication that the convexity implies that 1 is increasing).
To prove that also the second difference 2 is increasing, we set and remark that, by (2.2) (see [26, formula (3.11) Then is an increasing function of t since by the assumption for > j. Therefore, 1 and 2 are increasing and we have thus proved that (5.11) is decreasing. This condition together with assumption (a) implies that There exists then A ≥ 1 such that and hence, for all k ∈ ℕ, Choosing, for every t ≥ 1 , the smallest k ∈ ℕ such that jk 1∕2 ∈ [t, (j + 1)t] , we finally have Since (5.12) is trivial for 0 < t ≤ 1 , we have proved that condition (ii) of Lemma 2 is satisfied for = (1∕j) and = (1∕ ) and hence, from (i) of Lemma 2, there exists Ã ≥ 1 such that Then, for all > 0 , choosing j ∈ ℕ so that 1 j ≤ , there exists = 1 < 1 j ≤ such that condition (b) holds. Proof It follows from Theorem 2 and, in particular, (5.9). Proof By the proof of Theorem 3 we have that Λ {M} is nuclear if and only if (5.10) is satisfied, and this is equivalent to (3.6) since, analogously as in Proposition 4, the following two conditions are equivalent: Indeed, (b) � implies (3.6) and hence (a) � , i.e. (5.10), in the one-dimensional case, by the proof of Theorem 3. Conversely, if (a)′ holds then for every fixed j ∈ ℕ , and > j as in (a)′, there exists A > such that since is decreasing, similarly as in the proof of Proposition 4. Then, for all k ∈ ℕ, If t ≥ 1 we can choose a smallest k ∈ ℕ such that k 1∕2 ∕ ∈ [t, (1 + 1 )t] and obtain that Since (5.13) is trivial for 0 < t ≤ 1 , we have that for A = j 1 + 1 ≥ 1 and B = log(A∕ ) > 0 . By Lemma 2 with = (j) and

Rapidly decreasing ultradifferentiable functions
We shall now consider weight functions defined as below:

Definition 1 A weight function is a continuous increasing function
It is not restrictive to assume | [0,1] ≡ 0 . As usual, we define the Young conjugate * of by which is an increasing convex function such that * * = and * (s)∕s is increasing [12,24]. We remark that condition ( ) and the stronger condition (t) = o(t 2 ) as t tends to infinity are needed in the Roumieu and Beurling cases for Corollary 5 and Theorem 6. On the other hand, condition ( ) guarantees that * is finite, so that, from the properties of * (see [12] or [8, Lemma A.1]) we easily obtain (cf. [37]):
Proof Let us first remark that condition ( ) of Definition 1 ensures that W ( )  Finally, (vi) is an immediate consequence of (iv). ◻ Let us now define the spaces of rapidly decreasing -ultradifferentiable functions, in the Roumieu case and in the Beurling case From Lemma 11(iv) and (vii) (see also [6,Thm. 4.8]): and We refer to [6,8,22] for more equivalent seminorms on We can also insert h | + | at the denominator (for some h > 0 in the Roumieu case and for all h > 0 in the Beurling case) by Lemma 11(v). In particular, we have the following Proposition 5 Let be a weight function and M the weight matrix defined in (6.1), (6.2). We have S {M } (ℝ d ) = S { } (ℝ d ) and S (M ) (ℝ d ) = S ( ) (ℝ d ) and the equalities are also topological.
Indeed, by [11,Example 20], E ( ) (ℝ) ≠ E (M p ) (ℝ) for any sequence (M p ) as considered just above, where E ( ) (ℝ) and E (M p ) (ℝ) are the spaces of ultradifferentiable functions defined by weights and sequences (for the definitions see [11]). We fix a sequence (M p ) and prove that S ( ) (ℝ) ≠ S (M p ) (ℝ) . Clearly, we can assume that (M p ) is non-quasianalytic since the weight is non-quasianalytic. In particular, (M p ) satisfies (M0) (see [11], condition (M3) � , and use also (M1)). If f ∈ E (M p ) (ℝ) ⧵ E ( ) (ℝ) , then there are a compact set K ⊆ ℝ and m ∈ ℕ such that ‖f ‖ ∞, ( ) Hence, Since K is compact we can assume that the sequence (x n ) converges to some but, for n sufficiently large, and hence g ∉ S ( ) (ℝ) (see the definition of S ( ) (ℝ) above). Analogously The same arguments are valid for the Roumieu case and for dimension bigger than one (considering always isotropic classes).
The following Lemma was proved in dimension 1 in [25, Lemma 2.5]; here we give a version of it in dimension d.

Lemma 12
Let be a weight function. Then there exists a constant B > 0 and, for every > 0 , there exists C > 0 , such that Proof For t = 0 the thesis is trivial, so we can consider t ≠ 0 . Since |t | ≤ |t| | | for every multi-index , we have so the first inequality of (  ∀n ∈ ℕ ∃x n ∈ K, j n ∈ ℕ such that |f (j Fix now t ∈ ℝ d with |t| ≥ e * ( )∕ and let j 0 be such that |t| ∞ = |t j 0 | ; for every M ∈ ℕ 0 , we then write M ∶= Me j 0 . We then have |t| M ∞ = |t M | , and so by (6.4) we obtain since M ∈ ℕ d 0,t due to the fact that t j 0 ≠ 0 (we are in fact considering t ∈ ℝ d such that |t| ≥ e * ( )∕ ). By (6.5) we then obtain for |t| ≥ e * ( )∕ . Then the second inequality of (6.3) holds for ◻ Lemma 13 Let be a weight function and consider the weight matrix M as defined in (6.1), (6.2). Then for r > 0 : Then Therefore, for every > 0 and j ∈ ℕ with j ≥ c r , choosing x = j and multiplying by 1∕ in (6. Conversely, if (6.7) holds then, proceeding as in (a), we have that for every , D > 0 there exists C > 0 such that (6.10) is valid and, therefore, by Lemma 12, (z) = o(|z| 1∕r ) as |z| → +∞ for z ∈ ℝ d , or, equivalently, (t) = o(t 1∕r ) as t → +∞ .  [35]). Under the above conditions the Hermite functions constitute a basis for the Gel'fand-Shilov spaces S s (ℝ d ) and Σ s (ℝ d ) . In fact, S s (ℝ d ) and Σ s (ℝ d ) are the subspaces of S(ℝ d ) consisting of those functions f that can be expressed through Hermite expansions with coefficients c satisfying for some c, r > 0 (for every r > 0 ), as was shown by Zhang [41] (see also [13,29,36]). The critical exponent s = 1∕2 for the Gel'fand-Shilov spaces is closely related *  Proof We consider the weight function 2 (t) = max(0, t 2 − 1) and its corresponding weighted matrix as defined in (