On Some Sampling-Related Frames in 𝑈-Invariant Spaces

Abstract

This paper is concerned with the characterization as frames of some sequences in 𝑈-invariant spaces of a separable Hilbert space ℋ where 𝑈 denotes an unitary operator defined on ℋ; besides, the dual frames having the same form are also found. This general setting includes, in particular, shift-invariant or modulation-invariant subspaces in 𝐿²(ℝ), where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In doing so, we need the unitary operator 𝑈 to belong to a continuous group of unitary operators.

1. Introduction

This paper is concerned with the study of some special frames in 𝑈-invariant spaces. Given an unitary operator 𝑈 on a separable Hilbert space ℋ, we consider closed subspaces having the form 𝒜_𝑎∶=span{𝑈^𝑛𝑎∶𝑛∈ℤ}, where 𝑎 denotes some fixed element in ℋ. In case that the sequence {𝑈^𝑛𝑎}_𝑛∈ℤ is a Riesz basis for 𝒜_𝑎, we have 𝒜_𝑎=_𝑛∈ℤ𝛼_𝑛𝑈^𝑛𝑎∶𝛼_𝑛∈ℓ²(ℤ). (1)

Recall that a Riesz basis in a separable Hilbert space is the image of an orthonormal basis by means of a bounded invertible operator. Any Riesz basis {𝑥_𝑛}_𝑛∈ℤ has a unique biorthogonal (dual) Riesz basis {𝑦_𝑛}_𝑛∈ℤ, that is, ⟨𝑥_𝑛,𝑦_𝑚⟩_ℋ=𝛿_𝑛,𝑚, such that the expansions 𝑥=_𝑛∈ℤ𝑥,𝑦_𝑛ℋ𝑥_𝑛=_𝑛∈ℤ𝑥,𝑥_𝑛ℋ𝑦_𝑛 (2) hold for every 𝑥∈ℋ. We state the definition by considering the integers set ℤ as the index set since throughout the paper most of sequences are indexed in ℤ. A Riesz sequence in ℋ is a Riesz basis for its closed span. A necessary and sufficient condition in order for the sequence {𝑈^𝑛𝑎}_𝑛∈ℤ to be a Riesz sequence in ℋ is given in Theorem 3 infra.

Given 𝑠 elements 𝑏_𝑗, 𝑗=1,2,…,𝑠, in 𝒜_𝑎, a challenging problem is to characterize the sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} as a frame (Riesz basis) in 𝒜_𝑎, where 𝑟≥1 denotes a positive integer. Besides, another interesting problem is to look for those dual frames having the same form {𝑈^𝑟𝑘𝑐_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} for some 𝑐_𝑗∈𝒜_𝑎, 𝑗=1,2,…,𝑠, so that, for any 𝑥∈𝒜_𝑎 the expansion 𝑥=^𝑠_{𝑗=1𝑘∈ℤ}𝑥,𝑈^𝑟𝑘𝑏_𝑗𝑈^𝑟𝑘𝑐_𝑗inℋ (3) holds.

At this point, we give an explanation about the expression sampling-related frames appearing in the title. Namely, 𝑈-invariant subspaces in ℋ are natural generalizations of shift-invariant or modulation-invariant subspaces of 𝐿²(ℝ); there, the unitary involved operators are, respectively, the translation operator 𝑇∶𝑓(𝑡)↦𝑓(𝑡−1) or the modulation operator 𝑀∶𝑓(𝑡)↦e^2𝜋i𝑡𝑓(𝑡). In the shift-invariant subspace 𝑉²_𝜑={∑_𝑛𝛼_𝑛𝜑(𝑡−𝑛)∶{𝛼_𝑛}∈ℓ²(ℤ)}⊂𝐿²(ℝ) generated by 𝜑∈𝐿²(ℝ), for any 𝑓∈𝑉²_𝜑, the inner products {⟨𝑥,𝑈^𝑟𝑘𝑏_𝑗⟩}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} are ^∞_−∞𝑓(𝑡)𝑏_𝑗(𝑡−𝑟𝑘)𝑑𝑡=𝑓∗ℎ_𝑗(𝑟𝑘),𝑘∈ℤ,𝑗=1,2,…,𝑠, (4) where ℎ_𝑗(𝑡)∶=𝑏_𝑗(−𝑡) for each 𝑗=1,2,…,𝑠. Thus, the above inner products are nothing but samples of some filtered versions 𝑓∗ℎ_𝑗 of the function 𝑓 itself; this is precisely the generalized sampling problem in the shift-invariant space 𝑉²_𝜑. Mathematically, it consists of the stable recovery of any 𝑓∈𝑉²_𝜑 from the above sequence of samples, that is, to obtain sampling formulas in 𝑉²_𝜑 having the following form: 𝑓(𝑡)=^𝑠_{𝑗=1𝑘∈ℤ}𝑓∗ℎ_𝑗(𝑟𝑘)𝑆_𝑗(𝑡−𝑟𝑘),𝑡∈ℝ, (5) such that the sequence of reconstruction functions {𝑆_𝑗(⋅−𝑟𝑘)}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a frame for the shift-invariant space 𝑉²_𝜑. As a consequence, expansions (3) and (5) have the same nature. Recall that a sequence {𝑥_𝑛}_𝑛∈ℤ is a frame for a separable Hilbert space ℋ if there exist constants 𝐴,𝐵>0 (frame bounds), such that 𝐴‖𝑥‖²≤_𝑛∈ℤ𝑥,𝑥_𝑛²≤𝐵‖𝑥‖²∀𝑥∈ℋ. (6)

A sequence {𝑥_𝑛}_𝑛∈ℤ in ℋ satisfying only the right hand inequality above is said to be a Bessel sequence for ℋ. Given a frame {𝑥_𝑛}_𝑛∈ℤ for ℋ, the representation property of any vector 𝑥∈ℋ as a series 𝑥=∑_𝑛∈ℤ𝑐_𝑛𝑥_𝑛 is retained, but, unlike the case of Riesz bases (exact frames), the uniqueness of this representation (for overcomplete frames) is sacrificed. Suitable frame coefficients 𝑐_𝑛 which depend continuously and linearly on 𝑥 are obtained by using the dual frames {𝑦_𝑛}_𝑛∈ℤ of {𝑥_𝑛}_𝑛∈ℤ; that is, {𝑦_𝑛}_𝑛∈ℤ is another frame for ℋ such that 𝑥=∑_𝑛∈ℤ⟨𝑥,𝑦_𝑛⟩𝑥_𝑛=∑_𝑛∈ℤ⟨𝑥,𝑥_𝑛⟩𝑦_𝑛 for each 𝑥∈ℋ. For more details on the frame theory see [1].

Sampling in shift-invariant spaces of 𝐿²(ℝ) has been profusely treated in the mathematical literature (see, for instance, [2–13]).

The existence of expansions like (3) in 𝑈-invariant subspaces was treated for the first time in [14]; see also [15, 16]. Following similar techniques to those in [14], we give, in Section 2, a complete characterization in 𝒜_𝑎 of sequences having the form {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} where 𝑏_𝑗∈𝒜_𝑎 for each 𝑗=1,2,…,𝑠. In other words, we carry out the study of the completeness, Bessel, frame, or Riesz basis properties of the sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠}. Whenever it is a frame for 𝒜_𝑎, we find a family of dual frames having the same form {𝑈^𝑟𝑘𝑐_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} for some 𝑐_𝑗∈𝒜_𝑎, 𝑗=1,2,…,𝑠. In Section 4, we also discuss the case where some 𝑏_𝑗∉𝒜_𝑎; although the sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is not contained in 𝒜_𝑎, something can be said in the light of the theory of pseudoframes (see [17, 18]).

All the obtained results in Section 2 involve the discrete group of unitary operators {𝑈^𝑛}_𝑛∈ℤ which is completely determined by 𝑈. If we want to deal with something similar to the time-jitter error version of (5), that is, the recovery of any 𝑓∈𝑉²_𝜑 from a perturbed sequence of samples {(𝑓∗ℎ_𝑗)(𝑟𝑘+𝜖_𝑘𝑗)}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} with errors 𝜖_𝑘𝑗∈ℝ, then the availability of a continuous group of unitary operators {𝑈^𝑡}_𝑡∈ℝ containing, in particular, the operator 𝑈 (say for instance 𝑈=𝑈¹) becomes essential. In Section 3, after a brief on groups of unitary operators, we deal with two types of perturbation problems. The first one concerns the study of sequences as {𝑈^{𝑟𝑘+𝜖𝑘𝑗}𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} in ℋ, for small enough error sequence 𝝐∶={𝜖_𝑘𝑗}ℓ²-norm, the sequence is a Riesz sequence in ℋ. The second one goes into the recovery of any 𝑥∈𝒜_𝑎 from the perturbed sequence of inner products {⟨𝑥,𝑈^{𝑟𝑘+𝜖𝑘𝑗}𝑏_𝑗⟩_ℋ}_{𝑘∈ℤ;𝑗=1,2,…,𝑠}, for small enough errors 𝜖_𝑚𝑗, there exists a frame expansion for 𝑥∈𝒜_𝑎 having the inner products {⟨𝑥,𝑈^{𝑟𝑘+𝜖𝑘𝑗}𝑏_𝑗⟩_ℋ}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} as coefficients.

2. 𝑈-Invariant Subspaces

In a Hilbert space ℋ, the 𝑈-invariant subspaces are intimately related to stationary sequences.

2.1. Some Preliminaries on Stationary Sequences

A sequence 𝐬={𝑠_𝑘}_𝑘∈ℤ in a separable Hilbert space ℋ is said to be stationary if 𝑠_𝑚+𝑘,𝑠_𝑛+𝑘ℋ=𝑠_𝑚,𝑠_𝑛ℋ∀𝑚,𝑛,𝑘∈ℤ. (7)

The function 𝑅_𝐬(𝑘)∶=⟨𝑠_𝑘,𝑠₀⟩_ℋ, for every 𝑘∈ℤ, is called the autocovariance function of the sequence 𝐬. Moreover, two stationary sequences 𝐬={𝑠_𝑘}_𝑘∈ℤ and 𝐰={𝑤_𝑘}_𝑘∈ℤ are said to be stationary correlated if 𝑠_𝑚+𝑘,𝑤_𝑛+𝑘ℋ=𝑠_𝑚,𝑤_𝑛ℋ∀𝑚,𝑛,𝑘∈ℤ, (8) and 𝑅_𝐬,𝐰(𝑘)∶=⟨𝑠_𝑘,𝑤₀⟩_ℋ, for every 𝑘∈ℤ defines the corresponding cross-covariance function. The following result is a well-known characterization of stationary sequences (see [19]).

Lemma 1. To every stationary sequence 𝐬={𝑠_𝑛}_𝑛∈ℤ in a Hilbert space ℋ there exists a unique unitary operator 𝑈∶ℋ→ℋ and 𝑠∈ℋ such that 𝑠_𝑛=𝑈^𝑛𝑠 for all 𝑛∈ℤ. Conversely, every pair (𝑈,𝑠) of a unitary operator 𝑈 and an 𝑠∈ℋ defines by 𝑠_𝑛=𝑈^𝑛𝑠, 𝑛∈ℤ, a stationary sequence 𝐬={𝑠_𝑛}_𝑛∈ℤ in ℋ.
Moreover, two stationary sequence 𝐬 and 𝐰 are stationary correlated if and only if they are generated by the same unitary operator 𝑈; that is, 𝑠_𝑛=𝑈^𝑛𝑠 and 𝑤_𝑛=𝑈^𝑛𝑤 for some 𝑠,𝑤∈ℋ.

The autocovariance 𝑅_𝐬 and the cross-covariance 𝑅_𝐬,𝐰 functions admit a spectral representation which is related to the integral representation of the unitary operator 𝑈 (see [19]).

Theorem 2. For every stationary sequence 𝐬={𝑠_𝑛}_𝑛∈ℤ in a Hilbert space ℋ, the autocovariance function 𝑅_𝐬 admits a spectral representation 𝑅_𝐬(𝑘)=𝑠_𝑘,𝑠_0ℋ=12𝜋^𝜋_−𝜋e^i𝑘𝜃𝑑𝜇_𝐬(𝜃),𝑘∈ℤ, (9) in the form of an integral with respect to a (positive) spectral measure 𝜇_𝐬.
For every two stationary correlated sequences 𝐬={𝑠_𝑛}_𝑛∈ℤ, 𝐰={𝑤_𝑛}_𝑛∈ℤ in a Hilbert space ℋ, the cross-covariance function admits a spectral representation 𝑅_𝐬,𝐰(𝑘)=𝑠_𝑘,𝑤_0ℋ=12𝜋^𝜋_−𝜋e^i𝑘𝜃𝑑𝜇_𝐬,𝐰(𝜃),𝑘∈ℤ, (10) in the form of an integral with respect to a (complex) spectral measure 𝜇_𝐬,𝐰.

2.1.1. The 𝑈-Invariant Subspace 𝒜_𝑎

For a fixed 𝑎∈ℋ, we consider the subspace of ℋ given by 𝒜_𝑎∶=span{𝑈^𝑛𝑎,𝑛∈ℤ}. In case that the sequence {𝑈^𝑛𝑎}_𝑛∈ℤ is a Riesz sequence in ℋ, we have 𝒜_𝑎=_𝑛∈ℤ𝛼_𝑛𝑈^𝑛𝑎∶𝛼_𝑛∈ℓ²(ℤ). (11)

A necessary and sufficient condition in order for the sequence 𝐚={𝑈^𝑛𝑎}_𝑛∈ℤ to be a Riesz sequence in ℋ can be stated in terms of the Lebesgue decomposition of the spectral measure 𝜇_𝐚 into an absolute and a singular part as 𝑑𝜇_𝐚(𝜃)=𝜙_𝐚(e^i𝜃)𝑑𝜃+𝑑𝜇^𝑠_𝐚(𝜃).

Theorem 3. Let 𝐚∶={𝑈^𝑛𝑎}_𝑛∈ℤ be a sequence obtained from an unitary operator in a separable Hilbert space ℋ with spectral measure 𝑑𝜇_𝐚(𝜃)=𝜙_𝐚(e^i𝜃)𝑑𝜃+𝑑𝜇^𝑠_𝐚(𝜃), and let 𝒜_𝑎 be the closed subspace spanned by 𝐚. Then, 𝐚 is a Riesz basis for 𝒜_𝑎 if and only if 𝜇^𝑠_𝐚≡0 and 0<𝐴_𝐚∶=essinf_𝜁∈𝕋𝜙_𝐚(𝜁)≤𝐵_𝐚∶=esssup_𝜁∈𝕋𝜙_𝐚(𝜁)<∞, (12) where 𝕋∶={e^i𝜃∶𝜃∈[−𝜋,𝜋)}.

Proof. For a fixed ℓ²-sequence 𝑐∶={𝑐_𝑛}_𝑛∈ℤ, we have _𝑘∈ℤ𝑐_𝑘𝑈^𝑘𝑎²=_{𝑚∈ℤ𝑛∈ℤ}𝑐_𝑚𝑐_𝑛𝑈^𝑚𝑎,𝑈^𝑛𝑎=_{𝑚∈ℤ𝑛∈ℤ}𝑐_𝑚𝑐_𝑛12𝜋^𝜋_−𝜋e^{i(𝑚−𝑛)𝜃}𝑑𝜇_𝐚(𝜃)=12𝜋^𝜋_{−𝜋𝑚∈ℤ𝑛∈ℤ}𝑐_𝑚𝑐_𝑛e^{i(𝑚−𝑛)𝜃}𝑑𝜇_𝐚(𝜃)=12𝜋^𝜋_{−𝜋𝑘∈ℤ}𝑐_𝑘e^i𝑘𝜃2𝑑𝜇_𝐚(𝜃), (13) if 𝜇_𝐚 is not absolutely continuous with respect to Lebesgue measure 𝜆, Lemma 4 below implies that there exists a bounded sequence {𝑐^𝑁}^∞_𝑁=1⊂ℓ²(ℤ) such that ‖∑_𝑘∈ℤ𝑐^𝑁_𝑘𝑈^𝑘𝑎‖² tends to infinity with 𝑁, so 𝐚 cannot be a Bessel sequence, therefore, not a Riesz basis. Assume now that 𝜇^𝑠_𝐚≡0 and 𝐚 is a Riesz basis, this implies that there exists 𝐵<∞ such that _𝑘∈ℤ𝑐_𝑘𝑈^𝑘𝑎²≤𝐵‖𝑐‖²_ℓ2 (14) for all 𝑐={𝑐_𝑘}_𝑘∈ℤ∈ℓ²(ℤ). Let Ω_𝑛∶={𝜁∈𝕋∶𝜙_𝐚(𝜁)≥𝑛}, assume that 𝐵_𝐚=∞, this implies that 𝜆(Ω_𝑛)>0 for all 𝑛. Now, (13), (14), and Parseval's theorem imply that 12𝜋^𝜋_{−𝜋𝑘∈ℤ}𝑐_𝑘e^i𝑘𝜃2𝜙_𝐚e^i𝜃𝐵𝑑𝜃≤2𝜋^𝜋_{−𝜋𝑘∈ℤ}𝑐_𝑘e^i𝑘𝜃2𝑑𝜃 (15) for all 𝑐={𝑐_𝑘}_𝑘∈ℤ∈ℓ²(ℤ). Introducing the Fourier expansion of the characteristic function 𝜒_Ω𝑛∈𝐿²(−𝜋,𝜋) in (15), we obtain 𝑛≤𝐵 for all 𝑛, which contradicts 𝐵<∞; thus, the assumption 𝐵_𝐚=∞ is false. In a similar way, it can be proved that 0<𝐴_𝐚.
For the sufficient condition, assume now that 𝜇^𝑠_𝐚≡0; then, (13) implies that _𝑘∈ℤ𝑐_𝑘𝑈^𝑘𝑎²=12𝜋^𝜋_{−𝜋𝑘∈ℤ}𝑐_𝑘e^i𝑘𝜃2𝜙_𝐚e^i𝜃𝑑𝜃, (16) if, in addition, condition (12) is satisfied, we get 𝐴_𝐚2𝜋^𝜋_{−𝜋𝑘∈ℤ}𝑐_𝑘e^i𝑘𝜃2𝑑𝜃≤_𝑘∈ℤ𝑐_𝑘𝑈^𝑘𝑎²≤𝐵_𝐚2𝜋^𝜋_{−𝜋𝑘∈ℤ}𝑐_𝑘e^i𝑘𝜃2𝑑𝜃, (17) which implies that 𝐴_𝐚‖𝑐‖²_ℓ2≤_𝑘∈ℤ𝑐_𝑘𝑈^𝑘𝑎²≤𝐵_𝐚‖𝑐‖²_ℓ2foreach𝑐∈ℓ²(ℤ). (18) Therefore, the sequence 𝐚 is a Riesz basis for 𝒜_𝑎.

Lemma 4. Let 𝜇 be a finite positive measure on (−𝜋,𝜋) which is not absolutely continuous with respect to Lebesgue measure 𝜆. Then, there exists a bounded sequence {𝑐^𝑁}^∞_𝑁=1⊂ℓ²(ℤ) such that lim_{𝑁→∞𝑘}𝑐^𝑁_𝑘e^i𝑘𝜃2_{𝐿2𝜇(−𝜋,𝜋)}=∞. (19)

Proof. If the measure 𝜇 is not absolutely continuous with respect to Lebesgue measure, then 𝜇(𝑀)>0 for a Lebesgue measurable set 𝑀⊂(−𝜋,𝜋) of Lebesgue measure zero. Thus, there exists a Borel set 𝐵 of Lebesgue measure zero such that 𝑀⊂𝐵⊂(−𝜋,𝜋). In fact, the set 𝐵 is an intersection of a countable collection of open sets (see [20, page 63]). Therefore, 𝜇(𝐵)>0 and 𝜆(𝐵)=0. On the other hand, every finite Borel measure on (−𝜋,𝜋) is inner regular (see [20, page 340]); that is, 𝜇(𝐵)=sup𝜇(𝐶)∶𝐶⊂𝐵,𝐶compact, (20) then there exists a compact set 𝐶⊂(−𝜋,𝜋) such that 𝜇(𝐶)>0,𝜆(𝐶)=0. (21)
For any 𝜀>0, there exists a sequence of disjoint open intervals 𝐼_𝑗⊂(−𝜋,𝜋) such that 𝐶⊂^∞_𝑗=1𝐼_𝑗,^∞_𝑗=1𝜆𝐼_𝑗≤𝜆(𝐶)+𝜀=𝜀, (22) (see [20, pages 58 and 42]). Since 𝐶 is compact, we may take the sequence to be finite. Hence, for every 𝑁∈ℕ, there exist open disjoint intervals 𝐼^𝑁₁,𝐼^𝑁₂,…,𝐼^𝑁_𝑖𝑁 in (−𝜋,𝜋) such that 𝐶⊂^𝑖𝑁_𝑖=1𝐼^𝑁_𝑖,^𝑖𝑁_𝑖=1𝜆𝐼^𝑁_𝑖≤13^𝑁. (23) Besides, ∑^𝑖𝑁_𝑖=1𝜇(𝐼^𝑁_𝑖)≥𝜇(𝐶).
Consider the function 𝑔_𝑁∶(−𝜋,𝜋)→ℝ, where 𝑔_𝑁=2^𝑁/2𝜒_⋃^𝑖_{𝑁𝑖=1𝐼𝑁𝑖}, that satisfies 𝑔_𝑁²₂=2^𝑁𝑖𝑁_𝑖=1𝜆𝐼^𝑁_𝑖≤2^𝑁3^𝑁<1. (24)
We modify and extend each 𝑔_𝑁 to obtain a 2𝜋-periodic function 𝑓_𝑁∶ℝ→ℝ such that 𝑓_𝑁 and its derivative are continuous on ℝ, ‖𝑓_𝑁‖²₂≤1 and 𝑓_𝑁(𝜃)=𝑔_𝑁(𝜃) for every 𝜃∈⋃^𝑖𝑁_𝑖=1𝐼^𝑁_𝑖. Let ∑_𝑘𝑐^𝑁_𝑘e^i𝑘𝜃 be the Fourier series of 𝑓_𝑁. First, by using Parseval's identity, we have 𝑐^𝑁_𝑘²_ℓ2=1𝑓2𝜋^𝑁22≤12𝜋forevery𝑁∈ℕ, (25) so that {𝑐^𝑁}^∞_𝑁=1 is a bounded sequence in ℓ²(ℤ). Besides, the regularity of each 𝑓_𝑁 ensures that each Fourier series converges uniformly to 𝑓_𝑁. Therefore, each series ∑_𝑘𝑐^𝑁_𝑘e^i𝑘𝜃 converges to 𝑓_𝑁 in 𝐿²_𝜇(−𝜋,𝜋) and consequently, _𝑘𝑐^𝑁_𝑘e^i𝑘𝜃2_{𝐿2𝜇()−𝜋,𝜋}=^𝜋_−𝜋𝑓_𝑁²𝑑𝜇≥^𝜋_−𝜋𝑔_𝑁²𝑑𝜇=2^𝑁𝑖𝑁_𝑖=1𝜇𝐼^𝑁_𝑖≥2^𝑁𝜇(𝐶), (26) from which we obtain the desired result.

The proof of Theorem 3 is similar to that of Theorem 6 in [16], except we do not exclude the case in which the singular measure is atomless. Recently, we have been aware that Theorem 3 was exposed in [21].

2.2. Studying the Sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} in 𝒜_𝑎

For 𝑏_𝑗∈𝒜_𝑎, 𝑗=1,2,…,𝑠, consider the sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠}. For every 𝑗=1,2,…,𝑠, the spectral measure 𝜇_{𝐚,𝐛𝑗} in the integral representation of the cross-covariance function of the sequences 𝐚∶={𝑈^𝑘𝑎}_𝑘∈ℤ and 𝐛_𝑗∶={𝑈^𝑘𝑏_𝑗}_𝑘∈ℤ has no singular part. Indeed, according to Theorem 3, the spectral measure associated with the autocovariance function of the sequence {𝑈^𝑘𝑎}_𝑘∈ℤ has no singular part; then by using the Cauchy-Schwarz type inequality in [22, page 125] we get the result. In the sequel, we will use the abridged notation 𝑏_𝑘,𝑗∶=𝑈^𝑟𝑘𝑏_𝑗; our goal in this section is to study the sequence {𝑏_𝑘,𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} in 𝒜_𝑎 in terms of an 𝑠×𝑟 matrix Ψ_𝐚,𝐛(e^i𝜃) which is introduced below. For the sake of completeness, we include some needed calculations which appear in [14].

First of all, we have 𝑈^𝑘𝑎,𝑏_𝑛,𝑗=12𝜋^𝜋_−𝜋e^{i(𝑘−𝑟𝑛)𝜃}𝜙_{𝐚,𝐛𝑗}e^i𝜃𝑑𝜃, (27) where 𝜙_{𝐚,𝐛𝑗} stands for the cross spectral density of the stationary correlated sequences 𝐚∶={𝑈^𝑘𝑎}_𝑘∈ℤ and 𝐛_𝑗∶={𝑈^𝑘𝑏_𝑗}_𝑘∈ℤ. Define Φ_𝐚,𝐛e^i𝜃∶=𝜙_𝐚,𝐛1e^i𝜃,𝜙_𝐚,𝐛2e^i𝜃,…,𝜙_{𝐚,𝐛𝑠}e^i𝜃⊤. (28)

In what follows, we will use the left-shift operator 𝑆 defined as 𝑆:𝐿²(𝕋)⟶𝐿²(𝕋)_𝑘∈ℤ𝑎_𝑘e^i𝑘𝜃⟼_𝑘∈ℤ𝑎_𝑘+1e^i𝑘𝜃, (29) or equivalently, by (𝑆𝑓)(e^i𝜃)=𝑓(e^i𝜃)e^−i𝜃, where 𝕋∶={e^i𝜃∶𝜃∈[−𝜋,𝜋)} denotes the unidimensional torus. Also, we will consider the decimation operator 𝐷_𝑟, 𝑟 is a positive integer, defined as 𝐷_𝑟:𝐿²(𝕋)⟶𝐿²(𝕋)_𝑘∈ℤ𝑎_𝑘e^i𝑘𝜃⟼_𝑘∈ℤ𝑎_𝑟𝑘e^i𝑘𝜃, (30) which can equivalently be written as 𝐷_𝑟𝑓e^i𝜃=1𝑟^𝑟−1_𝑘=0𝑓e^{i((𝜃+2𝑘𝜋)/𝑟)}. (31)

For each 𝑙=0,1,…,𝑟−1, set the 𝑠×1 matrix of functions on the torus 𝕋 as follows: Ψ^𝑙_𝐚,𝐛e^i𝜃∶=𝐷_𝑟𝑆^−𝑙Φ_𝐚,𝐛e^i𝜃, (32) and define the 𝑠×𝑟 matrix of functions on the torus 𝕋 as follows: Ψ_𝐚,𝐛e^i𝜃∶=Ψ⁰_𝐚,𝐛e^i𝜃Ψ¹_𝐚,𝐛e^i𝜃⋯Ψ^𝑟−1_𝐚,𝐛e^i𝜃. (33)

It is worth to mention that the matrix Ψ_𝐚,𝐛 was explicitly computed in [14] for the translation and modulation cases in 𝐿²(ℝ).

Next, for any 𝑥∈𝒜_𝑎, we obtain an expression for the inner products 𝛼_𝑛,𝑗∶=⟨𝑥,𝑈^𝑟𝑛𝑏_𝑗⟩, 𝑛∈ℤ and 𝑗=1,2,…,𝑠. Indeed, writing 𝑥=∑_𝑘∈ℤ𝑥_𝑘𝑈^𝑘𝑎 where {𝑥_𝑘}_𝑘∈ℤ∈ℓ²(ℤ), we have 𝛼_𝑛,𝑗=𝑥,𝑈^𝑟𝑛𝑏_𝑗=_𝑘∈ℤ𝑥_𝑘𝑈^𝑘𝑎,𝑈^𝑟𝑛𝑏_𝑗=_𝑘∈ℤ𝑥_𝑘12𝜋^𝜋_−𝜋e^{i(𝑘−𝑟𝑛)𝜃}𝜙_{𝐚,𝐛𝑗}e^i𝜃=1𝑑𝜃2𝜋^𝜋_{−𝜋𝑘∈ℤ}𝑥_𝑘e^i𝑘𝜃𝜙_{𝐚,𝐛𝑗}e^i𝜃e^{−i𝑟𝑛𝜃}𝑑𝜃, (34) that is, 𝜶_𝑛∶=𝛼_𝑛,1,𝛼_𝑛,2,…,𝛼_𝑛,𝑠^⊤=12𝜋^𝜋_−𝜋Φ_𝐚,𝐛e^i𝜃𝑋e^i𝜃e^{−i𝑟𝑛𝜃}𝑑𝜃, (35) where 𝑋(e^i𝜃)∶=∑_𝑘∈ℤ𝑥_𝑘e^i𝑘𝜃.

Now, for 𝑙=0,1,…,𝑟−1, define the sequence 𝑥^(𝑙)∶={𝑥_𝑘^(𝑙)∶=𝑥_{𝑘𝑟+𝑙}}_𝑘∈ℤ. Thus, we can write 𝑋e^i𝜃=^𝑟−1_{𝑙=0𝑘∈ℤ}𝑥_{𝑘𝑟+𝑙}e^{i(𝑘𝑟+𝑙)𝜃}=^𝑟−1_𝑙=0𝑋^(𝑙)e^i𝑟𝜃e^i𝑙𝜃, (36) where 𝑋^(𝑙)(e^i𝜃)=∑_𝑘∈ℤ𝑥_𝑘^(𝑙)e^i𝑘𝜃.

Using (36) in (35), we obtain 𝜶_𝑛=^𝑟−1_𝑙=012𝜋^𝜋_−𝜋Φ_𝐚,𝐛e^i𝜃𝑋^(𝑙)e^i𝑟𝜃e^i𝑙𝜃e^{−i𝑟𝑛𝜃}𝑑𝜃. (37)

After some easy calculations, we get 𝜶_𝑛=^𝑟−1_𝑙=012𝜋^𝜋_−𝜋𝑆^−𝑙Φ_𝐚,𝐛e^i𝜃𝑋^(𝑙)e^i𝑟𝜃e^{−i𝑟𝑛𝜃}=𝑑𝜃_{𝑟−1𝑙=0}12𝜋^𝑟𝜋_−𝑟𝜋𝑆^−𝑙Φ_𝐚,𝐛e^i(𝜃/𝑟)𝑟𝑋^(𝑙)e^i𝜃e^−i𝑛𝜃=𝑑𝜃_{𝑟−1𝑙=0𝑟−1𝑘=02𝜋(𝑘+1)2𝜋𝑘}𝑆^−𝑙Φ_𝐚,𝐛e^i(𝜃/𝑟)𝑋2𝜋𝑟^(𝑙)e^i𝜃e^−i𝑛𝜃=𝑑𝜃_{02𝜋𝑟−1𝑙=0𝑟−1𝑘=0}𝑆^−𝑙Φ_𝐚,𝐛e^{i((𝜃+2𝜋𝑘)/𝑟)}𝑋2𝜋𝑟^(𝑙)e^i𝜃e^−i𝑛𝜃=1𝑑𝜃2𝜋^𝜋_−𝜋^𝑟−1_𝑙=0𝐷_𝑟𝑆^−𝑙Φ_𝐚,𝐛e^i𝜃𝑋^(𝑙)e^i𝜃e^−i𝑛𝜃𝑑𝜃. (38)

Defining 𝐂(e^i𝜃)∶=∑_𝑘∈ℤ𝜶_𝑘e^i𝑘𝜃, (38) implies that 𝐂e^i𝜃=^𝑟−1_𝑙=0𝐷_𝑟𝑆^−𝑙Φ_𝐚,𝐛e^i𝜃𝑋^(𝑙)e^i𝜃, (39) which can be written in matrix form as 𝐂e^i𝜃=_𝑘∈ℤ𝛼_𝑘,1e^i𝑘𝜃,_𝑘∈ℤ𝛼_𝑘,2e^i𝑘𝜃,…,_𝑘∈ℤ𝛼_𝑘,𝑠e^i𝑘𝜃⊤=Ψ_𝐚,𝐛e^i𝜃𝑋⁽⁰⁾e^i𝜃,𝑋⁽¹⁾e^i𝜃,…,𝑋^(𝑟−1)e^i𝜃⊤=Ψ_𝐚,𝐛e^i𝜃𝐗e^i𝜃=𝐿_Ψ𝐚,𝐛𝐗e^i𝜃, (40) where 𝐿_Ψ𝐚,𝐛∶𝐿²_𝑟(𝕋)→𝐿²_𝑠(𝕋) denotes the multiplication operator by Ψ_𝐚,𝐛 and 𝐗e^i𝜃∶=𝑋⁽⁰⁾e^i𝜃,𝑋⁽¹⁾e^i𝜃,…,𝑋^(𝑟−1)e^i𝜃⊤. (41)

By 𝐿²_𝑟(𝕋) (resp., 𝐿²_𝑠(𝕋)), we denote the product space 𝐿²(𝕋)×⋯×𝐿²(𝕋) for 𝑟 times (resp., 𝑠 times). Besides, Ψ_𝐚,𝐛𝐗²_{𝐿2𝑠(𝕋)}=12𝜋^𝜋_−𝜋Ψ_𝐚,𝐛e^i𝜃𝐗e^i𝜃,Ψ_𝐚,𝐛e^i𝜃𝐗e^i𝜃_ℂ𝑟=1𝑑𝜃2𝜋^𝜋_−𝜋Ψ^∗_𝐚,𝐛e^i𝜃Ψ_𝐚,𝐛e^i𝜃𝐗e^i𝜃,𝐗e^i𝜃_ℂ𝑟𝑑𝜃. (42)

The above calculations let us prove the following result.

Theorem 5. Let 𝑏_𝑗∈𝒜_𝑎 for 𝑗=1,2,…,𝑠 and let Ψ_𝐚,𝐛 be the associated matrix given in (33). Then, the following results hold. (a)The sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a complete system in 𝒜_𝑎 if and only if the rank of the matrix Ψ_𝐚,𝐛(𝜁) is 𝑟 a.e. 𝜁 in 𝕋. (b)The sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a Bessel sequence for 𝒜_𝑎 if and only if there exists a constant 𝐵<∞ such that Ψ^∗_𝐚,𝐛(𝜁)Ψ_𝐚,𝐛(𝜁)≤𝐵𝕀_𝑟𝑎.𝑒.𝜁𝑖𝑛𝕋. (43)(c)The sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a frame for 𝒜_𝑎 if and only if there exist constants 0<𝐴≤𝐵<∞ such that 𝐴𝕀_𝑟≤Ψ^∗_𝐚,𝐛(𝜁)Ψ_𝐚,𝐛(𝜁)≤𝐵𝕀_𝑟𝑎.𝑒.𝜁𝑖𝑛𝕋. (44)(d)The sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a Riesz basis for 𝒜_𝑎 if and only if it is a frame and 𝑠=𝑟.

Proof. To prove (a), assume that there exists a set Ω⊆𝕋 with positive measure such that rank[Ψ_𝐚,𝐛(𝜁)]<𝑟 for each 𝜁∈Ω. Then, there exists a measurable function 𝑣(𝜁), 𝜁∈Ω, such that Ψ_𝐚,𝐛(𝜁)𝑣(𝜁)=0 and ‖𝑣(𝜁)‖_𝐿²_𝑟(𝕋)=1 in Ω. This function can be constructed as in [23, Lemma 2.4]. Define 𝐕∈𝐿²_𝑟(𝕋) such that 𝐕(𝜁)=𝑣(𝜁) if 𝜁∈Ω, and 𝐕(𝜁)=0 if 𝜁∈𝕋⧵Ω. Hence, from (40), we obtain that the system is not complete. Conversely, if the system is not complete, by using (40), we obtain a 𝐕(𝜁) different from 0 in a set with positive measure such that Ψ_𝐚,𝐛(𝜁)𝐕(𝜁)=0. Thus, rankΨ_𝐚,𝐛(𝜁)<𝑟 on a set with positive measure.
To prove (b), we keep in mind that {𝑈^𝑘𝑎}_𝑘∈ℤ is a Riesz basis for 𝒜_𝑎, the mapping 𝑇∶ℓ²(ℤ)→𝒜_𝑎, given by 𝑇{𝑥_𝑘}_𝑘∈ℤ=𝑥=∑_𝑘∈ℤ𝑥_𝑘𝑈^𝑘𝑎 is bijective and there exist two constants 0<𝑚_𝑎≤𝑀_𝑎<∞ such that 𝑚_𝑎{𝑥_𝑘}²_ℓ2≤𝑇𝑥_𝑘²_ℋ≤𝑀_𝑎𝑥_𝑘²_ℓ2. (45)
Assume first that (43) is satisfied. It follows from (40) and (42) that Ψ_𝐚,𝐛𝐗²_{𝐿2𝑠(𝕋)}≤𝐵𝐗²_{𝐿2𝑟(𝕋)}. (46)
By construction ‖Ψ_𝐚,𝐛𝐗‖²_{𝐿2𝑠(𝕋)}=∑^𝑠_𝑗=1∑_𝑘∈ℤ|⟨𝑥,𝑏_𝑘,𝑗⟩|² and ‖𝐗‖²_𝐿²_𝑟(𝕋)=‖{𝑥_𝑘}_𝑘∈ℤ‖²_ℓ2, using (45), it follows from (46) that ^𝑠_{𝑗=1𝑘∈ℤ}𝑥,𝑏_𝑘,𝑗²≤𝐵𝑚_𝑎‖𝑥‖²_ℋ. (47) Conversely, assume that {𝑏_𝑘𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a Bessel sequence for 𝒜_𝑎, then there exists 0<𝐵^′<∞ such that ^𝑠_{𝑗=1𝑘∈ℤ}𝑥,𝑏_𝑘,𝑗²≤𝐵^′‖𝑥‖²_ℋ. (48)
Using (45), this implies that Ψ_𝐚,𝐛𝐗²_{𝐿2𝑠(𝕋)}≤𝐵^′𝑀_𝑎𝐗²_{𝐿2𝑟(𝕋)} (49) for all 𝐗∈𝐿²_𝑟(𝕋). Inserting the right hand side of (42) for ‖Ψ_𝐚,𝐛𝐗‖²_{𝐿2𝑠(𝕋)}, it is straightforward to see that (43) holds with 𝐵=𝐵^′𝑀_𝑎. The proof of (c) is completed proceeding as in (b).
To prove (d), consider the following mapping: 𝑆:𝒜_𝑎⟶ℓ²_𝑠(ℤ)𝑥⟼𝑥,𝑏_{𝑘,𝑗𝑘∈ℤ;𝑗=1,2,…,𝑠}. (50)
According to (40), the mapping 𝑆 is isometric equivalent to 𝐿_Ψ𝐚,𝐛, and assuming that {𝑏_𝑘,𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a frame, it is a Riesz basis if and only if 𝑆 is surjective.
First, if {𝑏_𝑘,𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a Riesz basis, then it is a frame and 𝑆 is surjective. Applying (a) yields that 𝐿_Ψ𝐚,𝐛 is bijective, and therefore 𝐿^∗_Ψ𝐚,𝐛=𝐿_Ψ^∗_𝐚,𝐛 is bijective. Hence, rank[Ψ_𝐚,𝐛(𝜁)Ψ^∗_𝐚,𝐛(𝜁)] is 𝑠 for almost every 𝜁 in 𝕋, so 𝑟=rankΨ^∗_𝐚,𝐛(𝜁)Ψ_𝐚,𝐛(𝜁)=rankΨ_𝐚,𝐛(𝜁)Ψ^∗_𝐚,𝐛(𝜁)=𝑠, (51) and finally 𝑠=𝑟.
Conversely, if {𝑏_𝑘,𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a frame and 𝑠=𝑟, (a) implies that Ψ_𝐚,𝐛(𝜁) is invertible for almost every 𝜁 in 𝕋, which implies that 𝐿_Ψ𝐚,𝐛 is surjective; then 𝑆 is surjective and {𝑏_𝑘,𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a Riesz basis.

The following lemma will allow us to restate Theorem 5.

Lemma 6. Let 𝐆(𝜁) be an 𝑠×𝑟 matrix with entries in 𝐿²(𝕋), and consider the following constants: 𝐴_𝐆∶=essinf_𝜁∈𝕋𝜆_min𝐆^∗(𝜁)𝐆(𝜁,𝐵_𝐆)∶=esssup_𝜁∈𝕋𝜆_max𝐺^∗(𝜁)𝐆(𝜁), (52) where 𝜆_min (resp., 𝜆_max) denotes the smallest (resp., the largest) eigenvalue of the positive semidefinite matrix 𝐆^∗(𝜁)𝐆(𝜁). Then, (a)the matrix 𝐆(𝜁) has essentially bounded entries on 𝕋 if and only if 𝐵_𝐆<∞;(b)there exist constants 0<𝐴≤𝐵<∞ such that 𝐴𝕀_𝑟≤𝐆^∗(𝜁)𝐆(𝜁)≤𝐵𝕀_𝑟, a.e., 𝜁∈𝕋 if and only if 0<𝐴_𝐆≤𝐵_𝐆<∞.

Proof. The first part of lemma follows from that 𝜆_max[𝐆^∗(𝜁)𝐆(𝜁)]=‖𝐆(𝜁)‖²₂, and max_𝑖,𝑗𝑎_𝑖𝑗≤‖𝐀‖₂≤√𝑚𝑛max_𝑖,𝑗𝑎_𝑖𝑗foranymatrix𝐀=𝑎_{𝑖𝑗𝑖=1,2…,𝑚𝑗=1,2…,𝑛}, (53) where ‖𝐀‖₂ denotes the spectral norm of the matrix 𝐀 (see, for instance, [24]).
Now, we prove the second part of the lemma. Since 𝐆^∗(𝜁)𝐆(𝜁)≤𝐵𝕀_𝑟 means that ⟨𝐵𝑥−𝐆^∗(𝜁)𝐆(𝜁)𝑥,𝑥⟩≥0 for all 𝑥∈ℂ^𝑟, in particular, taking an eigenvector 𝑥 associated with the largest eigenvalue 𝜆_max of 𝐆^∗(𝜁)𝐆(𝜁) such that ‖𝑥‖=1, one has that 𝐵≥𝜆_max(𝐆^∗(𝜁)𝐆(𝜁)). Hence, 𝐵≥esssup_𝜁∈𝕋𝜆_max[𝐆^∗(𝜁)𝐆(𝜁)]. In a similar way, 𝐴𝕀_𝑟≤𝐆^∗(𝜁)𝐆(𝜁) implies that 𝐴≤essinf_𝜁∈𝕋𝜆_min[𝐆^∗(𝜁)𝐆(𝜁)].
Conversely, Rayleigh-Ritz theorem [24, page 176] yields that 𝜆_max𝐆^∗(𝜁)𝐆(𝜁)=max_{𝑥∈ℂ𝑟}𝑥^∗𝐆^∗(𝜁)𝐆(𝜁)𝑥^∗𝑥=max_{𝑥∈ℂ𝑟}𝐆^∗(𝜁)𝐆(𝜁)𝑥,𝑥⟨⟩.𝑥,𝑥 (54) Thus, esssup_𝜁∈𝕋𝜆_max[𝐆^∗(𝜁)𝐆(𝜁)]=𝐵_𝐆 implies that max_{𝑥∈ℂ𝑟}𝐆^∗(𝜁)𝐆(𝜁)𝑥,𝑥⟨𝑥,𝑥⟩≤𝐵_𝐆,a.e.𝜁∈𝕋. (55)
In other words, 𝐵_𝐆𝕀_𝑟≥𝐆^∗(𝜁)𝐆(𝜁); analogously, 𝐆^∗(𝜁)𝐆(𝜁)≥𝐴_𝐆𝕀_𝑟.
It is easy to deduce from the proof that 𝐴_𝐆 and 𝐵_𝐆 are the optimal constants 𝐴>0 and 𝐵<∞ satisfying the inequalities 𝐴𝕀_𝑟≤𝐆^∗(𝜁)𝐆(𝜁)≤𝐵𝕀_𝑟, a.e., 𝜁∈𝕋.

As a consequence of Lemma 6, statements (b) and (c) in Theorem 5 can be restated in terms of the following constants: 𝐴_Ψ∶=essinf_𝜁∈𝕋𝜆_minΨ^∗_𝐚,𝐛(𝜁)Ψ_𝐚,𝐛(𝜁;𝐵_Ψ)∶=esssup_𝜁∈𝕋𝜆_maxΨ^∗_𝐚,𝐛(𝜁)Ψ_𝐚,𝐛(𝜁), (56) as follows.

Theorem 7. Let 𝑏_𝑗∈𝒜_𝑎 for 𝑗=1,2,…,𝑠, and let Ψ_𝐚,𝐛 be the associated matrix given in (33) and its related constants (56). Then, the following results hold.(i)The sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a Bessel sequence for 𝒜_𝑎 if and only if the constant 𝐵_Ψ<∞. (ii)The sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a frame for 𝒜_𝑎 if and only if the constants 𝐴_Ψ and 𝐵_Ψ satisfy 0<𝐴_Ψ≤𝐵_Ψ<∞. In this case, 𝐴_Ψ and 𝐵_Ψ are the optimal frame bounds for {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠}.

2.3. The Frame Expansion

Define the 𝑟×𝑠 matrix of functions on the torus 𝕋 as follows: Γe^i𝜃∶=_𝑘∈ℤΓ_𝑘e^i𝑘𝜃=Ψ^∗_𝐚,𝐛e^i𝜃Ψ_𝐚,𝐛e^i𝜃−1Ψ^∗_𝐚,𝐛e^i𝜃. (57) The following expansion involving the inner products 𝛼_𝑛,𝑗=⟨𝑥,𝑈^𝑟𝑛𝑏_𝑗⟩ of 𝑥∈𝒜_𝑎 holds.

Lemma 8. Assume that the matrix Ψ_𝐚,𝐛(𝜁) has essentially bounded entries on 𝕋. For any 𝑥=∑_𝑘∈ℤ𝑥_𝑘𝑈^𝑘𝑎∈𝒜_𝑎, one has 𝐱_𝐧=_𝑘∈ℤΓ_𝑘𝜶_𝑛−𝑘, (58) where 𝐱_𝐧 denotes the 𝑛th Fourier coefficient of the function 𝐗(e^i𝜃) defined in (41), and the sequence {𝜶_𝑛}_𝑛∈ℤ is given in (35).

Proof. Indeed, 𝐱_𝐧=12𝜋^𝜋_−𝜋𝐗e^i𝜃e^−i𝑛𝜃=1𝑑𝜃2𝜋^𝜋_{−𝜋𝑘∈ℤ}Γ_𝑘e^i𝑘𝜃Ψ_𝐚,𝐛e^i𝜃𝐗e^i𝜃e^−i𝑛𝜃𝑑𝜃=_𝑘∈ℤΓ_𝑘12𝜋^𝜋_−𝜋Ψ_𝐚,𝐛e^i𝜃𝐗e^i𝜃e^{−i(𝑛−𝑘)𝜃}=𝑑𝜃_𝑘∈ℤΓ_𝑘𝜶_𝑛−𝑘. (59)

At this point, we are ready to prove the following expansion result.

Theorem 9. Let 𝑏_𝑗∈𝒜_𝑎 for 𝑗=1,2,…,𝑠, and assume that the associated matrix Ψ_𝐚,𝐛 given in (33) has essentially bounded entries on 𝕋, that is, 𝐵_Ψ<∞. The following statements are equivalent. (i)The constant 𝐴_Ψ>0. (ii)There exist 𝑐_𝑗∈𝒜_𝑎, 𝑗=1,2,…,𝑠, such that the sequence {𝑈^𝑟𝑘𝑐_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a frame for 𝒜_𝑎, yielding, for any 𝑥∈𝒜_𝑎, the following expansion: 𝑥=^𝑠_{𝑗=1𝑘∈ℤ}𝑥,𝑈^𝑟𝑘𝑏_𝑗𝑈^𝑟𝑘𝑐_𝑗𝑖𝑛ℋ. (60)
In case the equivalent conditions hold, {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} and {𝑈^𝑟𝑘𝑐_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} form a pair of dual frames in 𝒜_𝑎.

Proof. First, we prove that (i) implies (ii). Observe that 𝑥=∑_𝑘∈ℤ𝑥_𝑘𝑈^𝑘𝑎 can be written as ∑_𝑛∈ℤ𝐱_𝐧^⊤𝐚_𝐧 where 𝐚_𝐧=(𝑈^𝑛𝑟𝑎,𝑈^𝑛𝑟+1𝑎,…,𝑈^{𝑛𝑟+𝑟−1}𝑎)^⊤. Next, 𝑥=_𝑛∈ℤ𝐱_𝐧^⊤𝐚_𝐧=_{𝑛∈ℤ𝑘∈ℤ}Γ_𝑘𝜶_𝑛−𝑘^⊤𝐚_𝐧=_{𝑛∈ℤ𝑘∈ℤ}𝜶^⊤_𝑛−𝑘Γ^⊤_𝑘𝐚_𝐧=_{𝑛∈ℤ𝑘∈ℤ}𝜶^⊤_𝑛Γ^⊤_𝑘𝐚_𝐧+𝐤=_𝑛∈ℤ𝜶^⊤_{𝑛𝑘∈ℤ}Γ^⊤_𝑘𝐚_𝐧+𝐤. (61) For 𝑙∈ℤ and 𝑗=1,2,…,𝑠 define 𝑐_𝑙,𝑗∶=𝑈^𝑟𝑙𝑐_𝑗, where (𝑐₁,𝑐₂,…,𝑐_𝑠)^⊤=∑_𝑘∈ℤΓ^⊤_𝑘𝐚_𝐤, and 𝑏_𝑙,𝑗∶=𝑈^𝑟𝑙𝑏_𝑗. Then, (61) implies 𝑥=_𝑛∈ℤ𝜶^⊤_{𝑛𝑘∈ℤ}Γ^⊤_𝑘𝐚_𝐧+𝐤=_𝑛∈ℤ𝜶^⊤_𝑛𝑈^𝑛𝑟_𝑘∈ℤΓ^⊤_𝑘𝐚_𝐤=^𝑠_{𝑙=1𝑛∈ℤ}𝑥,𝑏_𝑛,𝑙𝑐_𝑛,𝑙inℋ. (62)
In order to be allowed to use [1, Lemma 5.6.2], we have to prove that the above constructed sequence {𝑐_𝑘,𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a Bessel sequence for 𝒜_𝑎. To this end, we compute the corresponding Ψ_𝐚,𝐜 matrix for 𝐜∶={𝑐_𝑘,𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠}. Setting [Γ^⊤_𝑘]_𝑖𝑗=𝑎^𝑘_𝑖𝑗, we obtain 𝑈^𝑘𝑎,𝑐_𝑛,𝑗=^𝑟_{𝑙∈ℤ𝑖=1}𝑈^𝑘𝑎,𝑈^𝑛𝑟𝑎^𝑙_𝑗𝑖𝑈^{𝑙𝑟+𝑖𝑟+𝑖−1}𝑎=^𝑟_{𝑙∈ℤ𝑖=1}𝑎^𝑙_𝑗𝑖𝑈^{𝑘−𝑛𝑟−𝑙𝑟−𝑖+1}𝑎,𝑎=^𝑟_{𝑙∈ℤ𝑖=1}𝑎^𝑙_𝑗𝑖12𝜋^𝜋_−𝜋e^{i(𝑘−𝑛𝑟−𝑙𝑟−𝑖+1)𝜃}𝜙_𝐚e^i𝜃𝑑𝜃. (63)
Now, 𝑈^𝑘𝑎,𝑐_𝑛,1𝑈^𝑘𝑎,𝑐_𝑛,2⋮𝑈^𝑘𝑎,𝑐_𝑛,𝑠=_𝑙∈ℤΓ^⊤_𝑙1∫2𝜋^𝜋−𝜋e^{i(𝑘−𝑛𝑟−𝑙𝑟)𝜃}𝜙_𝐚e^i𝜃1𝑑𝜃∫2𝜋^𝜋−𝜋e^{i(𝑘−𝑛𝑟−𝑙𝑟−1)𝜃}𝜙_𝐚e^i𝜃⋮1𝑑𝜃∫2𝜋^𝜋−𝜋e^{i(𝑘−𝑛𝑟−𝑙𝑟−𝑟+1)𝜃}𝜙_𝐚e^i𝜃=1𝑑𝜃2𝜋^𝜋_{−𝜋𝑙∈ℤ}Γ^⊤_𝑙e^{−i𝑙𝑟𝜃}e^{i(𝑘−𝑛𝑟)𝜃}e^{i(𝑘−𝑛𝑟−1)𝜃}⋮e^{i(𝑘−𝑛𝑟−𝑟+1)𝜃}𝜙_𝐚e^i𝜃=1𝑑𝜃2𝜋^𝜋_−𝜋e^{i(𝑘−𝑛𝑟)𝜃}Γ^∗e^i𝑟𝜃𝐞̃e^i𝜃𝜙_𝐚e^i𝜃𝑑𝜃, (64) where 𝐞̃(e^i𝜃)∶=(1,e^−i𝜃,…,e^{−i(𝑟−1)𝜃})^⊤. Hence, we have deduced that Φ_𝐚,𝐜e^i𝜃=Γ^∗e^i𝑟𝜃𝐞̃e^i𝜃𝜙_𝐚e^i𝜃. (65)
Therefore, for 𝑙=0,1,…,𝑟−1, we have Ψ^𝑙_𝐚,𝐜e^i𝜃∶=𝐷_𝑟𝑆^−𝑙Γ^∗e^i𝑟𝜃𝐞̃e^i𝜃𝜙_𝐚e^i𝜃, (66) and consequently, the 𝑠×𝑟 matrix Ψ_𝐚,𝐜(e^i𝜃)∶=(Ψ⁰_𝐚,𝐜(e^i𝜃),Ψ¹_𝐚,𝐜(e^i𝜃),…,Ψ^𝑟−1_𝐚,𝐜(e^i𝜃)) can be written as Ψ_𝐚,𝐜e^i𝜃=𝐷_𝑟𝜙_𝐚e^i𝜃Γ^∗e^i𝑟𝜃𝐄e^i𝜃, (67) where 𝐄e^i𝜃1∶=e^i𝜃⋯e^{i(𝑟−1)𝜃}e^−i𝜃1⋯e^{i(𝑟−2)𝜃}⋮⋮⋱⋮e^{−i(𝑟−1)𝜃}e^{−i(𝑟−2)𝜃}.⋯1 (68)
As a consequence of Theorem 7, the proof ends if we prove that the matrix Ψ_𝐚,𝐜(e^i𝜃) has essentially bounded entries. Clearly, the decimation operator 𝐷_𝑟 sends bounded functions into bounded functions; Theorem 3 implies that 𝜙_𝐚 is bounded so, taking into account (67), it remains to check that the matrix Γ^∗(e^i𝑟𝜃) has essentially bounded entries.
Now, Γ^∗(e^i𝑟𝜃)=Ψ_𝐚,𝐛(e^i𝑟𝜃)[Ψ^∗_𝐚,𝐛(e^i𝑟𝜃)Ψ_𝐚,𝐛(e^i𝑟𝜃)]⁻¹, the lower bound condition (c) in Theorem 5 and Lemma 6 imply that [Ψ^∗_𝐚,𝐛(e^i𝑟𝜃)Ψ_𝐚,𝐛(e^i𝑟𝜃)]⁻¹ has bounded entries, and therefore the matrix Γ^∗(e^i𝑟𝜃) has bounded entries. We have shown that Ψ_𝐚,𝐜(e^i𝜃) has bounded entries; then Theorem 7, part (a) and Lemma 6 guarantee that the sequence {𝑐_𝑘,𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a Bessel sequence; then, the sequences {𝑏_𝑘,𝑗}_{𝑘∈ℤ,𝑗=1,2,…,𝑠} and {𝑐_𝑘,𝑗}_{𝑘∈ℤ,𝑗=1,2,…,𝑠} form a pair of dual frames in 𝒜_𝑎 (see [1, Lemma 5.6.2]).
Finally, condition (ii) implies condition (i). According to [1, Lemma 5.6.2], the sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a frame for 𝒜_𝑎 since it is a Bessel sequence and the expansion in (ii) holds. By using Theorem 7 we obtain that 𝐴_Ψ>0.

It is worth to observe that the analysis done in Theorem 9 provides a whole family of dual frames for the sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠}. In fact, everything works if we replace Γ(e^i𝜃) in (57) by any matrix of the following form: Γ_𝕌e^i𝜃∶=Ψ^†_𝐚,𝐛e^i𝜃+𝕌e^i𝜃𝕀_𝑠−Ψ_𝐚,𝐛e^i𝜃Ψ^†_𝐚,𝐛e^i𝜃, (69) where 𝕌(e^i𝜃) is any 𝑟×𝑠 matrix with entries in 𝐿^∞(𝕋), and Ψ^†_𝐚,𝐛 denotes the Moore-Penrose left-inverse Ψ^†_𝐚,𝐛(e^i𝜃)∶=[Ψ^∗_𝐚,𝐛(e^i𝜃)Ψ_𝐚,𝐛(e^i𝜃)]⁻¹Ψ^∗_𝐚,𝐛(e^i𝜃). Note that we need essentially bounded entries in the matrix Γ_𝕌(e^i𝜃) since the multiplication operator 𝑀_𝐹∶𝑓↦𝐹𝑓 in 𝐿²(𝕋) is well-defined (and consequently bounded) if and only if 𝐹∈𝐿^∞(𝕋).

Notice that if 𝑠=𝑟, we have Ψ^†_𝐚,𝐛=Ψ⁻¹_𝐚,𝐛 which implies a unique Γ_𝕌, and we are in the presence of a pair of dual Riesz bases. In fact, the following result holds.

Corollary 10. Let 𝑏_𝑗∈𝒜_𝑎 for 𝑗=1,2,…,𝑟; that is, 𝑟=𝑠 in Theorem 9. Assume that the square matrix Ψ_𝐚,𝐛 given in (33) has entries essentially bounded on 𝕋, that is, 𝐵_Ψ<∞. The following statements are equivalent.(a) The constant 𝐴_Ψ>0. (b) There exists a Riesz basis {𝐶_𝑘,𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} such that for any 𝑥∈𝒜_𝑎, the expansion 𝑥=^𝑠_{𝑗=1𝑘∈ℤ}𝑥,𝑈^𝑟𝑘𝑏_𝑗𝐶_𝑘,𝑗𝑖𝑛ℋ (70) holds.
In case the equivalent conditions are satisfied, necessarily, there exist 𝑐_𝑗∈𝒜_𝑎, 𝑗=1,2,…,𝑟, such that 𝐶_𝑘,𝑗=𝑈^𝑟𝑘𝑐_𝑗 for 𝑘∈ℤ and 𝑗=1,2,…,𝑟. Moreover, the sequences {𝑈^𝑟𝑘𝑐_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} and {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} are dual Riesz bases in 𝒜_𝑎, and the interpolation property ⟨𝑐_𝑗,𝑈^𝑟𝑘𝑏_𝑗′⟩=𝛿_𝑗,𝑗′𝛿_𝑘,0, where 𝑘∈ℤ and 𝑗,𝑗^′=1,2,…,𝑟, holds.

Proof. To prove (a)⇒(b), we use Theorem 9; whenever 0<𝐴_Ψ≤𝐵_Ψ<∞, there exist 𝑐_𝑗∈𝒜_𝑎, 𝑗=1,2,…,𝑠, such that the sequence {𝑈^𝑟𝑘𝑐_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a frame for 𝒜_𝑎 and for any 𝑥∈𝒜_𝑎, the expansion 𝑥=^𝑠_{𝑗=1𝑘∈ℤ}𝑥,𝑈^𝑟𝑘𝑏_𝑗𝑈^𝑟𝑘𝑐_𝑗inℋ, (71) holds. Actually, from Theorem 5, we get that 𝑟=𝑠 implies that {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a Riesz basis, and consequently, {𝑈^𝑟𝑘𝑐_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is indeed its dual Riesz basis.
The converse follows easily from the fact that if {𝐶_𝑘,𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a Riesz basis, then (b) implies that {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is its dual Riesz basis; hence, Theorem 5 provides 𝐴_Ψ>0. The interpolation property comes out from the biorthogonal condition of a pair of dual Riesz bases.

Closing this section it is worth to mention that the results stated and proved in Sections 2.2 and 2.3 mathematically enrich some of the remarkable results concerning regular sampling contained in the interesting [14]. Here, we have assumed only one generator 𝑎∈ℋ and that 𝑏_𝑗∈𝒜_𝑎 for all 𝑗=1,2,…,𝑠. If 𝑏_𝑗∉𝒜_𝑎 for some 𝑗, see the additional remarks in Section 4. The case of several generators 𝑎_𝑙∈ℋ, 𝑙=1,2,…,𝐿, can be essentially treated in the same way.

3. Some Perturbation Results

In Section 2, we have only used the discrete group of unitary operators {𝑈^𝑛}_𝑛∈ℤ which is completely determined by the unitary operator 𝑈. In order to carry out the study of some perturbation results stated below, we need the availability of a continuous group of unitary operators {𝑈^𝑡}_𝑡∈ℝ which includes the unitary operator 𝑈, say, for instance, that 𝑈∶=𝑈¹.

3.1. On Continuous Groups of Unitary Operators

Let {𝑈^𝑡}_𝑡∈ℝ denote a continuous group of unitary operators in ℋ such that 𝑈∶=𝑈¹. Recall that {𝑈^𝑡}_𝑡∈ℝ is a family of unitary operators in ℋ satisfying (see [25, vol. 2; page 29]) the following:(1)𝑈^𝑡𝑈^𝑡′=𝑈^𝑡+𝑡′,(2)𝑈⁰=𝐼_ℋ,(3)⟨𝑈^𝑡𝑥,𝑦⟩_ℋ is a continuous function of 𝑡 for any 𝑥,𝑦∈ℋ.

Note that (𝑈^𝑡)⁻¹=𝑈^−𝑡, and since (𝑈^𝑡)^∗=(𝑈^𝑡)⁻¹, we have (𝑈^𝑡)^∗=𝑈^−𝑡.

Classical Stone's theorem [26] assures us the existence of a self-adjoint operator 𝑇 (possibly unbounded) such that 𝑈^𝑡≡e^i𝑡𝑇. This self-adjoint operator 𝑇, defined on the dense domain of ℋ as follows: 𝐷_𝑇∶=𝑥∈ℋsuchthat^∞_−∞𝑤²𝑑𝐸_𝑤𝑥²<∞, (72) admits the spectral representation   𝑇=∫^∞_−∞𝑤𝑑𝐸_𝑤 which means that 𝑇𝑥,𝑦=^∞_−∞𝑤𝑑𝐸_𝑤𝑥,𝑦forany𝑥∈𝐷_𝑇,𝑦∈ℋ, (73) where {𝐸_𝑤}_𝑤∈ℝ is the corresponding resolution of the identity, that is, a one-parameter family of projection operators 𝐸_𝑤 in ℋ such that(i)𝐸_−∞∶=lim_{𝑤→−∞}𝐸_𝑤=𝑂_ℋ, 𝐸_∞∶=lim_𝑤→∞𝐸_𝑤=𝐼_ℋ, (ii)𝐸_𝑤−=𝐸_𝑤 for every −∞<𝑤<∞, (iii)𝐸_𝑢𝐸_𝑣=𝐸_𝑤 where 𝑤=min{𝑢,𝑣}.

Recall that ‖𝐸_𝑤𝑥‖² and ⟨𝐸_𝑤𝑥,𝑦⟩, as functions of 𝑤, have bounded variation and define, respectively, a positive and a complex Borel measure on ℝ.

Furthermore, for any 𝑥∈𝐷_𝑇, we have that lim_𝑡→0((𝑈^𝑡𝑥−𝑥)/𝑡)=i𝑇𝑥 and the operator i𝑇 is said to be the infinitesimal generator of the group {𝑈^𝑡}_𝑡∈ℝ. For each 𝑥∈𝐷_𝑇, 𝑈^𝑡𝑥 is a continuous differentiable function of 𝑡. Notice that, whenever the self-adjoint operator 𝑇 is bounded, 𝐷_𝑇=ℋ and e^i𝑡𝑇 can be defined as the usual exponential series; in any case, 𝑈^𝑡≡e^i𝑡𝑇 means that 𝑈^𝑡𝑥,𝑦=^∞_−∞e^i𝑤𝑡𝑑𝐸_𝑤𝑥,𝑦,𝑡∈ℝ, (74) where 𝑥∈𝐷_𝑇 and 𝑦∈ℋ. A comment on the continuity of a group of unitary operators is in order: The group is said to be strongly continuous if, for each 𝑥∈ℋ and 𝑡₀∈ℝ, 𝑈^𝑡𝑥→𝑈^𝑡0𝑥 as 𝑡→𝑡₀. If ℋ is a separable Hilbert space, strong continuity can be deduced from continuity and even from weak measurability; that is, ⟨𝑈^𝑡𝑥,𝑦⟩_ℋ is a Lebesgue measurable function of 𝑡 for any 𝑥,𝑦∈ℋ.

The following result taken from [25, vol. 2; page 24] will be used later. For 𝑥∈𝐷_𝑇 and 𝑦∈ℋ, the inequality ^∞_−∞𝜑(𝑤)𝑑𝐸_𝑤𝑥,𝑦≤𝑦^∞_−∞𝜑(𝑤)²𝑑𝐸_𝑤𝑥,𝑥, (75) holds, where 𝜑 is a real or complex function which is continuous in ℝ with the possible exception of a finite number of points.

For the details on the theory of continuous groups of unitary operators, see [22, 25, 27, 28].

3.2. Studying the Perturbed Sequence {𝑈^{𝑟𝑘+𝜖𝑘𝑗}𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…𝑠}

Given an error sequence 𝝐∶={𝜖_𝑘𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠}, consider the corresponding perturbed sequence {𝑈^{𝑟𝑘+𝜖𝑘𝑗}𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠}. Consider the following perturbation result (see [1, page 354] for the proof).

Lemma 11. Let {𝑥_𝑛}^∞_𝑛=1 be a frame for the Hilbert space ℋ with frame bounds 𝐴,𝐵, and let {𝑦_𝑛}^∞_𝑛=1 be a sequence in ℋ. If there exists a constant 𝑅<𝐴 such that ^∞_𝑛=1𝑥_𝑛−𝑦_𝑛,𝑥²≤𝑅‖𝑥‖²𝑓𝑜𝑟𝑒𝑎𝑐ℎ𝑥∈ℋ, (76) then, the sequence {𝑦_𝑛}^∞_𝑛=1 is also a frame for ℋ with bounds 𝐴(1−𝑅/𝐴)² and 𝐵(1+𝑅/𝐵)². If {𝑥_𝑛}^∞_𝑛=1 is a Riesz basis, then {𝑦_𝑛}^∞_𝑛=1 is a Riesz basis.

Note that it cannot be directly applied to the sequences {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} and {𝑈^{𝑟𝑘+𝜖𝑘𝑗}𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} since the first one is not a frame for the entire Hilbert space ℋ, and its perturbed sequence does not necessarily belong to the subspace 𝒜_𝑎. However, something can be said in case {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…𝑟} is a Riesz sequence in ℋ.

Theorem 12. Assume that for some 𝑏_𝑗∈𝐷_𝑇; that is, ∫^∞_−∞𝑤²𝑑‖𝐸_𝑤𝑏_𝑗‖²<∞ for each 1≤𝑗≤𝑟, the sequence {𝑈^𝑘𝑟𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑟} is a Riesz basis for 𝒜_𝑎 with Riesz bounds 0<𝐴_Ψ≤𝐵_Ψ<∞. For a sequence 𝝐∶={𝜖_𝑘𝑗}_{𝑘∈ℤ,𝑗=1,2,…,𝑟} of errors, let 𝑅 be the constant given by 𝑅∶=‖𝝐‖²max_{𝑗=1,2,…,𝑟}^∞_−∞𝑤²𝑑𝐸_𝑤𝑏_𝑗², (77) where ‖𝝐‖ denotes the ℓ²_𝑟-norm of the sequence 𝝐.
If 𝑅<𝐴_Ψ, then the sequence {𝑈^{𝑘𝑟+𝜖𝑘𝑗}𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑟} is a Riesz sequence in ℋ with Riesz bounds 𝐴_Ψ(1−𝑅/𝐴_Ψ)² and 𝐵_Ψ(1+𝑅/𝐵_Ψ)².

Proof. By using (75), we have 𝑥,𝑈^𝑘𝑟𝑏_𝑗−𝑈^{𝑘𝑟+𝜖𝑘𝑗}𝑏_𝑗=^∞_−∞e^{−i𝑘𝑟𝑤}𝑑𝐸_𝑤𝑥,𝑏_𝑗−^∞_−∞e^{−i𝑘𝑟𝑤−i𝜖𝑘𝑗𝑤}𝑑𝐸_𝑤𝑥,𝑏_𝑗=^∞_−∞e^{−i𝑘𝑟𝑤}1−e^{−i𝜖𝑘𝑗𝑤}𝑑𝐸_𝑤𝑥,𝑏_𝑗=^∞_−∞e^{i𝑘𝑟𝑤}1−e^{i𝜖𝑘𝑗𝑤}𝑑𝐸_𝑤𝑏_𝑗≤‖𝑥‖,𝑥_∞−∞1−e^{i𝜖𝑘𝑗𝑤2}𝑑𝐸_𝑤𝑏_𝑗²≤‖𝑥‖^∞_−∞𝑤²𝜖_𝑘𝑗²𝑑𝐸_𝑤𝑏_𝑗²=𝜖_𝑘𝑗‖𝑥‖^∞_−∞𝑤²𝑑𝐸_𝑤𝑏_𝑗². (78)
Hence, ^𝑟_{𝑗=1𝑘∈ℤ}𝑥,𝑈^𝑘𝑟𝑏_𝑗−𝑈^{𝑘𝑟+𝜖𝑘𝑗}𝑏_𝑗²≤‖𝑥‖^2𝑟_{𝑗=1𝑘∈ℤ}^∞_−∞𝑤²𝑑𝐸_𝑤𝑏_𝑗²𝜖_𝑘𝑗²≤‖𝑥‖²max_{𝑗=1,2,…,𝑟}^∞_−∞𝑤²𝑑𝐸_𝑤𝑏_𝑗^2𝑟_{𝑗=1𝑘∈ℤ}𝜖_𝑘𝑗². (79) Hence, Lemma 11 and Theorem 15.3.2 in [1, page 356] give the desired results.

3.3. On the Perturbed Frame Expansion

Next, we deal with the problem of the recovery of any 𝑥∈𝒜_𝑎 in a stable way from the following perturbed sequence: 𝑥,𝑈^{𝑟𝑚+𝜖𝑚𝑗}𝑏_{𝑗ℋ𝑚∈ℤ;𝑗=1,2,…,𝑠}, (80) where 𝝐∶={𝜖_𝑚𝑗}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} denotes a sequence of real errors. In order to face this problem, we propose a possible strategy. Let 𝒯_𝑈,𝑎∶𝐿²(0,1)→𝒜_𝑎 be the isomorphism which maps the orthonormal basis {e^{2𝜋i𝑛𝑤}}_𝑛∈ℤ onto the Riesz basis {𝑈^𝑛𝑎}_𝑛∈ℤ for 𝒜_𝑎. In other words, 𝒯_𝑈,𝑎:𝐿²(0,1)⟶𝒜_{𝑎𝑛∈ℤ}𝛼_𝑛e^{2𝜋i𝑛𝑤}⟼𝑥=_𝑛∈ℤ𝛼_𝑛𝑈^𝑛𝑎. (81)

Thus, 𝑥,𝑈^𝑡𝑏_𝑗ℋ=_𝑛∈ℤ𝛼_𝑛𝑈^𝑛𝑎,𝑈^𝑡𝑏_𝑗ℋ=_𝑛∈ℤ𝛼_𝑛𝑈^𝑡𝑏_𝑗,𝑈^𝑛𝑎_ℋ=𝐹,_𝑛∈ℤ𝑈^𝑡𝑏_𝑗,𝑈^𝑛𝑎_ℋe^{2𝜋i𝑛𝑤}_𝐿2(0,1)=𝐹,𝐾^𝑡_{𝑗𝐿2(0,1)}, (82) where 𝒯_𝑈,𝑎𝐹=𝑥, and the function 𝐾^𝑡_𝑗(𝑤)∶=∑_𝑛∈ℤ⟨𝑈^𝑡𝑏_𝑗,𝑈^𝑛𝑎⟩_ℋe^{2𝜋i𝑛𝑤} belongs to 𝐿²(0,1) since the sequence {⟨𝑈^𝑡𝑏,𝑈^𝑛𝑎⟩_ℋ}_𝑛∈ℤ belongs to ℓ²(ℤ) for each 𝑡∈ℝ.

Hence, for any 𝑥∈𝒜_𝑎, we have the following expressions: 𝑥,𝑈^𝑟𝑚𝑏_𝑗ℋ=𝐹,𝑔_𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}_𝐿2(0,1),𝑥,𝑈^{𝑟𝑚+𝜖𝑚𝑗}𝑏_𝑗ℋ=𝐹,𝑔_𝑚,𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}_𝐿2(0,1), (83) where the functions 𝑔_𝑗(𝑤)∶=_𝑘∈ℤ𝑎,𝑈^𝑘𝑏_𝑗ℋe^{2𝜋i𝑘𝑤},𝑔_𝑚,𝑗(𝑤)∶=_𝑘∈ℤ𝑎,𝑈^{𝑘+𝜖𝑚𝑗}𝑏_𝑗ℋe^{2𝜋i𝑘𝑤} (84) belong to 𝐿²(0,1). Therefore, we can see the sequence {𝑔_𝑚,𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} as a perturbation of the sequence {𝑔_𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}}_{𝑚∈ℤ;𝑗=1,2,…,𝑠}. From [6, Lemma 3], we know that this sequence is a frame for 𝐿²(0,1) if and only if 0<𝛼_𝔾≤𝛽_𝔾<∞ where the constants 𝛼_𝔾 and 𝛽_𝔾 are given by 𝛼_𝔾∶=essinf_{𝑤∈(0,1/𝑟)}𝜆_min𝔾^∗(𝑤)𝔾(𝑤,𝛽_𝔾)∶=esssup_{𝑤∈(0,1/𝑟)}𝜆_max𝔾^∗(𝑤)𝔾(𝑤), (85) and 𝔾(𝑤) is the 𝑠×𝑟 matrix 𝔾(𝑤)∶=𝑔_𝑗𝑤+𝑘−1𝑟^{𝑗=1,2,…,𝑠}_{𝑘=1,2,…,𝑟}. (86) Besides, the optimal frame bounds for {𝑔_𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} are 𝛼_𝔾/𝑟 and 𝛽_𝔾/𝑟.

Given an error sequence 𝝐∶={𝜖_𝑚𝑗}_{𝑚∈ℤ;𝑗=1,2,…,𝑠}, we define 𝑑^(𝑗)_𝑚,𝑘∶=𝑎,𝑈^{𝑟𝑚−𝑘+𝜖𝑚𝑗}𝑏_𝑗−𝑎,𝑈^{𝑟𝑚−𝑘}𝑏_𝑗. (87) For any sequence 𝑐={𝑐_𝑘}_𝑘∈ℤ∈ℓ²(ℤ), we have ^𝑠_{𝑗=1𝑚∈ℤ𝑘∈ℤ}𝑑^(𝑗)_𝑚,𝑘𝑐_𝑘²≤^𝑠_{𝑗=1𝑚∈ℤ𝑙,𝑘∈ℤ}𝑑^(𝑗)_𝑚,𝑙𝑐_𝑙𝑑^(𝑗)_𝑚,𝑘𝑐_𝑘=^𝑠_{𝑗=1𝑙,𝑘∈ℤ}𝑐_𝑙𝑐_{𝑘𝑚∈ℤ}𝑑^(𝑗)_𝑚,𝑙𝑑^(𝑗)_𝑚,𝑘≤^𝑠_{𝑗=1𝑙,𝑘∈ℤ}𝑐_𝑙²+𝑐_𝑘²2_𝑚∈ℤ𝑑^(𝑗)_𝑚,𝑙𝑑^(𝑗)_𝑚,𝑘=^𝑠_{𝑗=1𝑙∈ℤ}𝑐_𝑙²_{𝑘,𝑚∈ℤ}𝑑^(𝑗)_𝑚,𝑙𝑑^(𝑗)_𝑚,𝑘. (88)

Now, for |𝛾|<1/2, define the following functions: 𝑀_{𝑎,𝑏𝑗}𝛾∶=_𝑘∈ℤmax_{𝑡∈−𝛾,𝛾}𝑎,𝑈^𝑘+𝑡𝑏_𝑗−𝑎,𝑈^𝑘𝑏_𝑗,𝑁_{𝑎,𝑏𝑗}𝛾∶=max_{𝑘=0,1,…,𝑟−1𝑚∈ℤ}max_{𝑡∈−𝛾,𝛾}𝑎,𝑈^{𝑟𝑚+𝑘+𝑡}𝑏_𝑗−𝑎,𝑈^{𝑟𝑚+𝑘}𝑏_𝑗. (89)

Notice that 𝑁_{𝑎,𝑏𝑗}(𝛾)≤𝑀_{𝑎,𝑏𝑗}(𝛾) and for 𝑟=1, the equality holds. Moreover, assuming that the continuous functions 𝜑_𝑗(𝑡)∶=⟨𝑎,𝑈^𝑡𝑏_𝑗⟩, 𝑗=1,2,…,𝑠, satisfy a decay condition as 𝜑_𝑗(𝑡)=𝑂(|𝑡|^{−(1+𝜂𝑗)}) when |𝑡|→∞ for some 𝜂_𝑗>0, we may deduce that the functions 𝑁_{𝑎,𝑏𝑗}(𝛾) and 𝑀_{𝑎,𝑏𝑗}(𝛾) are continuous near to 0.

Theorem 13. Assume that for the functions 𝑔_𝑗, 𝑗=1,2,…,𝑠, given in (84) one has 0<𝛼_𝔾≤𝛽_𝔾<∞. For an error sequence 𝝐∶={𝜖_𝑚𝑗}_{𝑚∈ℤ;𝑗=1,…,𝑠}, define the constant 𝛾_𝑗∶=sup_𝑚∈ℤ|𝜖_𝑚𝑗| for each 𝑗=1,2,…,𝑠. Then the following condition: ^𝑠_𝑗=1𝑀_{𝑎,𝑏𝑗}𝛾_𝑗𝑁_{𝑎,𝑏𝑗}𝛾_𝑗<𝛼_𝔾𝑟 (90) implies that there exists a frame {𝐶^𝝐_𝑚,𝑗}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} for 𝒜_𝑎 such that, for any 𝑥∈𝒜_𝑎, the following sampling expansion: 𝑥=^𝑠_{𝑗=1𝑚∈ℤ}⟨𝑥,𝑈^{𝑟𝑚+𝜖𝑚𝑗}𝑏_𝑗⟩_ℋ𝐶^𝝐_𝑚,𝑗𝑖𝑛ℋ (91) holds. Moreover, when 𝑟=𝑠, the sequence {𝐶^𝝐_𝑚,𝑗}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} is a Riesz basis for 𝒜_𝑎, and the interpolation property ⟨𝐶^𝝐_𝑛,𝑙,𝑈^{𝑟𝑚+𝜖𝑚𝑗}𝑏_𝑗⟩_ℋ=𝛿_𝑙,𝑗𝛿_𝑛,𝑚 holds.

Proof. The sequence {𝑔_𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} is a frame (a Riesz basis if 𝑟=𝑠) for 𝐿²(0,1) with frame (Riesz) bounds 𝛼_𝔾 and 𝛽_𝔾. For any 𝐹(𝑤)=∑_𝑙∈ℤ𝑎_𝑙e^{2𝜋i𝑙𝑤} in 𝐿²(0,1), we have ^𝑠_{𝑚∈ℤ𝑗=1}𝑔_𝑚,𝑗(⋅)e^{2𝜋i𝑟𝑚⋅}−𝑔_𝑗(⋅)e^{2𝜋i𝑟𝑚⋅},𝐹(⋅)_𝐿2(0,1)²=^𝑠_{𝑚∈ℤ𝑗=1𝑘∈ℤ}𝑎,𝑈^{𝑘+𝜖𝑚𝑗}𝑏_𝑗−𝑎,𝑈^𝑘𝑏_𝑗e^{2𝜋i(𝑟𝑚−𝑘)⋅},𝐹(⋅)_𝐿2(0,1)²=^𝑠_{𝑚∈ℤ𝑗=1𝑘∈ℤ}𝑎,𝑈^{𝑟𝑚−𝑘+𝜖𝑚𝑗}𝑏_𝑗−𝑎,𝑈^{𝑟𝑚−𝑘}𝑏_𝑗e^{2𝜋i𝑘⋅},𝐹(⋅)_𝐿2(0,1)²=^𝑠_{𝑚∈ℤ𝑗=1𝑘∈ℤ}𝑎,𝑈^{𝑟𝑚−𝑘+𝜖𝑚𝑗}𝑏_𝑗−𝑎,𝑈^{𝑟𝑚−𝑘}𝑏_𝑗𝑎_𝑘²=^𝑠_{𝑗=1𝑚∈ℤ𝑘∈ℤ}𝑑^(𝑗)_𝑚,𝑘𝑎_𝑘². (92)
From (88) and the definition of the functions 𝑀_{𝑎,𝑏𝑗} and 𝑁_{𝑎,𝑏𝑗}, we obtain ^𝑠_{𝑚∈ℤ𝑗=1}𝑔_𝑚,𝑗(⋅)e^{2𝜋i𝑟𝑚⋅}−𝑔_𝑗(⋅)e^{2𝜋i𝑟𝑚⋅},𝐹(⋅)_𝐿2(0,1)²≤^𝑠_𝑗=1𝑀_{𝑎,𝑏𝑗}𝛾_𝑗𝑁_{𝑎,𝑏𝑗}𝛾_𝑗𝑎_{𝑙𝑙∈ℤ}²≤^𝑠_𝑗=1𝑀_{𝑎,𝑏𝑗}𝛾_𝑗𝑁_{𝑎,𝑏𝑗}𝛾_𝑗‖𝐹‖²_𝐿2()0,1. (93) By using Lemma 11, we obtain that the sequence {𝑔_𝑚,𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} is a frame for 𝐿²(0,1) (a Riesz basis if 𝑟=𝑠). Let {ℎ^𝝐_𝑚,𝑗}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} be its canonical dual frame. Hence, for any 𝐹∈𝐿²(0,1), 𝐹=^𝑠_{𝑚∈ℤ𝑗=1}𝐹(⋅),𝑔_𝑚,𝑗(⋅)e^{2𝜋i𝑟𝑚⋅}_𝐿2(0,1)ℎ^𝝐_𝑚,𝑗=^𝑠_{𝑚∈ℤ𝑗=1}𝑥,𝑈^{𝑟𝑚+𝜖𝑚𝑗}𝑏_𝑗ℋℎ^𝝐_𝑚,𝑗. (94)
Applying the isomorphism 𝒯_𝑈,𝑎, one gets (91), where 𝐶^𝝐_𝑚,𝑗=𝒯_𝑈,𝑎(ℎ^𝝐_𝑚,𝑗). Since 𝒯_𝑈,𝑎 is an isomorphism between 𝐿²(0,1) and 𝒜_𝑎; the sequence {𝐶^𝝐_𝑚,𝑗}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} is a frame for 𝒜_𝑎 (a Riesz basis if 𝑟=𝑠). The interpolatory property in the case 𝑟=𝑠 follows from the uniqueness of the coefficients with respect to a Riesz basis.

3.4. A Frame Algorithm in ℓ²(ℤ)

Sampling formula (91) is useless from a practical point of view, it is impossible to determine the involved frame {𝐶^𝝐_𝑚,𝑗}_{𝑚∈ℤ;𝑗=1,2,…,𝑠}. As a consequence, in order to recover 𝑥∈𝒜_𝑎 from the sequence of inner products {⟨𝑥,𝑈^{𝑟𝑚+𝜖𝑚𝑗}𝑏_𝑗⟩_ℋ}_{𝑚∈ℤ;𝑗=1,2,…,𝑠}, we should implement a frame algorithm in ℓ²(ℤ). Another possibility is given in the recent reference [29].

Next we are going to implement a frame algorithm in the ℓ²(ℤ) setting. To this end, consider the canonical isometry 𝒰∶ℓ²(ℤ)⟶𝐿²(0,1)suchthat𝒰𝑐∶=_𝑘∈ℤ𝑐_𝑘e^{2𝜋i𝑘𝑤}for𝑐=𝑐_{𝑘𝑘∈ℤ}∈ℓ²(ℤ). (95)

For 𝑥=∑_𝑛∈ℤ𝑐_𝑛𝑈^𝑛𝑎∈𝒜_𝑎, denote by 𝔽 the following sequence: 𝔽∶=𝒰⁻¹𝒯⁻¹_𝑈,𝑎𝑥=𝑐_{𝑛𝑛∈ℤ}∈ℓ²(ℤ). (96)

The inner products {⟨𝑥,𝑈^{𝑟𝑚+𝜖𝑚𝑗}𝑏_𝑗⟩_ℋ}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} can be written as 𝑥,𝑈^{𝑟𝑚+𝜖𝑚𝑗}𝑏_𝑗ℋ=𝒯⁻¹_𝑈,𝑎𝑥,𝑔_𝑚,𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}_𝐿2(0,1)=⟨𝔽,𝕃_𝑚,𝑗⟩_ℓ²_(ℤ), (97) where, for 𝑗=1,2,…,𝑠 and 𝑚∈ℤ, 𝕃_𝑚,𝑗∶=𝒰⁻¹𝑔_𝑚,𝑗(⋅)e^{2𝜋i𝑟𝑚⋅}=𝒰⁻¹_𝑘∈ℤ⟨𝑎,𝑈^{𝑘+𝜖𝑚𝑗}𝑏_𝑗⟩_ℋe^{2𝜋i𝑘⋅}e^{2𝜋i𝑟𝑚⋅}=𝒰⁻¹_𝑘∈ℤ𝑎,𝑈^{𝑘+𝜖𝑚𝑗}𝑏_𝑗ℋe^{2𝜋i(𝑟𝑚−𝑘)⋅}=𝒰⁻¹_𝑘∈ℤ𝑎,𝑈^{𝑟𝑚−𝑘+𝜖𝑚𝑗}𝑏_𝑗ℋe^{2𝜋i𝑘⋅}=𝑎,𝑈^{𝑟𝑚−𝑘+𝜖𝑚𝑗}𝑏_{𝑗ℋ𝑘∈ℤ}. (98)

The sequence {𝕃_𝑚,𝑗}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} is a frame for ℓ²(ℤ). Indeed, assume that the error sequence 𝝐∶={𝜖_𝑚,𝑗}_{𝑚∈ℤ;𝑗=1,…,𝑠} satisfies the hypothesis of Theorem 13; that is, 𝐾_𝝐∶=^𝑠_𝑗=1𝑀_{𝑎,𝑏𝑗}𝛾_𝑗𝑁_{𝑎,𝑏𝑗}𝛾_𝑗<𝛼_𝔾𝑟, (99) where 𝛾_𝑗∶=sup_𝑚∈ℤ|𝜖_𝑚𝑗| for each 𝑗=1,2,…,𝑠. As a consequence, the sequence {𝑔_𝑚,𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} is a frame for 𝐿²(0,1) with bounds 𝐴_𝝐𝛼∶=^𝔾𝑟1−𝑟𝐾_𝝐𝛼_𝔾²,𝐵_𝝐𝛽∶=^𝔾𝑟1−𝑟𝐾_𝝐𝛽_𝔾². (100)

Since 𝒰⁻¹ is an isometry, the sequence {𝕃_𝑚,𝑗}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} is a frame for ℓ²(ℤ) with the same bounds. Hence, the recovery of the element 𝑥=𝒯_𝑈,𝑎(𝒰𝔽)∈𝒜_𝑎 from the samples {⟨𝑥,𝑈^{𝑟𝑚+𝜖𝑚𝑗}𝑏_𝑗⟩_ℋ}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} is reduced to recover 𝔽∈ℓ²(ℤ) from the following sequence: 𝔽,𝕃_{𝑚,𝑗ℓ2(ℤ)𝑚∈ℤ;𝑗=1,2,…,𝑠}. (101)

In doing so, the classical frame algorithm reads (see, for instance, [1]), let 𝒮 be the frame operator in ℓ²(ℤ) of {𝕃_𝑚,𝑗}_{𝑚∈ℤ;𝑗=1,2,…,𝑠}, and define 𝔽₀∶=2𝒮𝐴_𝝐+𝐵_𝝐𝔽=2𝐴_𝝐+𝐵_𝝐^𝑠_{𝑗=1𝑚∈ℤ}𝔽,𝕃_𝑚,𝑗𝕃_𝑚,𝑗=2𝐴_𝝐+𝐵_𝝐^𝑠_{𝑗=1𝑚∈ℤ}𝑥,𝑈^{𝑟𝑚+𝜖𝑚𝑗}𝑏_𝑗𝕃_𝑚,𝑗, (102) and recursively, 𝔽_𝑘+1=𝔽_𝑘+2𝒮𝐴_𝝐+𝐵_𝝐𝔽−𝔽_𝑘foreach𝑘∈ℕ. (103)

Then, the sequence {𝑥_𝑛}_𝑛∈ℕ in 𝒜_𝑎 given by 𝑥_𝑛∶=∑_𝑘∈ℤ(𝔽_𝑛)_𝑘𝑈^𝑘𝑎, satisfies 𝑥−𝑥_𝑛ℋ≤𝒯_𝑈,𝑎𝔽−𝔽_{𝑘ℓ2(ℤ)}≤𝒯_𝑈,𝑎𝛾_𝝐^𝑘+1‖𝔽‖_ℓ²_(ℤ)≤𝒯_𝑈,𝑎𝒯⁻¹_𝑈,𝑎𝛾_𝝐^𝑘+1‖𝑥‖_ℋ, (104) where 𝛾_𝝐∶=(𝐵_𝝐−𝐴_𝜖)/(𝐵_𝝐+𝐴_𝝐).

4. Some Additional Remarks

Given 𝑠 vectors 𝑏_𝑗∈𝒜_𝑎, 𝑗=1,2,…,𝑠 with 𝑠≥𝑟, we have proved in Section 2 that the sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a frame for 𝒜_𝑎 if and only if the constants 𝐴_Ψ and 𝐵_Ψ defined in (56) satisfy 0<𝐴_Ψ≤𝐵_Ψ<∞. Furthermore, we have obtained a family of dual frames having the same form. As it was mentioned in the introduction, now we deal with the case that some 𝑏_𝑗∉𝒜_𝑎.(i) We have assumed in Theorems 5 and 7 that 𝑏_𝑗 belongs to 𝒜_𝑎 for each 𝑗=1,2,…,𝑠 since we required the sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} to be contained in 𝒜_𝑎. In case that some 𝑏_𝑗∉𝒜_𝑎, the sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is not necessarily contained in 𝒜_𝑎. However, a close look into the proof of Theorem 5 shows that whenever 0<𝐴_Ψ≤𝐵_Ψ<∞, the following inequalities: 𝐴_Ψ‖𝑥‖²≤^𝑠_{𝑗=1𝑘∈ℤ}𝑥,𝑈^𝑟𝑘𝑏_𝑗²≤𝐵_Ψ‖𝑥‖²∀𝑥∈𝒜_𝑎 (105) hold, and conversely. Hence, the sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a pseudoframe for 𝒜_𝑎 (see [17, 18]).

Denoting by 𝑃_𝒜𝑎 the orthogonal projection onto 𝒜_𝑎, since for each 𝑥∈𝒜_𝑎, 𝑥,𝑈^𝑟𝑘𝑏_𝑗=𝑥,𝑃_𝒜𝑎𝑈^𝑟𝑘𝑏_𝑗,𝑘∈ℤ,𝑗=1,2,…,𝑠. (106)

Theorems 5 and 7 can be reformulated in terms of {𝑃_𝒜𝑎(𝑈^𝑟𝑘𝑏_𝑗)}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} which is a sequence in 𝒜_𝑎.(ii) An analysis of the proof of Theorem 9 shows that, even if not all of the 𝑏_𝑗 belong to 𝒜_𝑎, there exist 𝑐_𝑗∈𝒜_𝑎, 𝑗=1,2,…,𝑠, such that the sequence {𝑈^𝑟𝑘𝑐_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a frame for 𝒜_𝑎, and the expansion (60) holds for each 𝑥∈𝒜_𝑎. Therefore, in case that some 𝑏_𝑗∉𝒜_𝑎, the sequence {𝑈^𝑟𝑘𝑏_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} is a pseudodual frame of the frame {𝑈^𝑟𝑘𝑐_𝑗}_{𝑘∈ℤ;𝑗=1,2,…,𝑠} for 𝒜_𝑎 (see [17]).(iii) In Section 3.3, having in mind the isomorphism 𝒯_𝑈,𝑎, for 𝑥=𝒯_𝑈,a𝐹∈𝒜_𝑎, we have obtained the following expressions: 𝑥,𝑈^𝑟𝑚𝑏_𝑗ℋ=𝐹,𝑔_𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}_𝐿2(0,1),where𝑚∈ℤ,𝑗=1,2,…,𝑠. (107)

Furthermore, we know that the sequence {𝑔_𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} is a frame for 𝐿²(0,1) if and only if the constants 𝛼_𝔾 and 𝛽_𝔾 given in (85) satisfy 0<𝛼_𝔾≤𝛽_𝔾<∞; besides, the optimal frame bounds for {𝑔_𝑗(𝑤)e^{2𝜋i𝑟𝑚𝑤}}_{𝑚∈ℤ;𝑗=1,2,…,𝑠} are 𝛼_𝔾/𝑟 and 𝛽_𝔾/𝑟. Hence, we obtain 𝛼_𝔾𝑟𝒯_𝑈,𝑎⁻²‖𝑥‖²≤^𝑠_{𝑗=1𝑘∈ℤ}𝑥,𝑈^𝑟𝑘𝑏_𝑗²≤𝛽_𝔾𝑟𝒯⁻¹_𝑈,𝑎²‖𝑥‖²,𝑥∈𝒜_𝑎. (108)

As a consequence, since we are dealing with optimal frame bounds, from (105) and (108), we derive the following equalities: 𝐴_Ψ=𝛼_𝔾𝑟𝒯_𝑈,𝑎⁻²,𝐵_Ψ=𝛽_𝔾𝑟𝒯⁻¹_𝑈,𝑎². (109)

Acknowledgments

The authors are indebted to the anonymous referee for her/his valuable suggestions. This work has been supported by Grant MTM2009-08345 from the Spanish Ministerio de Ciencia e Innovación (MICINN).