https://doi.org/10.1007/s40840-020-01070-y
Localization of Fréchet Frames and Expansion of Generalized Functions
Stevan Pilipovi´c1·Diana T. Stoeva2,3
Received: 30 September 2019 / Revised: 4 December 2020 / Accepted: 21 December 2020 / Published online: 23 February 2021
© The Author(s) 2021
Abstract
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 ultradistri- butions.
Keyword Localized frame·Banach frame·Fréchet frame·Tempered distributions· Ultradistributions·Frame expansions
Mathematics Subject Classification 42C15·46A13· 46B15·46F05
1 Introduction, Motivation and Main Aims
Localized frames were introduced independently by Gröchenig [23] and Balan, Casazza, Heil, and Landau [2,3]. The localization conditions in [23] are related to off-diagonal decay (of polynomial or exponential type) of the matrix determined by the inner products of the frame elements and the elements of a given Riesz basis. A
Communicated by Rosihan M. Ali.
B
Diana T. Stoevadiana.stoeva@univie.ac.at; dstoeva@kfs.oeaw.ac.at Stevan Pilipovi´c
pilipovic@dmi.uns.ac.rs
1 Faculty of Sciences, Department of Mathematics and Informatics, University of Novi Sad, Novi Sad, Serbia
2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, Vienna 1090, Austria 3 Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, Vienna 1040,
Austria
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 appli- cations of localized frames were considered. The localization and self-localization, considered independently in [1–3], 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. [31–34]).
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 Section5have an essential role in our investi- gations. 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 impor- tant impact in applications is related to the Hermite basis which is almost always used for the global expansion ofL2-functions or tempered generalized functions overRn. 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 Sections3 and4the 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 Section2the notation, basic defini- tions, and the needed known results. In Section3, we consider matrices with column decay and possible row increase. For such type of matrices, we obtain in Section4 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 Section5. The use of Jaffard’s Theorem and [23, Theorems 11 and 13] is intrinsically connected with the sub-exponential local- ization. Section6is devoted to Fréchet frames and series expansions in certain classes of Fréchet spaces based on polynomial, exponential, and sub-exponential localiza- tion. In particular, we obtain frame expansions in the Schwartz spaceS 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 basishn,n∈N, and construct a Riesz basis which is polynomially and exponentially localized tohn,n ∈ N. Finally, in “Appendix”, we add some details in the proof of the Jaffard’s theorem.
2 Notation, Definitions, and Preliminaries
Throughout the paper,(H,·,·)denotes a separable Hilbert space andG(resp. E) denotes the sequence(gn)∞n=1(resp.(en)∞n=1) with elements fromH. Recall thatGis called:
– frame forH[15] if there exist positive constantsAandB (calledframe bounds) so thatAf2≤∞
n=1|f,gn|2≤Bf2for every f ∈H;
– Riesz basis forH[4] if its elements are the images of the elements of an orthonor- mal basis under a bounded bijective operator onH.
Recall (see, e.g., [12]), ifGis a frame forH, then there exists a frame(fn)∞n=1for Hso that
f = ∞ n=1
f, fngn= ∞ n=1
f,gnfn, f ∈H.
Such(fn)∞n=1is called adual frame of(gn)∞n=1. Furthermore, theanalysis operator UG, given byUGf =(f,gn)∞n=1, is bounded fromHinto2; thesynthesis operator TG, given byTGf = ∞
n=1cngn, is bounded from2 into H; theframe operator SG := TGUG is a bounded bijection ofHontoHwith unconditional convergence of the series SGf =∞
n=1f,gngn. The sequence(SG−1gn)∞n=1is a dual frame of (gn)∞n=1, called thecanonical dual of (gn)∞n=1, and it will be denoted by(gn)∞n=1or G. When Gis a Riesz basis ofH(and thus a frame forH), then onlyGis a dual frame ofG, it is the unique biorthogonal sequence toG, and it is also a Riesz basis forH. A frameGwhich is not a Riesz basis has other dual frames in addition to the canonical dual and in that case we use notationGdor(gnd)∞n=1for a dual frame ofG.
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, thenis called aC B-space. AC B- space is clearly aB K-space.
Given aB K-spaceand a frameGforHwith a dual frameGd=(gnd)∞n=1, one associates withthe Banach space
HG,Gd :=
f ∈H : (f,gnd)∞n=1∈,fH
G,Gd := |(f,gnd)∞n=1|
.
WhenGis a Riesz basis forH, then we use notationHG forHG,G. 2.1 Localization of Frames
In this paper, we consider polynomially and exponentially localized frames in the way defined in [23], and furthermore, sub-exponential localization. LetGbe a Riesz basis for the Hilbert spaceH. A frameEforHis called:
– polynomially localized with respect to G with decayγ >0 (in short,γ-localized wrt(gn)∞n=1) if there is a constantCγ >0 so that
max{|em,gn|,|em,gn|} ≤Cγ(1+ |m−n|)−γ, m,n ∈N;
– exponentially localized with respect to Gif for someγ > 0 there is a constant Cγ >0 so that
max{|em,gn|,|em,gn|} ≤Cγe−γ|m−n|, m,n∈N.
– β-sub-exponentially localized with respect to G(forβ∈(0,1)) if for someγ >0 there isCγ >0 so that
max{|em,gn|,|em,gn|} ≤Cγe−γ|m−n|β, m,n∈N.
2.2 Fréchet Frames
We consider Fréchet spaces which are projective limits of Banach spaces as follows.
Let{Yk,| · |k}k∈N0 be a sequence of separable Banach spaces such that
{0} = ∩k∈N0Yk ⊆. . .⊆Y2⊆Y1⊆Y0, | · |0≤ | · |1≤ | · |2≤. . . (1) YF := ∩k∈N0Yk is dense inYk, k∈N0(N0=N∪ {0}). (2) Under the conditions (1)–(2),YFis a Fréchet space andYF∗is the inductive limit of the spacesYk∗,k∈ N. We will use such type of sequences in two cases: 1.Yk =Xk
with norm · k,k∈N0;2.Yk =kwith norm|·|k,k∈N0.
Let{k,|·|k}k∈N0 be a sequence ofC B-spaces satisfying (1). Then (2) holds, because every sequence (cn)∞n=1 ∈ F can be written as∞
n=1cnδn with the con- vergence inF, whereδn denotes then-th canonical vector,n ∈ N. Furthermore, ∗F can be identified with the sequence spaceF := {(Uδn)∞n=1 : U ∈∗F}with convergence naturally defined in correspondence with the convergence in∗F.
We use the termoperatorfor a linear mapping, and byinvertible operator on X, we mean a bounded bijective operator on X. Given sequences of Banach spaces, {Xk}k∈N0and{k}k∈N0, which satisfy (1)-(2), an operatorT :F →XFis calledF - boundedif for everyk∈N0, there exists a constantCk >0 such thatT(cn)∞n=1k ≤ Ck|(cn)∞n=1|kfor all(cn)∞n=1∈F.
Definition 2.1 [34] Let{Xk, · k}k∈N0 be a sequence of Banach spaces satisfying (1)-(2) and let {k,|·|k}k∈N0 be a sequence of B K-spaces satisfying (1)-(2). A sequence(φn)∞n=1with elements fromX∗Fis called aGeneral Fréchet frame(in short, General F -frame) forXFwith respect toFif there exist sequences{sk}k∈N0 ⊆N0, {sk}k∈N0 ⊆ N0, which increase to∞with the propertysk ≤sk,k ∈ N0, and there exist constants 0<Ak≤ Bk<∞,k∈N0, satisfying
(φn(f))∞n=1∈F, f∈XF, (3)
Akfsk ≤ |(φn(f))∞n=1|k≤ Bkfsk, f∈XF,k∈N0, (4) and there exists a continuous operatorV :F → XF so that V((φn(f))∞n=1)= f for every f∈XF.
Whensk =sk = k,k ∈ N0, and the continuity ofV is replaced by the stronger condition ofF-boundedness ofV, then the above definition reduces to the definition of aFréchet frame(in short,F -frame) forXFwith respect toFintroduced in [32].
Although we will use in the sequel this simplified definition, Definition2.1is the most general one, interesting in itself, and can be considered as a non-trivial generalization of Banach frames.
In the particular case whenXk =X, andk =,k∈N0, a Fréchet frame forXF
with respect toF becomes aBanach 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.
2.3 Sequence and Function Spaces
Recall that a positive continuous functionμonRis called: ak-moderate weightif k≥0 and there exists a constantC >0 so thatμ(t+x)≤C(1+|t|)kμ(x),t,x∈R;
aβ-sub-exponential(resp.exponential) weight, ifβ ∈(0,1)(resp.β =1) and there exist constants C > 0, γ > 0, so that μ(t +x) ≤ Ceγ|t|βμ(x), t,x ∈ R. If β is clear from the context, we will write justsub-exponential weight. Letμ be ak- moderate, sub-exponential, or exponential weight so thatμ(n)≥1 for everyn ∈N, andp∈ [1,∞). Then the Banach space
μp:=
⎧⎨
⎩(an)∞n=1 : |(an)n|p,μ:=
∞
n=1
|an|pμ(n)p 1/p
<∞
⎫⎬
⎭
is aC 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 forHand let Gd=(gnd)∞n=1be a dual frame of G. Let μk be k-moderate (resp. sub-exponential or exponential) weights, k∈N0,so that
1=μ0(x)≤μ1(x)≤μ2(x)≤..., for every x∈R. (5) Then the spacesk :=2μk, k ∈N0, satisfy(1)–(2). Denote M := {(f,gdn)∞n=1 : f ∈H}. The assumption that M∩Fis dense in M ∩k = {0}with respect to the
|·|k-norm for every k ∈ N, leads to the conclusion that the spaces Xk :=HGk,Gd, k∈N, satisfy(1)–(2).
If G is a Riesz basis forH, then the density assumption of M∩Fin M∩k= {0}, k∈N, is fulfilled and in addition one has that gn∈ XF for every n∈N.
Throughout the paper, we also consider specific weights, relevant to the function spaces of interest and the corresponding sequence spaces. Letk ∈N0andμk(x)= (1+ |x|)k (resp.μk(x)=ek|x|β, β∈(0,1]),x ∈R. Then, withk :=μpk,k∈N0,
the projective limit∩k∈N0kis the spacesof rapidly (resp.sβ of sub-exponentially whenβ < 1 and exponentially whenβ = 1) decreasing sequences determined by {(an)∞n=1 ∈ CN : (∞
n=1|anμk(n))|p)1/p <∞, ∀k ∈ N0}, which is the same set for anyp ∈ [1,∞). The spaces(resp.sβ) can also be derived as the projective limit of the Banach spacessk (resp.sβk) defined as{(an)∞n=1 ∈CN : |(an)∞n=1|sup,k :=
supn∈N|an|nk <∞}, (resp.|·|βsup,k :=supn∈N|an|eknβ <∞}),k ∈ N0; note that here instead ofk∈ {0,1,2,3, . . .}one can also use any strictly increasing sequence of nonnegative numbersk∈ {0,q1,q2,q3, . . .}.
Recall that the well-known Schwartz spaceSis the intersection of Banach spaces Sk(R):=
f ∈L2(R): ||f||k = k m=0
||(1+ | · |2)k/2f(m)||L2(R)
,k∈N.
The dualS(R)is the space of tempered distributions.
The space of sub-exponentially decreasing functions of order 1/α,α >1/2,isα :=
XF = ∩k∈N0k,αwherek,αare Banach spaces ofL2−functions with finite norms
||f||αk = sup
n∈N0
knek|x|1/α|f(n)(x)|
n!α
L2(R)
<∞,k∈N.
Its dual(α(R))is the space of Beurling tempered ultradistributions, cf. [19,30].
Remark 2.3 The caseα=1/2 leads to the trivial space1/2= {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 forα <1/2 (cf. [10,11,19,29]). We will not treat these cases in the current paper.
In the sequel,(hn)∞n=1is the Hermite orthonormal basis ofL2(R)re-indexed from 1 to∞instead of from 0 to∞. Recall thathn ∈α,α >1/2,n ∈ N. Moreover, we know [36]:
– If f ∈ S, then(f,hn)∞n=1 ∈ s; conversely, if (an)∞n=1 ∈ s, then∞
n=1anhn
converges inSto some f with(f,hn)∞n=1=(an)∞n=1. – IfF ∈S, then(bn)∞n=1:=(F(hn))∞n=1∈sandF(f)=∞
n=1f,hnbn, f ∈S;
conversely, if(bn)∞n=1 ∈s, then the mapping F : f → ∞
n=1f,hnbnis well defined onS, it determinesFas an element ofSand(F(hn))∞n=1=(bn)∞n=1. The above two statements also hold whenS,S,s, andsare replaced byα,(α), s1/(2α), and(s1/(2α))withα >1/2, respectively [19,29,30].
We can considerS andα as the projective limit of Hilbert spacesHk,k ∈ N0, with elements f =
nanhn,in the first case with norms fHk := |(annk)n|2 <∞},k∈N0, and in the second case with norms
fHk := |(anekn1/(2α))n|2 <∞},k∈N0.
Thus,(hn)nis an F-frame forS(R)with respect tosas well as anF-frame forα with respect tos1/2α,α >1/2, (F- boundedness is trivial).
3 Matrix-Type Operators
Papers [14,21,23] concern matrices with off-diagonal decay of the form: for some γ >0 there isCγ >0 such that
|Am,n| ≤ Cγ
(1+ |m−n|)γ (resp.|Am,n| ≤Cγe−γ|m−n|), ∀n,m∈N. (6) As we noted in the introduction, matrix operators in this section and the next one are not essentially related to Sections5and6. But they significantly illuminate such operators in our main results. Moreover, we refer in Section4to results of Section3 and in Remark6.4we refer to Section4.
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 Propositions3.2and3.6) 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(Amn)m,n∈N, the letterAwill denote the mapping (cn)∞n=1 → (am)∞m=1determined by am = ∞
n=1Am,ncn (assuming convergence), m∈N; conversely, for a given mappingAdetermined on a sequence space containing the canonical vectorsδn,n ∈ N, the corresponding matrix(Amn)m,n∈N is given by Am,n = Aδn, δm. We will sometimes useAwith the meaning of(Amn)m,n∈N and vice-verse.
3.1 Polynomial-Type Conditions
Let us begin with some comparison of polynomial type of off-diagonal decay:
Lemma 3.1 Letγ >0. Consider the following conditions:
(∗) |Am,n| ≤C
(1+m)γ
(1+n)2γ, n≥m,
(1+n)γ
(1+m)2γ,n≤m, for some C>0.
(∗∗) |Am,n| ≤C(1+ |n−m|)−γ, for some C>0.
(∗ ∗ ∗) |Am,n| ≤C mγ
nγ,n≥m,
nγ
mγ,n≤m, for some C>0.
Then, the implications(∗)⇒(∗∗)⇒(∗ ∗ ∗)hold. The converse implications are not valid.
Proof Implications(∗)⇒(∗∗)⇒(∗ ∗ ∗)follow from the inequalities ((11++maxmi n((mm,,nn))))2γγ
≤(1+ |n−m|)−γ ≤ ((maxmi n((mm,,nn))))γγ,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 Am,n which satisfies|Am,n| = Cnmγγ, n ≤ m, for someγ > 0 and some positive constantC,and assume that there existγ1(γ ) ∈ Nand a positive constant K so that form ≥ n one has Cnmγγ ≤ (1+mK−n)γ1; then takingm = 2n, one obtains 0<C·2−γ ≤ (1+Kn)γ1 →0 asn→ ∞, 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(Amn)m,n∈Nsatisfies the condition
|Am,n| ≤
C0nγ0, n>m, C1nγ1m−γ1,n≤m,
for someγ0 ≥ 0, γ1 > 0,C0 >0,C1 >0. ThenAis a continuous operator from sγ1+γ0+1+εintosγ1for anyε∈(0,1].
Proof Letε∈(0,1]and let(cn)∞n=1∈sγ0+γ1+1+ε. For everyn>m,
|Am,ncn| ≤C0|cn|nγ0 ≤C0
sup
j
(|cj|jγ0+γ1+1+ε)
1 nγ1+1+ε. Next,
| m n=1
Am,ncn| ≤C1m−γ1 m n=1
|cn|nγ1
≤C1m−γ1|(cn)∞n=1|sup,γ0+γ1+1+ε m n=1
1 nγ0+1+ε. Therefore,
|am| ≤ | m n=1
Am,ncn| + | ∞
n=m+1
Am,ncn|
≤C1m−γ1|(cn)∞n=1|sup,γ0+γ1+1+ε∞
n=1
1 nγ0+1+ε
+C0|(cn)∞n=1|sup,γ0+γ1+1+ε ∞
n=m+1
1 nγ1+1+ε.
Since∞
n=m+1 1
nγ1+1+ε ≤m−γ1∞
n=m+1 1
n1+ε,the assertion follows.
A direct consequence of Proposition3.2is:
Corollary 3.3 Assume that the matrix(Amn)m,n∈N satisfies: there existγ0 ≥ 0 and C0>0, and for everyγ >0there is Cγ >0so that
|Am,n| ≤
C0nγ0, n >m Cγnγm−γ,n ≤m.
ThenAis a continuous operator fromsintos.
In order to determineAas a mapping from a spacesγ1 into the same space, we have to change the decay condition.
Proposition 3.4 Let(Amn)m,n∈Nsatisfy:
(∃ε >0, γ1∈N)(∃C0>0,C1>0)such that
|Am,n| ≤
C0n−1−ε, n>m,
C1nγ1m−γ1−1−ε,n≤m. (7) ThenAis a continuous operator fromsγ1 intosγ1.
Remark 3.5 For the same conclusion as above, one has in [23] another condition non- comparable to (7):
|Am,n| ≤C(1+ |n−m|)−γ1−1−ε. (8) 3.2 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 whileβ ∈ (0,1)corresponds to the pure sub-exponential growth order.
Proposition 3.6 Assume that the matrix(Amn)m,n∈N satisfies the condition: There exist positive constants C0,C1andγ0≥0,γ1>0, so that
|Am,n| ≤
C0eγ0nβ, n>m,
C1e−γ1(mβ−nβ),n≤m. (9) ThenAis a continuous operator fromsβγ1+γ0+εintosβγ1 for anyε∈(0,1).
Proof Letε∈(0,1)and let(cn)∞n=1∈sβγ1+γ0+ε. Then forn >m,
|Am,ncn| ≤C0|cn|eγ0nβ ≤C0(sup
j |cj|e(γ1+γ0+ε)jβ)e−(γ1+ε)nβ.
Further on,
| m n=1
Am,ncn| ≤C1
m n=1
|cn|eγ1(nβ−mβ)
≤C1e−γ1mβ(sup
j∈N|cj|e(γ1+γ0+ε)jβ) m n=1
e−(γ0+ε)nβ.
Therefore,
|am| ≤e−γ1mβ
C1
∞ n=1
e−(γ0+ε)nβ +C0
∞ n=1
e−εnβ
|(cn)n|βsup,γ1+γ0+ε.
This completes the proof.
Remark 3.7 Sincee−γ (m−n)β ≤e−γ (mβ−nβ) forn ≤m(β ∈(0,1], γ ∈(0,∞)), in (9) we considere−γ (mβ−nβ)instead ofe−γ (m−n)β.
As a consequence of Proposition3.6, we have:
Corollary 3.8 Assume that the matrix(Amn)m,n∈Nsatisfies the condition: There exist constants C0>0andγ0≥0, and for everyγ >0, there is a positive constant Cγ so that
|Am,n| ≤
C0eγ0nβ, n>m, Cγeγ (nβ−mβ),n≤m.
ThenAis a continuous operator fromsβ intosβ.
Proposition 3.9 Let(Amn)m,n∈Nsatisfy the condition: There exist positive constants ε, γ1,C0,C1, so that
|Am,n| ≤
C0e−εnβ, n >m, C1eγ1nβe−(γ1+ε)mβ,n ≤m.
ThenAis a continuous operator fromsβγ1 intosβγ1.
Remark 3.10 One can simply show that the assumption|Am,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.
4 Continuity of the Frame-Related Operators Under Relaxed “Decay”
Conditions
We now determine weaker localization conditions which are still sufficient to imply continuity of the frame-related operators.
Proposition 4.1 Let G be a frame forH, Gd be a dual frame of G, andμk(x) = (1+ |x|)k, k∈N0. Under the notations in Lemma2.2, assume that M∩F is dense in M∩k = {0}with respect to the|·|k-norm for every k∈Nand let E =(en)∞n=1 be a sequence with elements from XF which is a frame forH. Then the following statements hold.
(i) Assume that there exist s0∈N, C>0and for every k∈Nthere exists Ck >0 such that
|em,gn| ≤
Cns0, n>m, Cknkm−k,n≤m.
Then the analysis operator f →UEf =(f,em)∞m=1is continuous from XF
intos.
(ii) Assume that there exists0∈N0,C>0and for every k∈Nthere existsCk >0 such that
|em,gdn| ≤ Cms0, m>n, Ckmkn−k,m≤n.
Then the synthesis operator(cn)n→TE(cn)=
cnenis continuous fromsinto XF.
(iii) Under the assumptions of(i)and(ii), the frame operator TEUE is continuous from XF into XF.
Proof Note that under the given assumptions,F is the spaces.
(i) Let Am,n = gn,em,m,n ∈ N, andAbe the corresponding operator for the matrixA. Let f ∈ XF. Then(f,gdn)∞n=1∈sand
A(f,gdn)∞n=1= ∞
n=1
gn,emf,gnd ∞
m=1
=(f,em)∞m=1.
By Corollary3.3, it follows that(f,em)∞n=1∈s. Furthermore, by Proposition 3.2, for everyk∈N, there is a constantKs0,k,C,Ck so that
|(f,em)m|sup,k = |A(f,gnd)n|sup,k≤ Ks0,k,C,Ck|(f,gnd)n|sup,s0+k+2
≤ Ks0,k,C,Ck|(f,gdn)n|s0+k+2 =Ks0,k,C,Ckfs0+k+2. Therefore, the analysis operatorUE is continuous fromXFintos.
(ii) Let (cn) ∈ s. First we show that∞
n=1cnen converges in XF and then the continuity of TE. Since(cn)∞n=1 ∈ 2, we have x =
ncnen ∈ H. Denote Am,n= en,gdmand consider the corresponding operatorA. Then(x,gmd)m= (
nAm,ncn)m =A(cn)∈s(by Corollary3.3), which implies thatx∈ XF, and furthermore, for everyk∈N, one hasTE(cn)nk = xk= |(x,gdm)m|k.
For everyk∈N, there is a constantRksuch that|(dn)|k ≤Rk|(dn)|sup,k+2
for every(dn)∈sk+2. By Proposition3.2, we conclude that
TE(cn)nk ≤ Rk|(x,gdm)m|sup,k+2=Rk|A(cn)|sup,k+2
≤ RkK(s0,k,C,Ck)|(cn)|sup,s0+k+4.
Thus, the synthesis operatorTE is well defined and continuous fromsintoXF.
(iii) follows from (i) and (ii).
It is of interest to consider the case whenXFisS.
Corollary 4.2 Let(en)∞n=1be a frame of L2(R)with elements inS(R). Assume that for every k∈Nthere are constants Ck,Cksuch that
|em,hn| ≤
Ckmkn−k,n>m,
Cknkm−k,n≤m. (10) Then the analysis operator UEis continuous fromSintos, the synthesis operator TE
is continuous fromsintoS, and the frame operator TEUE is continuous fromSinto S.
Now, we consider sub-exponential weights.
Proposition 4.3 Letβ ∈(0,1)and let the assumptions of the first part of Lemma2.2 hold with the weightsμk(x)=ek|x|β, k ∈N0. Let E =(en)∞n=1be a sequence with elements from XFwhich is a frame forH. Then the following statements hold.
(i) Assume that there exist constantsγ0∈N, C>0such that for every k∈Nthere exists Ck >0such that
|em,gn| ≤
Ceγ0nβ, n>m,
Ckek(nβ−mβ),n≤m,k∈N. (11) Then the analysis operator f →UEf =(f,em)∞m=1is continuous from XF
intosβ.
(ii) Assume that there exist constantsγ˜0∈N,C˜ >0such that for every k∈Nthere existsC˜k >0such that
|em,gdn| ≤
Ce˜ γ˜0mβ, m>n,
C˜kek(mβ−nβ),m≤n. (12)
Then the synthesis operator(cn)n → TE(cn)=
cnenis continuous fromsβ into XF.
(iii) If(11)and(12)hold, then the frame operator TEUEis continuous from XFinto XF.
