Localization of Fréchet Frames and Expansion of Generalized Functions

Matrix-type operators with the off-diagonal decay of polynomial or sub-exponential types are revisited with weaker assumptions concerning row or column estimates, still giving the continuity results for the frame type operators. Such results are extended from Banach to Fréchet spaces. Moreover, the localization of Fréchet frames is used for the frame expansions of tempered distributions and a class of Beurling ultradistributions.

localized frame in this sense leads to the same type of localization of the canonical dual frame as well as to the convergence of the frame expansions in all associated Banach spaces. We refer to [5,13,14,20,21], where various interesting properties and applications of localized frames were considered. The localization and self-localization, considered independently in [1][2][3], are directed to the over-completeness of frames and the relations between frame bounds and density with applications to Gabor frames. For the present paper, we have chosen to stick to the localization concept from [23], because the results obtained for a family of Banach spaces there can naturally be related to Fréchet frames (cf. [31][32][33][34]).
Our main aim in this paper is to present in Section 6 the frame expansions of tempered distributions and tempered ultradistributions of Beurling type by the use of localization. Matrix-type operators of Section 5 have an essential role in our investigations. The important novelty is the analysis related to sub-exponential off-diagonal decay without assumption of the exponential off-diagonal decay as it was considered in [23]. More precisely, in [23] the presumed exponential off-diagonal decay of matrices implies the analysis of sub-exponentially weighted spaces. Probably the most important impact in applications is related to the Hermite basis which is almost always used for the global expansion of L 2 -functions or tempered generalized functions over R n . Our results by the use of localization, show that the same is true if one uses a kind of perturbation of Hermite functions through localization.
As particular results, not directly involved in the main ones, we extend in Sections 3 and 4 the continuity results on matrix-type operators acting on elements of a Banach or Fréchet spaces expanded by frames. We consider relaxed version of the classical off-diagonal decay conditions, assuming the column-decay and allowing row-increase in a matrix.
The paper is organized as follows. We recall in Section 2 the notation, basic definitions, and the needed known results. In Section 3, we consider matrices with column decay and possible row increase. For such type of matrices, we obtain in Section 4 continuity results for the frame related operators using less restrictive conditions in comparison with the localization conditions known in the literature. Sub-exponential localization is introduced and analyzed in Section 5. The use of Jaffard's Theorem and [23,Theorems 11 and 13] is intrinsically connected with the sub-exponential localization. Section 6 is devoted to Fréchet frames and series expansions in certain classes of Fréchet spaces based on polynomial, exponential, and sub-exponential localization. In particular, we obtain frame expansions in the Schwartz space S of rapidly decreasing functions and its dual, the space of tempered distributions, as well as in the spaces α , α > 1/2, and their duals, spaces of tempered ultradistributions. In order to illustrate some results, we provide examples with the Hermite orthonormal basis h n , n ∈ N, and construct a Riesz basis which is polynomially and exponentially localized to h n , n ∈ N. Finally, in "Appendix", we add some details in the proof of the Jaffard's theorem.

Notation, Definitions, and Preliminaries
Throughout the paper, (H, ·, · ) denotes a separable Hilbert space and G (resp. E) denotes the sequence (g n ) ∞ n=1 (resp. (e n ) ∞ n=1 ) with elements from H. Recall that G is called: frame for H [15] if there exist positive constants A and B (called frame bounds) so that A f 2 ≤ ∞ n=1 | f , g n | 2 ≤ B f 2 for every f ∈ H; -Riesz basis for H [4] if its elements are the images of the elements of an orthonormal basis under a bounded bijective operator on H.
Recall (see, e.g., [12]), if G is a frame for H, then there exists a frame ( n=1 , is bounded from H into 2 ; the synthesis operator T G , given by T G f = ∞ n=1 c n g n , is bounded from 2 into H; the frame operator n=1 , called the canonical dual of (g n ) ∞ n=1 , and it will be denoted by ( g n ) ∞ n=1 or G. When G is a Riesz basis of H (and thus a frame for H), then only G is a dual frame of G, it is the unique biorthogonal sequence to G, and it is also a Riesz basis for H. A frame G which is not a Riesz basis has other dual frames in addition to the canonical dual and in that case we use notation G d or (g d n ) ∞ n=1 for a dual frame of G. Next, (X , · ) denotes a Banach space and ( , |· |) denotes a Banach sequence space; is called a B K -space if the coordinate functionals are continuous. If the canonical vectors form a Schauder basis for , then is called a C B-space. A C Bspace is clearly a B K -space.
Given a B K -space and a frame G for H with a dual frame G d = (g d n ) ∞ n=1 , one associates with the Banach space When G is a Riesz basis for H, then we use notation H G for H G, G .

Localization of Frames
In this paper, we consider polynomially and exponentially localized frames in the way defined in [23], and furthermore, sub-exponential localization. Let G be a Riesz basis for the Hilbert space H. A frame E for H is called: polynomially localized with respect to G with decay γ > 0 (in short, γ -localized wrt (g n ) ∞ n=1 ) if there is a constant C γ > 0 so that max{| e m , g n |, | e m , g n |} ≤ C γ (1 + |m − n|) −γ , m, n ∈ N; exponentially localized with respect to G if for some γ > 0 there is a constant C γ > 0 so that max{| e m , g n |, | e m , g n |} ≤ C γ e −γ |m−n| , m, n ∈ N.

Fréchet Frames
We consider Fréchet spaces which are projective limits of Banach spaces as follows.
Let {Y k , | · | k } k∈N 0 be a sequence of separable Banach spaces such that Under the conditions (1)-(2), Y F is a Fréchet space and Y * F is the inductive limit of the spaces Y * k , k ∈ N. We will use such type of sequences in two cases: Let { k , |· | k } k∈N 0 be a sequence of C B-spaces satisfying (1). Then (2) holds, because every sequence (c n ) ∞ n=1 ∈ F can be written as ∞ n=1 c n δ n with the convergence in F , where δ n denotes the n-th canonical vector, n ∈ N. Furthermore, * F can be identified with the sequence space F := {(U δ n ) ∞ n=1 : U ∈ * F } with convergence naturally defined in correspondence with the convergence in * F . We use the term operator for a linear mapping, and by invertible operator on X , we mean a bounded bijective operator on X . Given sequences of Banach spaces, {X k } k∈N 0 and { k } k∈N 0 , which satisfy (1)-(2), an operator T : and there exists a continuous operator V : When s k = s k = k, k ∈ N 0 , and the continuity of V is replaced by the stronger condition of F-boundedness of V , then the above definition reduces to the definition of a Fréchet frame (in short, F-frame) for X F with respect to F introduced in [32]. Although we will use in the sequel this simplified definition, Definition 2.1 is the most general one, interesting in itself, and can be considered as a non-trivial generalization of Banach frames.
In the particular case when X k = X , and k = , k ∈ N 0 , a Fréchet frame for X F with respect to F becomes a Banach frame for X with respect to as introduced in [22].
For another approach to frames in Fréchet spaces, we refer to [6]. For more on frames for Banach spaces, see, e.g., [8,9,37] and the references therein.

Sequence and Function Spaces
Recall that a positive continuous function μ on R is called: a k-moderate weight if k ≥ 0 and there exists a constant C > 0 so that μ(t + x) ≤ C(1+|t|) k μ(x), t, x ∈ R; a β-sub-exponential (resp. exponential) weight, if β ∈ (0, 1) (resp. β = 1) and there exist constants C > 0, γ > 0, so that μ(t + x) ≤ Ce γ |t| β μ(x), t, x ∈ R. If β is clear from the context, we will write just sub-exponential weight. Let μ be a kmoderate, sub-exponential, or exponential weight so that μ(n) ≥ 1 for every n ∈ N, and p ∈ [1, ∞). Then the Banach space ⎭ is a C B-space. We refer, for example, to [28,Ch. 27] for the so-called Köthe sequence spaces. We will need the following lemma, which can be easily proved by the use of [32,Theor. 4.2]. Lemma 2.2 Let G be a frame for H and let G d = (g d n ) ∞ n=1 be a dual frame of G. Let μ k be k-moderate (resp. sub-exponential or exponential) weights, k ∈ N 0 , so that Then the spaces k : If G is a Riesz basis for H, then the density assumption of M ∩ F in M ∩ k = {0}, k ∈ N, is fulfilled and in addition one has that g n ∈ X F for every n ∈ N.
Recall that the well-known Schwartz space S is the intersection of Banach spaces The dual S (R) is the space of tempered distributions. The space of sub-exponentially decreasing functions of order 1/α, α > 1/2, is α := Its dual ( α (R)) is the space of Beurling tempered ultradistributions, cf. [19,30].

Remark 2.3
The case α = 1/2 leads to the trivial space 1/2 = {0}. There is another way in considering the test space which corresponds to that limiting Beurling case α = 1/2 and can be considered also for α < 1/2 (cf. [10,11,19,29]). We will not treat these cases in the current paper.
We can consider S and α as the projective limit of Hilbert spaces H k , k ∈ N 0 , with elements f = n a n h n , in the first case with norms f H k := |(a n n k ) n | 2 < ∞}, k ∈ N 0 , and in the second case with norms f H k := |(a n e kn 1/(2α) ) n | 2 < ∞}, k ∈ N 0 .
Thus, (h n ) n is an F-frame for S(R) with respect to s as well as an F-frame for α with respect to s 1/2α , α > 1/2, (F-boundedness is trivial).

Matrix-Type Operators
Papers [14,21,23] concern matrices with off-diagonal decay of the form: for some As we noted in the introduction, matrix operators in this section and the next one are not essentially related to Sections 5 and 6. But they significantly illuminate such operators in our main results. Moreover, we refer in Section 4 to results of Section 3 and in Remark 6.4 we refer to Section 4.
We will consider matrices with more general off-diagonal type of decay (see ( * * * ) below which is weaker condition compare to the polynomial type condition in (6)). Moreover, we consider matrices which have column decrease but allow row increase (see Propositions 3.2 and 3.6) allowing sub-exponential type conditions as well. For such more general matrices, we generalize some results from [23] with respect to certain Banach spaces and, furthermore, proceed to the Fréchet case.
In the sequel, for a given matrix (A mn ) m,n∈N , the letter A will denote the mapping (c n ) ∞ n=1 → (a m ) ∞ m=1 determined by a m = ∞ n=1 A m,n c n (assuming convergence), m ∈ N; conversely, for a given mapping A determined on a sequence space containing the canonical vectors δ n , n ∈ N, the corresponding matrix (A mn ) m,n∈N is given by A m,n = Aδ n , δ m . We will sometimes use A with the meaning of (A mn ) m,n∈N and vice-verse.

Polynomial-Type Conditions
Let us begin with some comparison of polynomial type of off-diagonal decay: Consider the following conditions: Then, the implications ( * ) ⇒ ( * * ) ⇒ ( * * * ) hold. The converse implications are not valid. γ (max(m,n)) γ , n, m ∈ N, which are easy to be verified. To show that ( * * * ) does not imply ( * * ) even up to a multiplication with a constant, take a matrix A m,n which satisfies |A m,n | = Cn γ m γ , n ≤ m, for some γ > 0 and some positive constant C, and assume that there exist γ 1 (γ ) ∈ N and a positive constant K so that for m ≥ n one has Cn γ m γ ≤ K (1+m−n) γ 1 ; then taking m = 2n, one obtains 0 < C · 2 −γ ≤ K (1+n) γ 1 → 0 as n → ∞, which leads to a contradiction. In a similar spirit, one can show that ( * * ) does not imply ( * ).
Below we show that the relaxed polynomial-type conditions, as well as conditions allowing row-increase, still lead to continuous operators.

Proposition 3.2 Assume that the matrix (A mn ) m,n∈N satisfies the condition
1 n 1+ε , the assertion follows.
A direct consequence of Proposition 3.2 is: Corollary 3.3 Assume that the matrix (A mn ) m,n∈N satisfies: there exist γ 0 ≥ 0 and C 0 > 0, and for every γ > 0 there is C γ > 0 so that Then A is a continuous operator from s into s.
In order to determine A as a mapping from a space s γ 1 into the same space, we have to change the decay condition.
Then A is a continuous operator from s γ 1 into s γ 1 .

Remark 3.5
For the same conclusion as above, one has in [23] another condition noncomparable to (7):

Sub-exponential-and Exponential-Type Conditions
Up to the end of the paper β will be a fixed number of the interval (0, 1]; β = 1 is related to the exponential growth order while β ∈ (0, 1) corresponds to the pure sub-exponential growth order.
Further on, Therefore, This completes the proof.
As a consequence of Proposition 3.6, we have:

Remark 3.10
One can simply show that the assumption |A m,n | ≤ Ce −γ |m−n| β , m, n ∈ N, leads to similar continuity results. We will consider this condition later in relation to the invertibility of such matrices and the Jaffard theorem.

Proposition 4.1 Let G be a frame for H, G d be a dual frame of G, and μ k (x) =
(1 + |x|) k , k ∈ N 0 . Under the notations in Lemma 2.2, assume that M ∩ F is dense in M ∩ k = {0} with respect to the |· | k -norm for every k ∈ N and let E = (e n ) ∞ n=1 be a sequence with elements from X F which is a frame for H. Then the following statements hold.
(i) Assume that there exist s 0 ∈ N, C > 0 and for every k ∈ N there exists C k > 0 such that | e m , g n | ≤ Cn s 0 , n > m, C k n k m −k , n ≤ m.
Then the analysis operator f → U E f = ( f , e m ) ∞ m=1 is continuous from X F into s.
(ii) Assume that there exist s 0 ∈ N 0 , C > 0 and for every k ∈ N there exists C k > 0 such that

Then the synthesis operator (c n ) n → T E (c n ) = c n e n is continuous from s into X F . (iii) Under the assumptions of (i) and (ii), the frame operator T E U E is continuous from X F into X F .
Proof Note that under the given assumptions, F is the space s.
(i) Let A m,n = g n , e m , m, n ∈ N, and A be the corresponding operator for the matrix A. Let f ∈ X F . Then ( f , g d n ) ∞ n=1 ∈ s and By Corollary 3.3, it follows that ( f , e m ) ∞ n=1 ∈ s. Furthermore, by Proposition 3.2, for every k ∈ N, there is a constant K s 0 ,k,C,C k so that Therefore, the analysis operator U E is continuous from X F into s. (ii) Let (c n ) ∈ s. First we show that ∞ n=1 c n e n converges in X F and then the continuity of T E . Since (c n ) ∞ n=1 ∈ 2 , we have x = n c n e n ∈ H. Denote A m,n = e n , g d m and consider the corresponding operator A. Then ( x, g d m ) m = ( n A m,n c n ) m = A(c n ) ∈ s (by Corollary 3.3), which implies that x ∈ X F , and furthermore, for every k ∈ N, one has T E (c n ) n k = x k = |( x, g d m ) m | k .
For every k ∈ N, there is a constant R k such that |(d n ) | k ≤ R k |(d n ) | sup,k+2 for every (d n ) ∈ s k+2 . By Proposition 3.2, we conclude that Thus, the synthesis operator T E is well defined and continuous from s into X F . (iii) follows from (i) and (ii).
It is of interest to consider the case when X F is S.
n=1 be a frame of L 2 (R) with elements in S(R). Assume that for every k ∈ N there are constants C k , C k such that

Then the analysis operator U E is continuous from S into s, the synthesis operator T E is continuous from s into S, and the frame operator T E U E is continuous from S into S.
Now, we consider sub-exponential weights.

Proposition 4.3
Let β ∈ (0, 1) and let the assumptions of the first part of Lemma 2.2 hold with the weights μ k (x) = e k|x| β , k ∈ N 0 . Let E = (e n ) ∞ n=1 be a sequence with elements from X F which is a frame for H. Then the following statements hold.
(i) Assume that there exist constants γ 0 ∈ N, C > 0 such that for every k ∈ N there exists C k > 0 such that | e m , g n | ≤ Ce γ 0 n β , n > m, C k e k(n β −m β ) , n ≤ m, k ∈ N.
Then the analysis operator f Then the synthesis operator (c n ) n → T E (c n ) = c n e n is continuous from s β into X F . (iii) If (11) and (12) hold, then the frame operator T E U E is continuous from X F into X F .
Proof Under the given assumptions, F is the space s β . The rest of the proof can be done in a similar way as the proof of Proposition 4.1, using Corollary 3.8 instead of Corollary 3.3.
n=1 be a sequence with elements from α which is a frame for L 2 (R) and such that for every k ∈ N there are constants C k , C k such that Then the analysis operator U E is continuous from α into s 1/(2α) , the synthesis operator T E is continuous from s 1/(2α) into α , and the frame operator T E U E is continuous from α into α .

Boundedness and Banach Frames Derived from Sub-exponential Localization of Frames
In this section we extend statements from [23] for polynomially and exponentially localized frames to the case of sub-exponentially localized frames (Theorem 5.4 below). We will use the Jaffard's theorem [27] given there for the sub-exponential and exponential case (see Theorem 5.2 below). First recall the Schur's test: If (A m,n ) m,n∈N is an infinite matrix satisfying sup m∈N n∈N |A m,n | ≤ K 1 and sup n∈N m∈N |A m,n | ≤ K 2 , then the corresponding matrix type operator A is well defined and bounded from p into p for 1 ≤ p ≤ ∞ and the operator norm Let β ∈ (0, 1] and γ ∈ (0, ∞). Define E γ,β to be the space of matrices (A m,n ) m,n∈N satisfying the following condition: By the Schur's test, when (A m,n ) m,n∈N ∈ E γ,β , then the corresponding matrix type operator A is well defined and bounded from 2 into 2 , and for the operator norm one has that A 2 → 2 ≤ 2C γ P γ,β , where C γ is the constant from (13) and P γ,β denotes the sum of the convergent series ∞ j=0 e −γ j β . We also need the following statements, an extension of [23, Lemmas 2 and 3] with a sketch of a proof in the spirit of that paper. It should be noted that the statements can be traced back to [18] and [24,Secs. 2,7]. 1 Lemma 5.1 For every γ ∈ (0, ∞) and β ∈ (0, 1], the following holds. (a) There exists a positive number C so that k∈N e −γ |m−k| β e −γ |k−n| β ≤ Ce −(γ /2)|m−n| β for every m, n ∈ N.
In "Appendix" we will give a sketch of the Jaffard's proof.
Here we consider the more general case intrinsically related to β ∈ (0, 1).

Theorem 5.4 Let p ∈ [1, ∞) and G be a Riesz basis for H, and let E be a frame for
H which is β-sub-exponentially or exponentially localized (respectively, (k + 1 + ε)localized for some ε > 0) with respect to G. Let μ be β μ -sub-exponential weight and let β μ < β in the case of β-sub-exponentially localized frame E (respectively, let μ be a k-moderate weight) with μ(n) ≥ 1 for every n ∈ N and p μ ⊂ 2 . Then for every p ∈ [1, ∞) the following statements hold. (vi) There is norm equivalence between f In the cases of polynomial and exponential localization, the assertions are given in [23,Prop. 8 and Prop. 10]. For the sub-exponential case, one can proceed in a similar way, but using Lemma 5.1 and Theorem 5.2. For the sake of completeness, we sketch a proof.
Let γ > 0 and C > 0 come from the sub-exponential localization of E with respect to G, i.e., max{| e m , g n |, | e m , g n |} ≤ Ce −γ |m−n| β , m, n ∈ N. Consider the matrix (A m,n ) m,n∈N determined by A m,n = e −γ |m−n| β for m, n ∈ N.
Therefore, ( f , e m ) ∞ m=1 also belongs to p μ and Then the series ∞ n=1 c n e n converges in H and let us denote its sum by y. Since A(|c n |) ∞ n=1 ∈ p μ by Lemma 5.1(b), and since | y, g m | ≤ C ∞ n=1 |c n || e n , g m | ≤ C ∞ n=1 A m,n |c n | for every m ∈ N, it follows that U G y ∈ p μ and therefore the element T G U G y = y belongs to H (iv) Use the operator V, determined in (iii), and observe that for m, n ∈ N we have e m , g n = ∞ j=1 e m , g j (V −1 ) jn and e m , g n = ∞ j=1 e m , g j (V −1 ) jn . Since V −1 ∈ E γ 1 ,β for some γ 1 ∈ (0, γ /2), one can apply Lemma 5.1(a) appropriately to conclude.
(v) follows from (iii) and for (vi) one can use the representations f = S −1 E S E = S E −1 S E f and the already proved (i)-(iv).

Expansions in Fréchet Spaces Via Localized Frames
Our goal is expansion of elements of a Fréchet space and its dual via localized frames and coefficients in a corresponding Fréchet sequence space. First we present in the next theorem general results related to frames localized with respect to a Riesz basis. In the next section, we will apply this theorem using frames localized with respect to the Hermite orthonormal basis in order to obtain frame expansions in the spaces S and α , α > 1/2, and their duals. To clarify notation, for an element e ∈ H, its corresponding element in H * by the Rieszs representation theorem will be denoted by the bold-style letter e. Note that in the setting of Lemma 2.2, for an element e from X 0 , one can conclude that e belongs to X * 0 ; thus, for e ∈ X F (⊆ X 0 ), we can consider e as an element of X * F . Theorem 6.1 Let G be a Riesz basis for H, k ∈ N 0 , and μ k be a β k -sub-exponential (resp. k-moderate) weight so that (5) holds. Let the spaces k and X k be as in Lemma 2.2. Assume that E = (e n ) ∞ n=1 is a sequence with elements in X F forming a frame for H which is β-sub-exponentially localized with β > β k for all k ∈ N 0 or exponentially localized (respectively, s-localized for every s ∈ N) with respect to G. Then, e n ∈ X F , n ∈ N, and the following statements hold:

(i) The analysis operator U E is F-bounded from X F into F , the synthesis operator T E is F-bounded from F into X F , and the frame operator S E is F-bounded and bijective from X F onto X F with unconditional convergence of the series in
f , e n e n = ∞ n=1 f , e n e n (with convergence in X F ) (15) with ( f , e n ) ∞ n=1 ∈ F and ( f , e n ) ∞ n=1 ∈ F . (iii) If X F and F have the following property with respect to (g n ) ∞ n=1 : then X F and F also have the properties P (e n ) and P ( e n ) .
(iv) Both sequences (e n ) ∞ n=1 and ( e n ) ∞ n=1 form Fréchet frames for X F with respect to F .
g(e n ) e n = ∞ n=1 g( e n ) e n (with convergence in X * F ) (16) with (g(e n )) ∞ n=1 ∈ * F and (g( e n )) ∞ n=1 ∈ * F . (vi) If (a n ) ∞ n=1 ∈ * F , then ∞ n=1 a n e n (resp. ∞ n=1 a n e n ) converges in X * F , i.e., the mapping f → ∞ n=1 f , e n a n (resp. f → ∞ n=1 f , e n a n ) determines a continuous linear functional on X F . Proof (i) The properties for U E , T E , and S E follow easily using Theorem 5.4(i)-(iii).
Further, the bijectivity of S E on X F implies that e n ∈ X F for every n ∈ N. (ii) By Theorem 5.4(v), for every k ∈ N and every f ∈ X k we have that f = ∞ n=1 f , e n e n = ∞ n=1 f , e n e n with convergence in X k . This implies that for every f ∈ X F , one has that f = ∞ n=1 f , e n e n = ∞ n=1 f , e n e n with convergence in X F . For every k ∈ N and every f ∈ X k , by Theorem 5.4(i), we have that ( f , e n ) ∞ n=1 ∈ k . Therefore, ( f , e n ) ∞ n=1 ∈ F for every f ∈ X F . Furthermore, by Theorem 5.4(iv), ( e n ) ∞ n=1 has the same type of localization with respect to G as (e n ) ∞ n=1 . Thus, applying Theorem 5.4(i) with ( e n ) ∞ n=1 as a starting frame, we get that ( f , e n ) ∞ n=1 ∈ F for f ∈ X F . (iii) If f ∈ X F , it is already proved in (i) that ( f , e n ) ∞ n=1 ∈ F and ( f , e n ) ∞ n=1 ∈ F . To complete the proof of P (e n ) , assume that f ∈ H is such that ( f , e n ) ∞ n=1 ∈ F . Consider Let k ∈ N. Since ( f , e j ) ∞ j=1 ∈ 2 μ k and by Theorem 5.4(iv), ( e n ) ∞ n=1 has the same type of localization with respect to G as (e n ) ∞ n=1 , it follows from Lemma 5.1(b) (for the case of sub-exponential localization) and from the way of the proof of [23, Lemma 3] (for the case of polynomial and exponential localization) that ( ∞ j=1 e j , g n f , e j ) ∞ n=1 ∈ 2 μ k . Therefore, ( f , g n ) ∞ n=1 ∈ F and thus, by P (g n ) , it follows that f ∈ X F . For completing the proof of P ( e n ) , if f ∈ H is such that ( f , e n ) ∞ n=1 ∈ F , it follows in a similar way as above that f ∈ X F . (iv) By (i), (e n ( f )) ∞ n=1 ∈ F for f ∈ X F , and by Theorem 5.4(vi), for k ∈ N and f ∈ X k , the norms |( f , e n ) ∞ n=1 | k and f X k are equivalent. Furthermore, it follows from Theorem 5.4 that the operator V := S −1 E T E | F maps F into X F and it is F-bounded. Clearly, V (e n ( f )) ∞ n=1 = f , f ∈ X F . Therefore, (e n ) ∞ n=1 is an F-frame for X F with respect to F . In an analogue way, ( e n ) ∞ n=1 is also an F-frame for X F with respect to F .
(v) The representations in (i) can be re-written as f = ∞ n=1 e n ( f )e n = ∞ n=1 e n ( f ) e n , f ∈ X F , which implies validity of (16) for g ∈ X * F . For the rest of the proof, consider the F-bounded (and hence continuous) operator V from the proof of (iv) and observe that e n = V δ n , n ∈ N. This implies that for g ∈ X * F we have (g( e n )) ∞ n=1 = (gV (δ n )) ∞ n=1 ∈ F . With similar arguments, considering the operator V = S −1 E T E | F , it follows that (g(e n )) ∞ n=1 ∈ F . (vi) Let (a n ) ∞ n=1 ∈ * F and thus there is k 0 ∈ N so that (a n ) ∞ n=1 ∈ * k 0 , i.e., C := ∞ n=1 |a n | 2 |μ k 0 (n)| −2 < ∞. By Theorem 5.4(vi), there is a positive constant Therefore, ∞ n=1 f , e n a n converges and furthermore, which implies continuity of the linear mapping f → ∞ n=1 f , e n a n . In a similar way, it follows that f → ∞ n=1 f , e n a n determines a continuous linear functional on X F .

Remark 6.2
Note that in the setting of the above theorem, when G is an orthonormal basis of H or more generally, when G is a Riesz basis for H satisfying any of the following two conditions: (P 1 ): ∀s ∈ N ∃C s > 0 : | g m , g n | ≤ C s (1 + |m − n|) −s , m, n ∈ N, (P 2 ): ∃s > 0 ∃C s > 0 : | g m , g n | ≤ C s e −s|m−n| , m, n ∈ N, then the property P (g n ) is satisfied.

Frame Expansions of Tempered Distributions and Ultradistributions
Here we apply Theorem 6.1 to obtain series expansions in the spaces S and α (α > 1/2), and their dual spaces, via frames which are localized with respect to the Hermite basis. Theorem 6.3 Assume that the sequence (e n ) ∞ n=1 with elements from S(R) (resp. in α ) is a frame for L 2 (R) which is polynomially (resp. sub-exponentially or exponentially) localized with respect to the Hermite basis (h n ) ∞ n=1 with decay γ for every γ ∈ N. Let (g n ) ∞ n=1 = (h n ) ∞ n=1 . Then P (g n ) and the conclusions in Theorem 6.1 hold with X F replaced by S (resp. α ) and F replaced by s (resp. s 1/(2α) ).
Proof For k ∈ N 0 , consider the k-moderate weight μ k (x) = (1 + |x|) k . The spaces k := 2 μ k , k ∈ N 0 , satisfy (1)-(2) and their projective limit F is the space s. Consider the spaces X k := H k (h n ) , k ∈ N 0 , which satisfy (1)- (2). As observed after Theorem 6.1, the property P (h n ) is satisfied. Since for f ∈ L 2 (R) one has that f ∈ S if and only if ( f , h n ∞ n=1 ) ∈ s, it now follows that X F = S. Then the conclusions of Theorem 6.3 follow from Theorem 6.1.
Now we give some details for the Jaffard's proof of Theorem 5.2, providing explicit estimates for the bounds.
Denote by n 0 the highest natural number such that n 0 ∈ |m−n| . Then For n > n 0 we have r n < r n 0 = e n 0 ln r and hence, Therefore, ∞ k=0 |(R k ) m,n | ≤ e −γ 1 |m−n| β 1 + r 1 − r · 1 2P + 1 1 − r . Now using the representation A −1 = A * (A A * ) −1 and Lemma 7.1, we can conclude that where C A is a positive constant such that |A m,n | ≤ C A e −γ |m−n| β for m, n ∈ N.