A Balian–Low type theorem for Gabor Riesz sequences of arbitrary density

Gabor systems are used in fields ranging from audio processing to digital communication. Such a Gabor system (g,Λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(g,\varLambda )$$\end{document} consists of all time-frequency shifts π(λ)g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi (\lambda ) g$$\end{document} of a window function g∈L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g \in L^2({\mathbb {R}})$$\end{document} along a lattice Λ⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda \subset {\mathbb {R}}^2$$\end{document}. We focus on Gabor systems that are also Riesz sequences, meaning that one can stably reconstruct the coefficients c=(cλ)λ∈Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c = (c_\lambda )_{\lambda \in \varLambda }$$\end{document} from the function ∑λ∈Λcλπ(λ)g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{\lambda \in \varLambda } c_\lambda \, \pi (\lambda ) g$$\end{document}. In digital communication, a function of this form is used to transmit the digital sequence c. It is desirable for g to be well localized in time and frequency, since the transmitted signal will then be almost compactly supported in time and frequency if the sequence c has finite support. In this paper, we study what additional structural properties the signal space G(g,Λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(g,\varLambda )$$\end{document}, i.e., the span of the Gabor system, satisfies in addition to being a closed subspace of L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb {R}})$$\end{document}. The most well-known result in this direction—the Balian–Low theorem—states that if g is well localized in time and frequency and if (g,Λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(g,\varLambda )$$\end{document} is a Riesz sequence, then G(g,Λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(g,\varLambda )$$\end{document} is necessarily a proper subspace of L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb {R}})$$\end{document}. We prove a generalization of this result related to the invariance of G(g,Λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}(g,\varLambda )$$\end{document} under time-frequency shifts. Precisely, we show that if (g,Λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(g,\varLambda )$$\end{document} is a Riesz sequence with g being well localized in time and frequency (precisely, g should belong to the so-called Feichtinger algebra), then π(μ)G(g,Λ)⊂G(g,Λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi (\mu ) \mathcal {G}(g,\varLambda ) \subset \mathcal {G}(g,\varLambda )$$\end{document} holds if and only if μ∈Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in \varLambda $$\end{document}. For lattices of rational density, this was already known, with the proof based on Zak transform techniques. These methods do not generalize to arbitrary lattices, however. Instead, our proof for lattices of irrational density relies on combining methods from time-frequency analysis with properties of a special C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra, the so-called irrational rotation algebra.


Introduction
The central idea of digital signal processing is to represent continuous signals in terms of discrete sequences of coefficients. This is usually achieved by taking the inner products of the signal with respect to some well-structured system of functions. In the area of applied harmonic analysis, many different such systems have been proposed, with wavelet and Gabor systems being the most popular among them. The most important property of such a system is the ability to stably represent arbitrary signals; in this case, the system is called a frame [8].
Formally, a Gabor system (g, Λ) consists of all time-frequency shifts π(λ)g of the window function g ∈ L 2 (R) along a lattice Λ ⊂ R 2 , where [π(x, ω)g](y) = e 2πiωy · g(y − x); see [16]. When working with such a Gabor frame, the window function g should have a good time-frequency localization, so that the frame coefficients faithfully reflect the timefrequency behavior of the analyzed function. The Feichtinger algebra S 0 (R d ) [13,21] is a particularly popular window class. Among other advantages, choosing a window from S 0 ensures that the canonical dual window also belongs to the Feichtinger algebra [17], so that for example the membership of a function f ∈ L 2 (R d ) in the modulation space M p,q (R d ) can be characterized in terms of the decay properties of its frame coefficients. One crucial obstruction, however, is that a Gabor system with window belonging to S 0 (R d ) can not form an orthonormal basis-in fact not even a Riesz basis-for L 2 (R d ). We call this phenomenon the S 0 Balian-Low theorem (or the Feichtinger algebra Balian-Low theorem); it is a consequence of the Amalgam Balian-Low theorem [2,Theorem 3.2]. The same no-go type result holds for the case where g belongs to the space H 1 (R d ) consisting of functions in the L 2 -Sobolev space H 1 (R d ) whose Fourier transform also belongs to H 1 (R d ). This is the classical Balian-Low theorem; see [16,Theorem 8.4.5] for the case of orthonormal bases, and [10, Theorem 2.3] for the general case.
Yet, even though a Gabor system with g ∈ S 0 (R d ) cannot form a Riesz basis for all of L 2 (R d ), it might still be a Riesz sequence, that is, a Riesz basis for its closed linear span G(g, Λ), at least if G(g, Λ) is a proper subspace of L 2 (R d ). In this case, one might wonder about further properties-in addition to being a proper subspace of L 2 (R d )-that the Gabor space G(g, Λ) has to have. One important property in time-frequency analysis is the invariance of G(g, Λ) under time-frequency shifts T a M b . For lattices Λ of rational density and for dimension d = 1, it was observed in [4] that if (g, Λ) is a Riesz sequence and if g ∈ S 0 (R), then the set of parameters (a, b) ∈ R 2 such that G(g, Λ) is invariant under the time-frequency shift T a M b is exactly equal to Λ. A multi-dimensional variant of this was derived in [5].
These results generalize the S 0 Balian-Low theorem to subspaces of L 2 (R). Indeed, to derive the S 0 Balian-Low theorem from the above result, note that if G(g, Λ) = L 2 (R) then G(g, Λ) is invariant under all time-frequency shifts, even under those with (a, b) / ∈ Λ; hence, g cannot belong to S 0 . A corresponding generalization of the classical Balian-Low theorem was proved in [6]; a quantitative version can be found in [7].
We emphasize that in all articles [4][5][6][7] it is assumed that the generating lattice Λ has rational density. This restriction is needed in order to utilize the Zak transform which is used extensively in [4][5][6][7]. It is thus natural to ask whether the results in [4] and [6] still hold for lattices with irrational density.
In a sense, this question has analogies with the research concerning the regularity of the canonical dual window of a Gabor frame. In 1997 it was shown (see [14,Theorem 3.4]) that if g ∈ S 0 (R d ) generates a Gabor frame for L 2 (R d ) over a lattice of rational density, then the canonical dual window also belongs to S 0 (R d ). It was conjectured in the same article that this property continues to hold for general lattices. Six years later, this conjecture was confirmed by Gröchenig and Leinert [17] by using C * -algebra methods.
Here, we likewise extend the result in [4] from lattices of rational density to arbitrary lattices: Theorem 1 If g ∈ S 0 (R) and Λ ⊂ R 2 is a lattice such that the Gabor system (g, Λ) is a Riesz basis for its closed linear span G(g, Λ), then the time-frequency shifts T a M b that leave In other words, the following conditions cannot hold simultaneously: One can easily find examples where only two of these conditions hold simultaneously. For instance, conditions (i) and (ii) hold for the Gaussian g(x) = e −π x 2 and Λ = Z × 2Z; conditions (i) and (iii) hold for g( 1 4 ] (x) and Λ = Z × 2Z; conditions (ii) and (iii) hold for all the cases in Example 6 below. Here, 1 S denotes the characteristic function of S ⊂ R.
As indicated above, the Zak transform is a powerful tool for analyzing Gabor systems generated by lattices with rational density; yet, it is not of much use in the case of irrational density lattices. Consequently, the methods used in the present paper differ substantially from those in [4][5][6][7]. Instead of applying the Zak transform and thus dealing with functions on R 2 , we work directly with the given objects and exploit the rich theory of time-frequency analysis. Along the way, we obtain several new statements related to time-frequency shift invariance that are interesting in their own right.
The proof of Theorem 1 consists of several steps. First, for g ∈ S 0 (R) and only assuming that (g, Λ) is a frame sequence-that is, a frame for its closed linear span-we prove the following dichotomy: Either (g, Λ) spans all of L 2 (R), or the set of (a, b) ∈ R 2 for which T a M b leaves G(g, Λ) invariant is a lattice containing Λ as a sublattice; (D) see Theorem 8. This result significantly reduces the range of parameters (a, b) that we need to consider. Next, we give a characterization for the invariance of G(g, Λ) under a time-frequency shift T a M b with (a, b) / ∈ Λ in terms of the adjoint system of (g, Λ); see Theorem 11. This characterization holds for general g ∈ L 2 (R), not only for g ∈ S 0 (R). Combining this characterization with a deep existing result about traces of projections in the so-called irrational rotation algebra (see [26,27]), we arrive at the conclusion of Theorem 1.
With Theorem 1 established for g in the Feichtinger algebra, it is natural to ask whether the same statement holds in the setting of the classical Balian-Low theorem, that is, when g has finite uncertainty product x 2 |g(x)| 2 dx · ω 2 | g(ω)| 2 dω < ∞, a condition which we simply write as g ∈ H 1 . Unfortunately, we were not able to prove a full-fledged version of Theorem 1 for g ∈ H 1 ; the best we could do is to show that the dichotomy (D) described above for g ∈ S 0 still holds for g ∈ H 1 .
The outline of the paper is as follows: After recalling the necessary background on Janssen's representation, time-frequency shift invariance, symplectic operators, and the two spaces S 0 (R) and H 1 in Sect. 2, the paper proper starts in Sect. 3, where we prove the dichotomy (D) described above, for g lying in a broad function space which contains both S 0 (R) and H 1 . Next, in Sect. 4 we show that one can reduce to the case of a separable lattice Λ = αZ × βZ, with an additional time-frequency shift of the form T α/ν for some ν ∈ N ≥2 . For this setting, we then derive a characterization in terms of the adjoint Gabor system. Throughout Sect. 4, the generating function g is only assumed to be in L 2 (R). The paper culminates in Sect. 5, where we prove Theorem 1 using an auxiliary result which relies on irrational rotation algebras. Appendix A contains a proof of the auxiliary result including a short treatise on irrational rotation algebras.

Preliminaries
For a, b ∈ R and f ∈ L 2 (R) we define the operators of translation by a and modulation by b as respectively. Both T a and M b are unitary operators on L 2 (R) and hence so is the timefrequency shift A (full rank) lattice Λ ⊂ R 2 is any set of the form Λ = AZ 2 with an invertible matrix A ∈ R 2×2 . The density of Λ is defined by d(Λ) = |det A| −1 . Note that AZ 2 = Z 2 if and only if A ∈ Z 2×2 and det A = ±1. This will be used heavily in the proof of Proposition 5 below.
A lattice Λ is called separable if A can be chosen to be diagonal, i.e., Λ = αZ × βZ with α, β > 0. The next lemma shows that every lattice can be transformed into a separable one by means of a symplectic matrix; this will be used frequently.
We use the normalization F f (ω) = f (ω) = R f (x) e −2πi xω dx for the Fourier transform of f ∈ L 1 (R). It is well-known that F extends to a unitary map F : if it ceases to be a frame when an arbitrary element is removed.

Bessel vectors and Janssen's representation
Let Λ = αZ × βZ be a separable lattice with α, β > 0. The adjoint lattice of Λ is defined as Λ • = 1 β Z × 1 α Z. We say that g ∈ L 2 (R) is a Bessel vector for Λ if the system (g, Λ) is a Bessel system in L 2 (R), meaning that the analysis operator C Λ,g corresponding to (g, Λ) is bounded as an operator from L 2 (R) to 2 (Z 2 ). It is defined by We denote the set of Bessel vectors for Λ by B Λ . This is a linear subspace of L 2 (R) which is dense because it contains the Schwartz space S(R); see [ Equation (1) implies that The series in Equations (2) and (3) both converge unconditionally in L 2 (R).

Time-frequency shift invariance
For a closed linear subspace G ⊂ L 2 (R), we denote by I(G) the set of all pairs (a, b) ∈ R 2 such that G is invariant under the time-frequency shift π(a, b); that is, If G = G(g, Λ) for some g ∈ L 2 (R) and a lattice Λ ⊂ R 2 , then clearly Λ ⊂ I(G). Any time-frequency shift π(z) with z ∈ I(G)\Λ will be called an additional time-frequency shift for G(g, Λ). For Gabor spaces G = G(g, Λ), the set I(G) has some additional structure: Lemma 3 shows that z ∈ I(G) implies −z ∈ I(G), i.e., π(z)G ⊂ G and π(z) −1 G ⊂ G. Hence, we have π(z)G = G whenever z ∈ I(G).
The next lemma characterizes the case when G is invariant under all time-frequency shifts.

Lemma 4 For a nonempty closed linear subspace
To prove the converse, assume that I(G) = R 2 and let f ∈ G ⊥ and g ∈ G\{0}. Then f , π(z)g = 0 for all z ∈ R 2 , so that the short-time Fourier transform V g f of f with window g satisfies V g f ≡ 0. By [16, Corollary 3.2.2] and since g = 0, this implies f = 0. We have thus shown G ⊥ = {0}, whence G = L 2 (R), since G is a closed subspace of L 2 (R).

Symplectic operators
It is often useful to reduce a statement involving a non-separable lattice to one that involves a separable lattice, since separable lattices are usually easier to handle. For this reduction, we will use so-called symplectic operators (see [16,Sect. 9.4] where (as in [16, Page 185 and Eq. (9.25)]) In the sequel, we fix for each B ∈ R 2×2 with det B = 1 one choice of the operator U B , and for functions g ∈ L 2 (R), closed subspaces G ⊂ L 2 (R), and sets Λ ⊂ R 2 we write As shown in [16,Page 197 Note that (4) implies Therefore, (g, Λ) is a frame (Riesz basis, resp.) for its closed linear span G if and only if (g B , Λ B ) is a frame (Riesz basis, resp.) for its closed linear span G B . Thanks to Lemma 3, the equivalence (5) also implies that

The Feichtinger algebra
We denote by S 0 (R) the Feichtinger algebra, which is the space of functions f ∈ L 2 (R) such that f , π(·)ϕ ∈ L 1 (R 2 ) for some (and hence every) for several characterizations of S 0 (R) and their proofs.  (3) and from the density where the series converges absolutely in operator norm.

The space H 1
Let H 1 (R) denote the space of all functions f in L 2 (R) for which the weak derivative f exists and belongs to L 2 (R). In other words, It is well known (see [23,Theorem 7.16]) that each f ∈ H 1 (R) has a representative that is absolutely continuous on R and whose classical derivative exists and coincides with the weak derivative f almost everywhere. By H 1 we denote the space of all functions f ∈ H 1 (R) whose Fourier transform f also belongs to It is easily seen that both S 0 (R) and H 1 contain the Schwartz space S(R). To help the readers' understanding, we provide some examples of functions in

Examples. The characteristic function
is continuous but not smooth, hence, does not belong to S(R). However, h ∈ S 0 (R) ∩ H 1 . Indeed, one can directly check h, π(·)h ∈ L 1 (R 2 ) to show that h ∈ S 0 (R) (see e.g., [12,Chapter 16]). Also, we have However, using [15,Theorem 3.2.13] and the fact that h ∈ S 0 (R), we have g ∈ S 0 (R), hence, g ∈ S 0 (R)\H 1 . On the other hand, a function in

Time-frequency shift invariance: a closer look
In this section, we first establish a certain trichotomy concerning the set of invariant timefrequency shifts. We then show that one of the three cases of the trichotomy is excluded if the generator function g is "sufficiently nice".
The next proposition establishes the trichotomy: the invariance set I(G) either fills the whole space R 2 , or it consists of equispaced lines that are aligned with the lattice, or it is a refinement of Λ (and in particular a lattice itself). Note that this holds regardless of the regularity of the generator g or the (frame) properties of the Gabor system (g, Λ).

Proposition 5 Let H be a closed additive subgroup of R 2 and suppose that H
and m, n ∈ N ≥1 such that exactly one of the following conditions holds: In particular, if Λ ⊂ R 2 is a lattice and g ∈ L 2 (R), then one of the above cases holds for H = I(G(g, Λ)).
Let us consider the case (ii): Note that μ, λ are linearly independent, and hence sn − tm = 0, so that d = (sn − tm) −1 is well-defined. Furthermore, we see (2) of the statement of the proposition holds.
Assume now that Case (iii) holds: H is a lattice, i.e., |n| . This completes the proof of the proposition, since the conditions (1)-(3) are clearly mutually exclusive.
We now provide some examples for the three cases in Proposition 5. In particular, we show that all three cases can occur for H = I(G(g, Λ)) with g ∈ L 2 (R) and a lattice Λ ⊂ R 2 such that (g, Λ) is a Riesz sequence.
Note that all the functions g in Example 6 are well localized in time but not in frequency.
In the remainder of this section, we show that Case (2) in Proposition 5 cannot occur if (g, Λ) is a frame sequence with a sufficiently nice window g. In this case, the trichotomy from Proposition 5 becomes a dichotomy. By g being "sufficiently nice" we mean that g ∈ W(C, 2 ), where 2 ) for a suitable θ B,C ∈ C. This shows that U B g ∈ W(C, 2 ) whenever g ∈ W(C, 2 ) and det B = 1.
The following lemma shows that the function classes considered in this paper (namely, S 0 (R) and H 1 ) are contained in W(C, 2 ).
We now show that Case (2) in Proposition 5 cannot occur if (g, Λ) is a frame sequence with generator g ∈ W(C, 2 ) \ {0}. a lattice such that (g, Λ) is a frame for G = G(g, Λ). Then either I(G) = R 2 or there exist λ 1 , λ 2 ∈ Λ and m, n ∈ N ≥1 such that Proof The claim is equivalent to the statement that either Case (1) or (3) in Proposition 5 holds for H = I(G). Since Proposition 5 shows the trichotomy into Cases (1)- (3), it is enough to show that Case (2) in Proposition 5 cannot hold for H . Therefore, we assume towards a contradiction that Case (2) holds, i.e., there are λ 1 , λ 2 ∈ Λ and n ∈ N ≥1 such that Λ = Z · λ 1 + Z · λ 2 and I(G) = Z · λ 1 n + R · λ 2 .
Our goal is to show that E = R, up to null-sets. This will then imply G = L 2 (E) = L 2 (R) and hence I(G) = R 2 , providing the desired contradiction. Towards proving E = R, let us consider for given f ∈ L 2 (R) the continuous function Γ f : R → R defined by where S : L 2 (R) → G denotes the frame operator of (g, Λ). By [16, Proposition 7.1.1], the operator S has the Walnut representation where only finitely many terms of the sum do not vanish, and where The fact that g ∈ W (C, 2 ) easily implies that the series defining G n converges locally uniformly, and that the G n are continuous functions. Since G n is also α-periodic, this means that each G n is bounded. Now, since multiplication with G n commutes with the modulation M ω , using the identity T n/β M ω = e −2πi n β ω M ω T n/β , we get where L ∞ c (R) denotes the space of functions f ∈ L ∞ (R) with compact support, and where there are only finitely many n ∈ Z (depending only on f , but not on the choice of ω) for which G n · T n β f , f = 0. Since (g, Λ) is a frame for G and since M ω f ∈ G for all ω ∈ R and all f ∈ G, there exists and apply Equation (10) to see

Using standard arguments, this implies that h(x) ≥ β A for almost all x ∈ E.
Since T mα g ∈ G = L 2 (E) and thus T mα g(x) = 0 for almost all x ∈ R\E and arbitrary m ∈ Z, it follows that h(x) = 0 for almost all x ∈ R\E. Recall from above that h(x) ≥ β A for almost all x ∈ E; thus, h(x) ∈ {0} ∪ [β A, ∞) almost everywhere. Also recall from above that h = G 0 is continuous. Hence, the open set h −1 ((0, β A)) has measure zero and is thus empty; therefore, we see h(x) ∈ {0} ∪ [β A, ∞) for all x ∈ R. By the intermediate value theorem, this implies that h(x) ≥ β A for all x ∈ R (since h ≥ |g| 2 and g ≡ 0) and thus, indeed, E = R (up to null-sets), since h(x) = 0 a.e. on R\E.
By combining Theorem 8 and Lemma 7, we obtain the following corollary.

Corollary 9
Let g ∈ S 0 (R) \ {0} or g ∈ H 1 \ {0} and let Λ ⊂ R 2 be a lattice such that (g, Λ) is a Riesz basis for G : = G(g, Λ). Then I(G) is a refinement of Λ as in (9). shift, meaning I(G) = Λ. In view of Theorem 1 it is our goal to show that this is impossible, at least if g ∈ S 0 . To make the situation more accessible, we first reduce to the case where Λ = αZ × βZ is separable, and where the additional time-frequency shift is of the form ( α ν , 0) for some ν ∈ N ≥2 , meaning that T α/ν g ∈ G. After that, we provide a characterization of this simplified condition in terms of the adjoint Gabor system. It is this characterization that we will use to prove our main result, Theorem 1, in the next section.
In what follows, fix g ∈ L 2 (R), α, β > 0, Λ = αZ × βZ, and ν ∈ N ≥2 , and assume that (g, Λ) is a frame for G = G(g, Λ). The adjoint system is then a frame for its closed linear span K by [ It is natural to ask what an additional time-frequency shift invariance of the form Tα ν g ∈ G means for the adjoint system F . To answer this question, we set Again by [28, Theorem 2.2 (c)], F 0 is a frame sequence if and only if the system (g, α ν Z × βZ) is a frame sequence. In this case, each F s is a frame sequence because M s α F 0 is, and multiplying the vectors of a frame sequence by unimodular constants results in a frame sequence. For s ∈ {0, . . . , ν − 1}, we set L s := span F s = M s α L 0 . Note that Indeed, the inclusion "⊃" is trivial. Conversely, since F is a frame sequence, each f ∈ K satisfies f = k, ∈Z c k, T k/β M /α g with a suitable sequence c = (c k, ) k, ∈Z ∈ 2 (Z 2 ). Since F is a Bessel sequence, the function is well-defined for s ∈ {0, . . . , ν − 1}, and f = f 0 + · · · + f ν−1 ∈ L 0 + · · · + L ν−1 .
In the sequel, the symbol denotes the direct (not necessarily orthogonal) sum of subspaces, whereas ⊕ is used to denote an orthogonal sum.
The next theorem provides several equivalent conditions for the additional time-frequency invariance Tα ν g ∈ G in terms of properties of the adjoint system F . The-for our purposesmost useful statement from Theorem 11 is that an additional time-frequency shift of the form Tα ν g ∈ G implies that the scaled frame operator (αβ) −1 S 1 β Z× ν α Z,γ ,g is a projection.

Theorem 11
Let g ∈ L 2 (R) and α, β > 0, and assume that (g, αZ×βZ) is a frame sequence with canonical dual window γ ∈ G, where G = G(g, αZ × βZ). Let ν ∈ N ≥2 , and define the systems F s and the spaces K, L s as above, and let S 1 β Z× ν α Z,γ ,g be the operator defined as in Equation (2).
Then the following are equivalent: for all k ∈ Z and all ∈ Z\νZ.
If one of (i)-(iv) holds, then for each s = 0, . . . , ν − 1 the system F s is a frame for L s and the operator P s :
Furthermore, L 0 is the set of functions f ∈ L 2 (R) whose restrictions to the intervals [2k, 2k + 1], k ∈ Z, only have non-zero Fourier coefficients with even index. The space L 1 is described similarly with "even" replaced by "odd". Hence K = L 2 (R) = L 0 ⊕ L 1 .
(ii)⇒(iii): Since Pg = g, it is a consequence of (12) that P| L 0 = id L 0 . Furthermore, for s ∈ {1, . . . , ν − 1} and k, ∈ Z, Eq. (12) implies which shows P| L s = 0. Using these observations and noting that L r = M r /α L 0 , we see for r , s ∈ {0, . . . , ν − 1} that P r | L r = M r /α P M −r /α | L r = id L r and furthermore P r | L s = M r /α P M −r /α | L s = 0 for s = r . Therefore, the sum K = L 0 · · · L ν−1 is direct, and P s | K = M s/α P M −s/α | K is the projection onto L s with respect to this decomposition. Finally, since γ ∈ K and since K is invariant under T k/β M /α , it follows by definition of the operator Finally, we show that F s is a frame for L s , where it clearly suffices to show this for s = 0. Since F 0 is a Bessel sequence, [8,Corollary 5.5.2] shows that we only need to prove that the synthesis operator we see directly from the definition of S 1 β Z× ν α Z,γ ,g that ran P ⊂ L 0 . Hence, Pg − g ∈ L 0 . On the other hand, again as a consequence of ran P ⊂ L 0 we see that M s/α P M −s/α g ∈ L s , so that Equation (14) implies and thus Pg = g since the sum L 0 + · · · + L ν−1 is direct. Similarly, for any s ∈{1, . . . , ν−1} we get because of ran P ⊂ L 0 that M s/α P M −s/α g ∈ L s ; but this implies as in Equation (16) that Again, since L 0 +· · ·+L ν−1 is a direct sum, this implies P M − s α g = 0 for s = 1, . . . , ν − 1. Since P commutes with M ±ν/α (see Equation (12)), we have P M (ν−s)/α g = 0 and therefore P M s/α g = 0 for s = 1, . . . , ν − 1.
Note that with P :

Proof of the main theorem
In this section, we prove our main result, Theorem 1, which we state here once more for the convenience of the reader.
Theorem 1 If g ∈ S 0 (R) and Λ ⊂ R 2 is a lattice such that the Gabor system (g, Λ) is a Riesz basis for its closed linear span G(g, Λ), then the time-frequency shifts T a M b that leave G(g, Λ) invariant satisfy (a, b) ∈ Λ.
A crucial ingredient for the proof of Theorem 1 is the following auxiliary statement which relies on a deep result concerning the structure of the irrational rotation algebra A θ (see [11,26,27]). We postpone its proof to Appendix A.

Theorem 14 Let H = {0} be a Hilbert space and let U , V ∈ B(H) be unitary operators on
If a = (a k, ) k, ∈Z ∈ 1 (Z 2 ) is such that the operator P a := k, ∈Z a k, V k U satisfies P 2 a = P a , then a 0,0 ∈ Z + θ Z.

Remark 15
On a first look, it might appear as if the proof of Theorem 1 would also apply in case of g ∈ H 1 : First, the classical Balian-Low theorem implies that G := G(g, Λ) L 2 (R), so that Lemma 10 allows the reduction to a Gabor Riesz sequence (h, Λ) with h ∈ H 1 , a separable lattice Λ = αZ×βZ, and an additional time-frequency shift of the form T α/ν h ∈ G. One can then apply Theorem 11 to see that that L 2 (R) = L 0 · · · L ν−1 . In the S 0 -case, we then employed Janssen's representation (17) for the projection P 0 = (αβ) −1 S 1 β Z× ν α Z,γ ,h , which then led to success in the proof of Theorem 1, thanks to existing results concerning the structure of the irrational rotation algebra. However, in the case h ∈ H 1 the series in (17) might not converge in operator norm, so that one does not know whether P 0 belongs to the irrational rotation algebra. Thus, the proof breaks down at this point. acknowledges support by the DFG Grant PF 450/9-2. F. Philipp was funded by the Carl Zeiss Foundation within the project DeepTurb-Deep Learning in und von Turbulenz. F. Voigtlaender acknowledges support by the DFG in the context of the Emmy Noether junior research group VO 2594/1-1.

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A Appendix-Proof of Theorem 14
As a foreword, it bears to point out that the intent of this appendix is to give a minimal and self-contained 'translation' of a deep result from the theory of C * -algebras (namely that the trace of a projection operator on an irrational rotation algebra has a very specific form, Theorem 1.2 in [27]) to the standard language of Gabor analysis literature. As such, within this paper, the result will essentially be treated like a 'black box' and we will only show how it applies to our present problem. For more details, we refer the interested reader to [27] for a proof of the result or to [11] for more background.
The proof will make use of some parts of the theory of C * -algebras, which we recall here for the convenience of the reader, based on [25]. Readers familiar with C * -algebras will probably want to skip this part-except possibly Lemma 16.
A C * -algebra is a (complex) Banach algebra (A, · ), additionally equipped with a map A → A, x → x * (called the involution on A), satisfying the following properties: -(x + y) * = x * + y * , (λ x) * = λ x * , and (x y) * = y * x * and (x * ) * = x for all x, y ∈ A and λ ∈ C; x * = x and x * x = x 2 for all x ∈ A.
An element p ∈ A is called an idempotent if p 2 = p. An idempotent p is called a projection if additionally p = p * holds. A C * -algebra A is called unital if it contains a (necessarily unique) element 1 ∈ A satisfying 1 = 0 and x 1 = 1 x = x for all x ∈ A. In a unital C * -algebra A, an element x ∈ A is called unitary if x * x = 1 = x x * . If A is a unital C *algebra and a ∈ A, then σ (a * a) ⊂ [0, ∞); see [25,Theorem 2.2.4]. Here, σ (b) = {λ ∈ C : b − λ1 not invertible in A}.

Lemma 16
Any idempotent e in a unital C * -algebra A is similar to a projection p ∈ A. That is, there exist a projection p ∈ A and an invertible element a ∈ A such that e = a −1 pa. Consequently, ez −1 = z −1 e and, as z = z * , also e * z −1 = z −1 e * . Now, define the element p := ez −1 e * . We have p * = p. Furthermore, since we just saw that z −1 commutes with e and e * and that ee * e = ze, we also see that p 2 = z −2 (ee * e)e * = z −1 ee * = p. Hence, p is a projection. We further observe that ep = p and pe = ez −1 e * e = z −1 ee * e = z −1 ze = e. Set a := 1 − p + e. Then we see because of (1 ∓ p ± e)(1 ± p ∓ e) = 1 ± p ∓ e ∓ p − p + e ± e + p − e = 1 that a is invertible with a −1 = 1 + p − e. Hence, from ae = e − pe + e = e we obtain aea −1 = e(1 + p − e) = e + ep − e = ep = p, which proves the lemma.

Proof
A closed subspace B of a C * -algebra A is called a C * -subalgebra of A if it is closed under both multiplication and involution. It is clear that B is then itself a C * -algebra. As usual, given a subset S ⊂ A, there is a smallest (with respect to inclusion) C * -subalgebra of A containing S. We call it the C * -algebra generated by S, and denote it by C * (S).
A map ϕ : A → B between two C * -algebras A and B is called a * -homomorphism if it is linear and satisfies ϕ(x y) = ϕ(x) ϕ(y) as well as ϕ(x * ) = [ϕ(x)] * for all x, y ∈ A. A bijective * -homomorphism is called a * -isomorphism. Any * -homomorphism ϕ : A → B necessarily satisfies ϕ(x) B ≤ x A for all x ∈ A, and is hence continuous; see [25,Theorem 2.1.7]. 14 We will make use of the so-called irrational rotation algebra A θ with θ ∈ R\Q, as introduced for instance in [11,Chapter VI]. The actual definition of this algebra is not relevant for us; we will only need to know that it satisfies the following properties:

Proof of Theorem
-A θ is a unital C * -algebra; -The algebra A θ is universal among all unital C * -algebras generated by unitary elements trace τ : A θ → C. By definition of a trace, this means in particular that τ is linear and continuous, satisfying τ (1) = 1 and τ (x y) = τ (yx) for all x, y ∈ A θ . -For any projection p ∈ A θ , we have τ ( p) ∈ Z + θ Z; see [27,Theorem 1.2]. We remark that this result was originally proven in [26].
Let us define τ := τ • ϕ, and note that τ : A → C is continuous. It is easy to see that τ is linear with τ (id H ) = 1 and τ (AB) = τ (B A) for all A, B ∈ A; this is called the cyclicity of the trace. Next, from the relation U V = e 2πiθ V U , we immediately get for k, ∈ Z that Thus, noting that V k U ∈ A, we obtain τ (V k U ) = e −2πi θ τ (V k U ) = e −2πikθ τ (V k U ) by cyclicity. As θ is irrational, this implies τ (V k U ) = δ ,0 δ k,0 . Next, since we have V k U = 1 for all k, ∈ Z and since a ∈ 1 (Z 2 ), we see that P a = k, ∈Z a k, V k U ∈ A, with unconditional convergence of the defining series. Hence, τ (P a ) = k, ∈Z a k, · τ (V k U ) = a 0,0 .
Since P 2 a = P a and since ϕ : A → A θ is a * -homomorphism, we see that e := ϕ(P a ) ∈ A θ is an idempotent. By Lemma 16 there exist b, p ∈ A θ such that b is invertible, p is a projection, and e = b −1 pb. Thanks to the cyclicity of the trace, we thus see that a 0,0 = τ (P a ) = τ (e) = τ (b −1 pb) = τ ( p) ∈ Z + θ Z, as claimed.