A Note on the Invertibility of the Gabor Frame Operator on Certain

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d the Feichtinger algebra V = S0(R ); see [11] for separable lattices Λ and [1, Theorem 7] for irregular sets Λ. In the case of separable lattices, the question has been answered affirmatively also for the Schwartz space V = S(R) [15, Proposition 5.5] and for the Wiener amalgam space ∞ 1 V = W (L ,ℓv) with a so-called admissible weight v; see [18]. Similarly, the setting of the q spaces V = W (Cα,ℓv) (with the Hölder spaces Cα) is studied in [24]—but except in the case q = 1, some additional assumptions on the window function g are imposed. To the best of our knowledge, the question has not been answered for modulation spaces 1 1 Rd 2 Rd other than V = Mv , and in particular, not for any of the spaces V = H ( ), V = L1+|x|( ), and V = H1(Rd). In this note, we show that the answer is affirmative for all of these spaces: 1 Rd 2 Rd H1 Rd R2d Theorem 1.1. Let V ∈ {H ( ),L1+|x|( ), ( )}. Let g ∈ V and let Λ ⊂ be a lattice such that the Gabor system (g, Λ) is a frame for L2(Rd) with frame operator S. Then the canonical dual window S−1g belongs to V . Furthermore, (S−1/2g, Λ) is a Parseval frame for L2(Rd) with S−1/2g ∈ V .

d As mentioned above, the corresponding statement of Theorem 1.1 for V = S0(R ) with separable lattices Λ was proved in [11]. In addition to several deeper insights, the proof given in [11] relies on a simple but essential argument showing that the frame operator S = SΛ,g maps V boundedly into itself, which is shown in [11] based on Janssen’s representation of SΛ,g. d In our setting, this argument is not applicable, because—unlike in the case of V = S0(R )— there exist functions g ∈ H1 for which (g, Λ) is not an L2-Bessel system. In addition, the series in Janssen’s representation is not even guaranteed to converge unconditionally in the strong sense for H1-functions, even if (g, Λ) is an L2-Bessel system; see Proposition A.1. 1 2 H1 To bypass these obstacles, we introduce for each space V ∈ {H ,L1+|x|, } the associated subspace VΛ consisting of all those functions g ∈ V that generate a Bessel system over the given lattice Λ. We remark that most of the existing works concerning the regularity of the (canonical) dual window rely on deep results related to Wiener’s 1/f-lemma on absolutely convergent Fourier series. In contrast, our methods are based on elementary spectral theory (see Section 4) and on certain observations regarding the interaction of the Gabor frame operator with partial derivatives; see Proposition 3.2. The paper is organized as follows: Section 2 discusses the concept of Gabor Bessel vectors and introduces some related notions. Then, in Section 3, we endow the space VΛ (for each 1 2 H1 choice V ∈ {H ,L1+|x|, }) with a Banach space norm and show that the frame operator S 2 maps VΛ boundedly into itself, provided that the Gabor system (g, Λ) is an L -Bessel system and that the window function g belongs to V . Finally, we prove in Section 4 that for any 1 2 H1 V ∈ {H ,L1+|x|, } the spectrum of S as an operator on V coincides with the spectrum of S as an operator on L2. This easily implies our main result, Theorem 1.1.

2. Bessel vectors

For a, b ∈ Rd and f ∈ L2(Rd) we deﬁne the operators of translation by a and modulation by b as 2πib·x Taf(x) := f(x − a) and Mbf(x) := e · f(x), INVERTIBILITY OF THE GABOR FRAME OPERATOR ON CERTAIN MODULATION SPACES 3

2 d respectively. Both Ta and Mb are unitary operators on L (R ) and hence so is the time- frequency shift −2πia·b π(a, b) := TaMb = e MbTa. 1 Rd 2 Rd −2πix·ξ The Fourier transform F is defined on L ( )∩L ( ) by Ff(ξ)= f(ξ)= Rd f(x)e dx and extended to a unitary operator on L2(Rd). For z = (z , z ) ∈ Rd × Rd =∼ R2d and 1 2 R f ∈ L2(Rd), a direct calculation shows that b F[π(z)f]= e−2πiz1·z2 · π(Jz)f, (2.1) where 0 I b J = . −I 0 A (full rank) lattice in R2d is a set of the form Λ = AZ2d, where A ∈ R2d×2d is invertible. The volume of Λ is defined by Vol(Λ) := |det A| and its density by d(Λ) := Vol(Λ)−1. The adjoint lattice of Λ is denoted and defined by Λ◦ := JA−⊤Z2d. The Gabor system generated by a window function g ∈ L2(Rd) and a lattice Λ ⊂ R2d is given by (g, Λ) := π(λ)g : λ ∈ Λ . 2 Rd We say that g ∈ L ( ) is a Bessel vectorwith respect to Λ if the system (g, Λ) is a Bessel 2 d system in L (R ), meaning that the associated analysis operator CΛ,g defined by 2 Rd CΛ,gf := hf,π(λ)gi λ∈Λ, f ∈ L ( ), (2.2) is a bounded operator from L2(Rd) to ℓ2(Λ). We define 2 d BΛ := g ∈ L (R ) : (g, Λ) is a Bessel system , which is a dense linear subspace of L2(Rd) because each Schwartz function is a Bessel vector with respect to any lattice; see [8, Theorem 3.3.1]. It is well-known that BΛ = BΛ◦ (see, e.g., [8, Proposition 3.5.10]). In fact, we have for g ∈BΛ that 1/2 CΛ◦,g = Vol(Λ) · CΛ,g ; (2.3) see [17, proof of Theorem 2.3.1]. The cross frame operator SΛ,g,h with respect to Λ and two functions g, h ∈BΛ is defined by ∗ SΛ,g,h := CΛ,hCΛ,g.

In particular, we write SΛ,g := SΛ,g,g which is called the frame operator of (g, Λ). The 2 d system (g, Λ) is called a frame if SΛ,g is bounded and boundedly invertible on L (R ), that is, if A IdL2(Rd) ≤ SΛ,g ≤ B IdL2(Rd) for some constants 0 < A ≤ B < ∞ (called the frame bounds). In particular, a frame with frame bounds A = B = 1 is called a Parseval frame. In our proofs, the so-called fundamental identity of Gabor analysis will play an essential role. This identity states that hf,π(λ)gihπ(λ)γ, hi = d(Λ) · hγ,π(µ)gihπ(µ)f, hi. (2.4) ◦ λX∈Λ µX∈Λ 1 d d 2 d It holds, for example, if f, h ∈ M (R )= S0(R ) (the Feichtinger algebra) and g, γ ∈ L (R ); see [8, Theorem 3.5.11]. We will use the following version of the fundamental identity: 4 D.G.LEE,F.PHILIPP,ANDF.VOIGTLAENDER

Lemma 2.1. The fundamental identity (2.4) holds if g, h ∈BΛ or f, γ ∈BΛ.

Proof. In [16, Subsection 1.4.1], the claim is shown for separable lattices in R2. Here, we provide a short proof for the general case. If g, h ∈BΛ, then Equation (2.3) shows that both 2 sides of Equation (2.4) depend continuously on f, γ ∈ L . Similarly, if f, γ ∈ BΛ then both 2 sides of Equation (2.4) depend continuously on g, h ∈ L . Therefore, and because BΛ is dense 2 d 2d in L (R ), it is no restriction to assume that f, g, h, γ ∈ BΛ. LetΛ= AZ and deﬁne the function G(x)= f,π(A(n − x))g π(A(n − x))γ, h , x ∈ R2d. n∈Z2d X d d Writing Ax = ((Ax)1, (Ax)2) ∈ R × R , a direct computation shows that hf,π(A(n − x))gi = e2πi(Ax)2·((Ax)1−(An)1) · hπ(Ax)f,π(An)gi.

2d Therefore, and because of g, γ ∈ BΛ and since z 7→ π(z)u is continuous on R for each u ∈ L2(Rd), the function G is continuous. Furthermore, we have

≤ kCΛ,gk · kCΛ,γ k · kπ(Ax)fkL2 · kπ(Ax)hkL2

= kCΛ,gk · kCΛ,γ k · kfkL2 · khkL2 , which will justify the application of the dominated convergence theorem in the following calculation. Indeed, G is Z2d-periodic and the k-th Fourier coeﬃcient of G (for k ∈ Z2d) is given by

−2πikx 2πik(n−x) ck = G(x)e dx = f,π(A(n − x))g π(A(n − x))γ, h e dx Q Q Z n∈Z2d Z X 2πikx 1 2πiA−⊤k·y = f,π(Ax)g π(Ax)γ, h e dx = Vgf(y)Vγh(y) · e dy R2d | det A| R2d Z Z

= d(Λ) Vπ(zk)g[π(zk)f](y)Vγ h(y) dy = d(Λ) · hπ(zk)f, hihγ,π(zk)gi, R2d Z 2d −⊤ ◦ 2 Rd where Q := [0, 1] , zk := −JA k ∈ Λ , and for f1, g1 ∈ L ( ), Vg1 f1(z) = hf1,π(z)g1i 2d for z ∈ R is the short-time Fourier transform of f1 with respect to g1. Here, we used the orthogonality relation for the short-time Fourier transform (see [10, Theorem 3.2.1]) and the 2πihJz,·i identity Vπ(z)g[π(z)f]= e · Vgf ([17, Lemma 1.4.4(b)]). Now, as also f, g ∈BΛ = BΛ◦ , 1 Z2d Z2d we see that (ck)k∈Z2d ∈ ℓ ( ). Since G is continuous and -periodic, this implies that the Fourier series of G converges uniformly and coincides pointwise with G. Hence,

2πikx 2d G(x)= ck e for all x ∈ R , Z2d kX∈ and setting x = 0 yields the claim. INVERTIBILITY OF THE GABOR FRAME OPERATOR ON CERTAIN MODULATION SPACES 5

3. Certain subspaces of modulation spaces invariant under the frame operator The L2-Sobolev-space H1(Rd)= W 1,2(Rd) is the space of all functions f ∈ L2(Rd) whose distributional derivatives ∂ f := ∂f , j ∈ {1,...,d}, all belong to L2(Rd). We will frequently j ∂xj use the well-known characterization H1(Rd)= f ∈ L2(Rd):(1+ | · |)f(·) ∈ L2 of H1(Rd) in terms of the Fourier transform. With the weight function w : Rd → R, x 7→ 1+ |x|, we 2 2 Rd Rd C 2 define the weighted L -space Lw( ) := f : → : w(·)f(·) ∈ L b which is equipped 2 d 1 d −1 1 d with the norm kfk 2 := kwfk 2 . It is then clear that L (R )= F[H (R )] = F [H (R )]. Lw L w H1 Rd 1 Rd 2 Rd 1 Rd Finally, we define ( )= H ( ) ∩ Lw( ) which is the space of all functions f ∈ H ( ) whose Fourier transform f also belongs to H1(Rd). Equivalently, H1(Rd) is the space of all functions g ∈ L2(Rd) with finite uncertainty product (1.1). b 1 Rd 2 Rd H1 Rd It is worth to note that each of the spaces H ( ), Lw( ), and ( ) can be expressed 2 Rd 2 Rd 2 2 as a modulation space Mm( ) = {f ∈ L ( ) : R2d |hf,π(z)ϕi| |m(z)| dz < ∞} for some weight function m : R2d → C, where ϕ ∈ S(Rd)\{0} is any fixed function1, for instance a R Gaussian. Indeed, we have 1 Rd 2 Rd 2 Rd 2 Rd H1 Rd 1 Rd 2 Rd 2 Rd H ( )= Mm1 ( ), Lw( )= Mm2 ( ), and ( )= H ( )∩Lw( )= Mm3 ( ), 2 2 with m1(x,ω)=1+ |ω|, m2(x,ω)=1+ |x|, and m3(x,ω) = 1+ |x| + |ω| , respectively; see [10, Proposition 11.3.1] and [13, Corollary 2.3]. p Our main goal in this paper is to prove for each of these spaces that if the window function g of a Gabor frame (g, Λ) belongs to the space, then so does the canonical dual window. In this section, we will mostly concentrate on the space H1(Rd), since this will imply the desired result for the other spaces as well. d The corresponding result for the Feichtinger algebra S0(R ) was proved in [11] by showing d the much stronger statement that the frame operator maps S0(R ) boundedly into itself and d is in fact boundedly invertible on S0(R ). However, the methods used in [11] cannot be directly transferred to the case of a window function in H1(Rd) (or H1(Rd)), since the proof in [11] leverages two particular properties of the Feichtinger algebra which are not shared by H1(Rd): d (a) Every function from S0(R ) is a Bessel vector with respect to any given lattice; (b) The series in Janssen’s representation of the frame operator converges strongly (even absolutely in operator norm) to the frame operator when the window function belongs d to S0(R ). Indeed, it is well-known that g ∈ L2(R) is a Bessel vector with respect to Z×Z if and only if the Zak transform of g is essentially bounded (cf. [2, Theorem 3.1]), but [2, Example 3.4] provides an example of a function g ∈ H1(R) whose Zak transform is not essentially bounded; this 1 d d indicates that (a) does not hold for H (R ) instead of S0(R ). Concerning the statement (b) for H1(Rd), it is easy to see that if Janssen’s representation converges strongly (with respect to some enumeration of Z2) to the frame operator of (g, Λ), then the frame operator must be bounded on L2(R) and thus the associated window function g is necessarily a Bessel vector.

1 2 The deﬁnition of Mm is known to be independent of the choice of ϕ; see e.g., [10, Proposition 11.3.2]. 6 D.G.LEE,F.PHILIPP,ANDF.VOIGTLAENDER

Therefore, the example above again serves as a counterexample: namely, the statement (b) fails for such a non-Bessel window functions g ∈ H1(R). Even more, we show in the Appendix that there exist Bessel vectors g ∈ H1(R) for which Janssen’s representation neither converges unconditionally in the strong sense nor conditionally in the operator norm. We mention ∞ 1 that in the case of the Wiener amalgam space W (L ,ℓv) with an admissible weight v, the convergence issue was circumvented by employing Walnut’s representation instead of ∞ 1 Janssen’s to prove the result for W (L ,ℓv) in [18]. Fortunately, it turns out that establishing the corresponding result for V = H1(Rd), 2 Rd H1 Rd Lw( ), and ( ) only requires the invertibility of the frame operator on a particular subspace of V . Precisely, given a lattice Λ ⊂ R2d, we deﬁne 1 Rd 1 Rd H1 Rd H1 Rd 2 Rd 2 Rd HΛ( ) := H ( ) ∩BΛ, Λ( ) := ( ) ∩BΛ, and Lw,Λ( ) := Lw( ) ∩BΛ. We equip the ﬁrst two of these spaces with the norms

kfk 1 := k∇fk 2 + kC k 2 2 and kfkH1 := k∇fk 2 + k∇fk 2 + kC k 2 2 , HΛ L Λ,f L →ℓ Λ L L Λ,f L →ℓ respectively, where b d

k∇fkL2 := k∂jfkL2 Xj=1 2 Rd and CΛ,f is the analysis operator deﬁned in (2.2). Finally, we equip the space Lw,Λ( ) with the norm

kfk 2 := kfk 2 + kC k 2 2 , where kfk 2 := kw · fk 2 . Lw,Λ Lw Λ,f L ,ℓ Lw L We start by showing that these spaces are Banach spaces. R2d 1 Rd 2 Rd H1 Rd Lemma 3.1. For a lattice Λ ⊂ , the spaces HΛ( ), Lw,Λ( ), and Λ( ) are Banach spaces which are continuously embedded in L2(Rd).

2 Rd 2 2 Proof. We naturally equip the space BΛ ⊂ L ( ) with the norm kfkBΛ := kCΛ,f kL →ℓ .

Then (BΛ, k · kBΛ ) is a Banach space by [12, Proposition 3.1]. Moreover, for f ∈BΛ, ∗ ∗ 2 2 2 kfkL = CΛ,f δ0,0 L2 ≤ kCΛ,f kℓ →L = kfkBΛ , (3.1) 2 Rd 1 Rd if (fn)n∈N is a Cauchy sequence in HΛ( ), then it ). Hence, which implies that BΛ ֒→ L ( 1 Rd is a Cauchy sequence in both H ( ) (equipped with the norm kfkH1 := kfkL2 + k∇fkL2 ) 1 Rd and in BΛ. Therefore, there exist f ∈ H ( ) and g ∈ BΛ such that kfn − fkH1 → 0 and 1 Rd 2 Rd 2 Rd kfn − gkBΛ → 0 as n → ∞. But as H ( ) ֒→ L ( ) and BΛ ֒→ L ( ), we have fn → f 2 d and f → g also in L (R ), which implies f = g. Hence, kf − fk 1 → 0 as n → ∞, which n n HΛ 1 Rd 2 Rd H1 Rd proves that HΛ( ) is complete. The proof for Lw,Λ( ) and Λ( ) is similar. R2d 1 Rd 1 Rd Proposition 3.2. Let Λ ⊂ be a lattice. If g, h ∈ HΛ( ), then SΛ,g,h maps HΛ( ) 1 d boundedly into itself with operator norm not exceeding kgk 1 khk 1 . For f ∈ H (R ) and HΛ HΛ Λ j ∈ {1,...,d} we have ∗ ∂j(SΛ,g,hf)= SΛ,g,h(∂jf)+ d(Λ) · CΛ◦,f dj,Λ◦,g,h, (3.2) INVERTIBILITY OF THE GABOR FRAME OPERATOR ON CERTAIN MODULATION SPACES 7

2 ◦ where dj,Λ◦,g,h ∈ ℓ (Λ ) is deﬁned by ◦ (dj,Λ◦,g,h)µ := ∂jh, π(µ)g + h, π(µ)(∂j g) , µ ∈ Λ . (3.3) 1 Rd Proof. Let f ∈ HΛ( ) and set u := SΛ,g,hf. First of all, we have u ∈ BΛ. Indeed, a direct ∗ computation shows that SΛ,g,h commutes with π(λ) for all λ ∈ Λ, and that SΛ,g,h = SΛ,h,g, which shows for v ∈ L2(Rd) that

(CΛ,uv)λ = hv,π(λ)ui = hv,π(λ)SΛ,g,hfi = hSΛ,h,gv,π(λ)fi = (CΛ,f ◦ SΛ,h,g v)λ, and therefore

kCΛ,uk ≤ kSΛ,h,gk · kCΛ,f k ≤ kCΛ,gk · kCΛ,hk · kCΛ,f k < ∞, (3.4) ∗ since SΛ,h,g = CΛ,gCΛ,h. We now show that u ∈ H1(Rd). To this end, note for v ∈ H1(Rd), a, b ∈ Rd, and j ∈ {1,...,d} that

∂j(Mbv) = 2πi · bj · Mbv + Mb(∂jv) and ∂j(Tav)= Ta(∂jv) and therefore ∂j(π(z)v) = 2πi · zd+j · π(z)v + π(z)(∂j v). Hence, setting cλ,j := 2πi · λd+j · hf,π(λ)gi for λ = (a, b) ∈ Λ, we see that

cλ,j = h∂jf,π(λ)gi + hf,π(λ)(∂j g)i. (3.5) 2 2 In particular, (cλ,j)λ∈Λ ∈ ℓ (Λ) for each j ∈ {1,...,d}, because f, g ∈BΛ and ∂jf, ∂jg ∈ L . 2 Rd ∞ Rd In order to show that ∂ju exists and is in L ( ), let φ ∈ Cc ( ) be a test function. Note ∞ Rd that Cc ( ) ⊂BΛ. Therefore, we obtain

− u, ∂j φ = − hf,π(λ)g π(λ)h, ∂j φ = hf,π(λ)g 2πiλd+j · π(λ)h + π(λ)(∂j h), φ λ∈Λ λ∈Λ X X = cλ,j · π(λ)h, φ + f,π(λ)g π(λ)(∂jh), φ λ∈Λ λ∈Λ X X (3.5) = hSΛ,g,h(∂jf), φi + hf,π(λ)(∂j g)ihπ(λ)h, φi + f,π(λ)g π(λ)(∂j h), φ λ∈Λ λ∈Λ X X (2.4) = hSΛ,g,h(∂jf), φi + d(Λ) h, π(µ)(∂j g) + ∂jh, π(µ)g π(µ)f, φ µ∈Λ◦ X h i

= SΛ,g,h(∂jf)+ d(Λ) h, π(µ)(∂j g) + ∂jh, π(µ)g π(µ)f , φ * ◦ + µX∈Λ h i ∗ = SΛ,g,h(∂jf)+ d(Λ) · CΛ◦,f dj , φ , 2 ◦ with dj = dj, Λ◦,g,h as in (3.3). Note that dj ∈ ℓ (Λ ) because g, h ∈ BΛ = BΛ◦ and 2 1 d ∂jh, ∂j g ∈ L . Since j ∈ {1,...,d} is chosen arbitrarily, this proves that u ∈ H (R ) with ∗ 2 Rd ∂ju = SΛ,g,h(∂jf)+ d(Λ) · CΛ◦,f dj ∈ L ( ) for j ∈ {1,...,d}, which is (3.2). Next, recalling Equation (2.3) we get 1/2 kdj kℓ2 ≤ kCΛ◦,hk·k∂j gkL2 +kCΛ◦,gk·k∂j hkL2 = Vol(Λ) kCΛ,hk·k∂j gkL2 +kCΛ,gk·k∂j hkL2 , 8 D.G.LEE,F.PHILIPP,ANDF.VOIGTLAENDER

∗ 1/2 and kCΛ◦,f k = Vol(Λ) kCΛ,f k. Therefore,

k∂jukL2 ≤ kSΛ,g,hk · k∂jfkL2 + kCΛ,hk · k∂jgkL2 + kCΛ,gk · k∂jhkL2 kCΛ,f k. Hence, with (3.4), we see d kS fk 1 = k∇uk 2 + kC k≤ k∂ uk 2 + kC k · kC k · kC k Λ,g,h HΛ L Λ,u j L Λ,g Λ,h Λ,f Xj=1 ≤ kSΛ,g,hk·k∇fkL2 + kCΛ,hk·k∇gkL2 +kCΛ,gk·k∇hkL2 +kCΛ,gk·kCΛ,hk kCΛ,f k

≤ kCΛ,gk · kCΛ,hk·k∇fkL2 + k∇gkL2 + kCΛ,gk k∇hkL2 + kCΛ,hk kCΛ,f k ≤ kgk 1 khk 1 · kfk 1 , HΛ HΛ HΛ and the proposition is proved.

4. Spectrum and dual windows

Let X be a Banach space. As usual, we denote the set of bounded linear operators from X into itself by B(X). The resolvent set ρ(T ) of an operator T ∈ B(X) is the set of all z ∈ C for which T − z := T − zI : X → X is bijective. Note that ρ(T ) is always open in C. The spectrum of T is the complement σ(T ) := C\ρ(T ). The approximate point spectrum σap(T ) is a subset of σ(T ) and is deﬁned as the set of points z ∈ C for which there exists a sequence (fn)n∈N ⊂ X such that kfnk = 1 for all n ∈ N and k(T − z)fnk→ 0 as n →∞. By [5, Proposition VII.6.7] we have ∂σ(T ) ⊂ σap(T ). (4.1) Lemma 4.1. Let (H, k · k) be a Hilbert space, let S ∈B(H) be self-adjoint, and let X ⊂H be a dense linear subspace satisfying S(X) ⊂ X. If k · kX is a norm on X such that (X, k · kX ) ,(is complete and satisﬁes X ֒→ H, then A := S|X ∈ B(X). If, in addition, σap(A) ⊂ σ(S then σ(A)= σ(S).

Proof. The fact that A ∈ B(X) easily follows from the closed graph theorem. Next, since X ֒→H, there exists C > 0 with kfk≤ C kfkX for all f ∈ X. Assume now that additionally σap(A) ⊂ σ(S) holds. Note that σ(S) ⊂ R, since S is self-adjoint. Since σ(A) ⊂ C is compact, the value r := maxw∈σ(A) |Im w| exists. Choose z ∈ σ(A) such that |Im z| = r. Clearly, z cannot belong to the interior of σ(A), and hence z ∈ ∂σ(A). In view of Equation (4.1), this implies z ∈ σap(A) ⊂ σ(S) ⊂ R, hence r = 0 and thus σ(A) ⊂ R. Therefore, σ(A) has empty interior in C, meaning σ(A)= ∂σ(A). Thanks to Equation (4.1), this means σ(A) ⊂ σap(A), and hence σ(A) ⊂ σ(S), since by assumption σap(A) ⊂ σ(S). For the converse inclusion it suﬃces to show that ρ(A) ∩ R ⊂ ρ(S). To see that this holds, let z ∈ ρ(A) ∩ R and denote by E the spectral measure of the self-adjoint operator S. Since R ∩ ρ(A) ⊂ R is open, there are a, b ∈ R and δ0 > 0 such that z ∈ (a, b) and [a − δ0, b + δ0] ⊂ ρ(A). By Stone’s formula (see, e.g., [20, Thm. VII.13]), the spectral projection of S with respect to (a, b] can be expressed as 1 b+δ E((a, b])f = lim lim (S − t − iε)−1f − (S − t + iε)−1f dt, f ∈H, δ↓0 ε↓0 2πi Za+δ where all limits are taken with respect to the norm of H. INVERTIBILITY OF THE GABOR FRAME OPERATOR ON CERTAIN MODULATION SPACES 9

Note for w ∈ C \ R that w ∈ ρ(S) ⊂ ρ(A). Furthermore, A − w = (S − w)|X , which easily −1 −1 implies (S − w) |X = (A − w) . Hence, for f ∈ X, 1 b+δ kE((a, b])fk≤ lim lim (S − t − iε)−1f − (S − t + iε)−1f dt δ↓0 ε↓0 2π Za+δ 1 b +δ ≤ C · lim lim (A − t − iε)−1f − (A − t + iε) −1f dt δ↓0 ε↓0 2π X Za+δ 1 b+δ = C · lim lim (A − t − iε)−1f − (A − t + iε)−1f dt δ↓0 2π ε↓0 X Za+δ

= 0, since the map ρ(A) → X, z 7→ (A − z)−1f is analytic and thus uniformly continuous on compact sets. This implies E((a, b])f = 0 for all f ∈ X and therefore E((a, b]) = 0 as X is dense in H. But this means that (a, b) ⊂ ρ(S) (see [20, Prop. on p. 236]) and thus z ∈ ρ(S).

1 2 H1 For proving the invertibility of SΛ,g on HΛ,Lw,Λ, and Λ, we ﬁrst focus on the space 1 Rd 1 Rd 1 Rd HΛ( ). Note that if g ∈ HΛ( ), then SΛ,g maps HΛ( ) boundedly into itself by Proposi- 1 Rd 1 Rd tion 3.2. For g ∈ HΛ( ), we will denote the restriction of SΛ,g to HΛ( ) by AΛ,g; that is, 1 d A := S | 1 Rd ∈B(H (R )). Λ,g Λ,g HΛ( ) Λ Z2d 1 Rd Theorem 4.2. Let Λ ⊂ be a lattice and let g ∈ HΛ( ). Then

σ(AΛ,g)= σ(SΛ,g).

Proof. For brevity, we set A := AΛ,g and S := SΛ,g. Due to Lemma 4.1, we only have to prove 1 Rd that σap(A) ⊂ σ(S). For this, let z ∈ σap(A). Then there exists a sequence (fn)n∈N ⊂ HΛ( ) such that kf k 1 = 1 for all n ∈ N and k(A − z)f k 1 → 0 as n → ∞. The latter means n HΛ n HΛ that, for each j ∈ {1,...,d},

∂j(Sfn) − z · (∂jfn) L2 → 0 and CΛ,(S−z)fn → 0. (4.2) Suppose towards a contradiction that z∈ / σ(S). Since S is self-adjoint, this implies z∈ / σ(S).

Furthermore, because S is self-adjoint and commutes with π(λ) for all λ ∈ Λ, we see for −1 f ∈ BΛ that CΛ,(S−z)f = CΛ,f ◦ (S − z) and hence CΛ,fn = CΛ,(S−z)fn ◦ (S − z) , which implies that kCΛ,fn k → 0. Hence, also kCΛ◦,fn k → 0 as n → ∞ (see Equation (2.3)). Now, by Equation (3.2), we have ∗ ∂j(Sfn) − z · (∂jfn) = (S − z)(∂jfn)+ CΛ◦,fn dj 2 ◦ with some dj ∈ ℓ (Λ ) which is independent of n. Hence, the ﬁrst limit in (4.2) combined 2 2 with kCΛ◦,fn k→ 0 implies that k(S − z)(∂j fn)kL → 0 and thus k∂jfnkL → 0 as n →∞ for d all j ∈ {1,...,d}, since z∈ / σ(S). Hence, kf k 1 = k∂ f k 2 +kC k→ 0 as n →∞, n HΛ j=1 j n L Λ,fn in contradiction to kfnk 1 = 1 for all n ∈ N. This proves that, indeed, σap(A) ⊂ σ(S). HΛ P 2 Rd We now show analogous properties to Proposition 3.2 and Theorem 4.2 for Lw,Λ( ). 10 D.G.LEE,F.PHILIPP,ANDF.VOIGTLAENDER

Z2d 2 Rd 2 Rd Corollary 4.3. Let Λ ⊂ be a lattice. If g, h ∈ Lw,Λ( ), then SΛ,g,h maps Lw,Λ( ) w 2 d boundedly into itself. If g = h and if A := SΛ,g| 2 Rd ∈ B(L (R )) denotes the Λ,g Lw,Λ( ) w,Λ 2 Rd restriction of SΛ,g to Lw,Λ( ), then w σ(AΛ,g)= σ(SΛ,g). 2 Rd Proof. We equip the space BΛ ⊂ L ( ) with the norm kfkBΛ := kCΛ,f kL2→ℓ2 , where we recall from Equation (3.1) that kfkL2 ≤ kfkBΛ . Equation (2.1) shows that the Fourier b transform is an isometric isomorphism from BΛ to BΛ, where Λ := JΛ. Furthermore, it is well-known (see for instance [9, Section 9.3]) that the Fourier transform F : L2 → L2 re- 2 Rd 1 Rd 1 stricts to an isomorphism of Banach spaces F : Lw( ) → H ( b ), where H is equipped with the norm kfkH1 := kfkL2 + k∇fkL2 . Taken together, we thus see that the Fourier 2 d 1 d transform restricts to an isomorphism F : L (R ) → Hb (R ); here, we implicitly used that w,Λ Λ kfkH1 ≍ kfkH1 + kfkBb , which follows from k · kL2 ≤k·kBb . Λb Λ Λ Plancherel’s theorem, in combination with Equation (2.1) shows for f ∈ L2(Rd) that \ \ b b F SΛ,g,hf = f, π(λ)g π(λ)h = hf,π(Jλ)g iπ(Jλ)h = hf,π(λ)g iπ(λ)h = SΛ,g,b hf. λ∈Λ λ∈Λ b X D E X λX∈Λ b 1 Rd 1 Rbd b b b b b b Since Ab b b = Sb b b|H1 : Hb ( ) → Hb( ) is well-defined and bounded by Proposition 3.2, Λ,g,h Λ,g,h Λb Λ Λ the preceding calculation combined with the considerations from the previous paragraph w 2 d 2 d shows that A = S | 2 Rd : L (R ) → L (R ) is well-defined and bounded, with Λ,g,h Λ,g,h Lw,Λ( ) w,Λ w,Λ w −1 b b AΛ,g,h = F ◦ AΛ,g,b h ◦ F. w Finally, if g = h, we see σ(AΛ,g,g)= σ(AΛb,g,b gb)= σ(SΛb,g,b gb)= σ(SΛ,g,g), where the second step −1 b b is due to Theorem 4.2, and the final step used the identity SΛ,g,h = F ◦ SΛ,g,b h ◦ F from above.

H1 Rd 1 Rd 2 Rd Finally, we establish the corresponding properties for Λ( )= HΛ( ) ∩ Lw,Λ( ). Z2d H1 Rd H1 Rd Corollary 4.4. Let Λ ⊂ be a lattice. If g, h ∈ Λ( ), then SΛ,g,h maps Λ( ) 1 d boundedly into itself. If g = h and A := S |H1 Rd ∈ B(H (R )) denotes the restriction Λ,g Λ,g Λ( ) Λ H1 Rd of SΛ,g to Λ( ), then σ(AΛ,g)= σ(SΛ,g). (4.3) H1 Proof. From the deﬁnition of Λ and the proof of Corollary 4.3 it is easy to see that 1 1 2 d H = H ∩ L (R ), and k · kH1 ≍ k·k 1 + k · k 2 . Therefore, Proposition 3.2 and Λ Λ w,Λ Λ HΛ Lw,Λ H1 Rd Corollary 4.3 imply that SΛ,g,h maps Λ( ) boundedly into itself. Lemma 4.1 shows that to prove (4.3), it suﬃces to show σap(AΛ,g) ⊂ σ(SΛ,g). Thus, let 1 d z ∈ σ (A ). Then there exists (f ) N ⊂ H (R ) with kf kH1 = 1 for all n ∈ N and ap Λ,g n n∈ Λ n Λ w k(AΛ,g − z)fnkH1 → 0 as n →∞. Thus, k(AΛ,g − z)fnk 1 → 0 and k(A − z)fnk 2 → 0 Λ HΛ Λ,g Lw,Λ as n → ∞. Furthermore, there is a subsequence (n ) N such that lim kf k 1 > 0 k k∈ k→∞ nk HΛ w or limk→∞ kfn k 2 > 0. Hence, z ∈ σ(AΛ,g) or z ∈ σ(A ). But Theorem 4.2 and k Lw,Λ Λ,g INVERTIBILITY OF THE GABOR FRAME OPERATOR ON CERTAIN MODULATION SPACES 11

w A Corollary 4.3 show σ(AΛ,g)= σ(AΛ,g)= σ(SΛ,g). We have thus shown σap( Λ,g) ⊂ σ(SΛ,g), so that Lemma 4.1 shows σ(AΛ,g)= σ(SΛ,g).

The next proposition shows that any operator obtained from SΛ,g through the holomorphic spectral calculus (see [21, Sections 10.21–10.29] for a definition) maps each of the spaces 1 Rd 2 Rd H1 Rd HΛ( ), Lw,Λ( ), and Λ( ) into itself. R2d 1 Rd 2 Rd H1 Rd Proposition 4.5. Let Λ ⊂ be a lattice, let V ∈ {HΛ( ),Lw,Λ( ), Λ( )}, and g ∈ V . Then for any open set Ω ⊂ C with σ(SΛ,g) ⊂ Ω, any analytic function F :Ω → C, and any f ∈ V , we have F (SΛ,g)f ∈ V . 1 Rd Proof. We only prove the claim for V = HΛ( ); the proofs for the other cases are similar, 1 Rd using Corollaries 4.3 or 4.4 instead of Theorem 4.2. Thus, let g ∈ HΛ( ) and set S := SΛ,g 1 Rd and A := AΛ,g. Let f ∈ HΛ( ) and define 1 h = − F (z) · (A − z)−1f dz ∈ H1 (Rd), 2πi Λ ZΓ where Γ ⊂ Ω \ σ(S) is a finite set of closed rectifiable curves surrounding σ(S) = σ(A) (existence of such curves is shown in [23, Theorem 13.5]). Note that the integral converges in 1 Rd 1 Rd 2 Rd 2 Rd ,HΛ( ). Since HΛ( ) ֒→ L ( ), it also converges (to the same limit) in L ( ) and hence by definition of the holomorphic spectral calculus, 1 F (S)f = − F (z) · (S − z)−1f dz = h ∈ H1 (Rd). 2πi Λ ZΓ Our main result (Theorem 1.1) is now an easy consequence of Proposition 4.5.

Proof of Theorem 1.1. Using the fact that SΛ,g commutes with π(λ) for all λ ∈ Λ, it is easily −1 −1/2 seen that (SΛ,g g, Λ) is the canonical dual frame of (g, Λ) and that (SΛ,g g, Λ) is a Parseval frame for L2(Rd); see for instance, [4, Theorem 12.3.2]. Note that since (g, Λ) is a frame for 2 d L (R ), we have σ(SΛ,g) ⊂ [A, B] where 0 < A ≤ B < ∞ are the optimal frame bounds −1 −1/2 for (g, Λ). Thus, we obtain SΛ,g g ∈ VΛ ⊂ V and SΛ,g g ∈ VΛ ⊂ V from Proposition 4.5 with F (z)= z−1 and F (z)= z−1/2 (with any suitable branch cut; for instance, the half-axis A R (−∞, 0]), respectively, on Ω = x + iy : x ∈ ( 2 , ∞),y ∈ . Finally, we state and prove a version of Theorem 1.1 for Gabor frame sequences. For com- pleteness, we briefly recall the necessary concepts. Generally, a (countable) family (hi)i∈I in a Hilbert space H is called a frame sequence, if (hi)i∈I is a frame for the subspace ′ H := span{hi : i ∈ I}⊂H. In this case, the frame operator S : H→H,f 7→ i∈I hf, hiihi, ′ ′ is a bounded, self-adjoint operator on H, and S|H′ : H → H is boundedly invertible; in particular, ran S = H′ ⊂ H is closed, so that S has a well-defined pseudo-inverseP S†, given by † −1 ′ S = (S|H′ ) ◦ PH′ : H→H , ′ where PH′ denotes the orthogonal projection onto H . The canonical dual system of (hi)i∈I is ′ † ′ ′ ′ then given by (hi)i∈I = (S hi)i∈I ⊂H , and it satisfies i∈I hf, hiihi = i∈I hf, hiihi = PH′ f for all f ∈H. P P 12 D.G.LEE,F.PHILIPP,ANDF.VOIGTLAENDER

Finally, in the case where (hi)i∈I = (g, Λ) is a Gabor family with a lattice Λ, it is easy to ′ ′ see that S ◦π(λ)= π(λ)◦S and π(λ)H ⊂H for λ ∈ Λ, which implies PH′ ◦π(λ)= π(λ)◦PH′ , and therefore S† ◦ π(λ) = π(λ) ◦ S† for all λ ∈ Λ. Consequently, setting γ := S†g, we have S†(π(λ)g) = π(λ)γ, so that the canonical dual system of a Gabor frame sequence (g, Λ) is the Gabor system (γ, Λ), where γ = S†g is called the canonical dual window of (g, Λ). Our next result shows that γ inherits the regularity of g, if one measures this regularity using one 1 2 H1 of the three spaces H ,Lw, or . 1 Rd 2 Rd H1 Rd R2d Proposition 4.6. Let V ∈ {H ( ),Lw( ), ( )}. Let Λ ⊂ be a lattice and let g ∈ V . If (g, Λ) is a frame sequence, then the associated canonical dual window γ satisﬁes γ ∈ V .

Proof. The frame operator S : L2(Rd) → L2(Rd) associated to (g, Λ) is non-negative and has closed range. Consequently, there exist ε > 0 and R > 0 such that σ(S) ⊂ {0}∪ [ε, R]; see for instance [3, Lemma A.2]. Now, with the open ball Bδ(0) := {z ∈ C: |z| < δ}, deﬁne ε ε ε C Ω := Bε/4(0) ∪ x + iy : x ∈ ( 2 , 2R),y ∈ (− 4 , 4 ) ⊂ , noting that Ω ⊂ C is open, with σ(S) ⊂ Ω. Furthermore, it is straightforward to see that

C 0, if z ∈ Bε/4(0), ϕ :Ω → , z 7→ −1 (z , otherwise is holomorphic. Since the functional calculus for self-adjoint operators is an extension of the holomorphic functional calculus, [3, Lemma A.6] shows that S† = ϕ(S). Finally, since † g ∈ VΛ, Proposition 4.5 now shows that γ = S g = ϕ(S)g ∈ VΛ ⊂ V as well.

Appendix A. (Non)-convergence of Janssen’s representation for H1 windows In this appendix we provide a counterexample showing that Janssen’s representation of the frame operator associated to a Bessel vector g ∈ H1 in general does not converge unconditionally with respect to the strong operator topology. We furthermore show that for convergence in operator norm, even conditional convergence fails in general. For simplicity, we only consider the setting d = 1 and the lattice Λ = Z × Z. Thus, given a function g ∈ H1 = H1(R), we say that g is a Bessel vector if the Gabor system 2 (TkMℓ g)k,ℓ∈Z ⊂ L (R) is a Bessel system. In this case, Janssen’s representation of the frame operator S := Sg := SZ×Z,g,g is (formally) given by

S = hg, TkMℓ gi TkMℓ. (A.1) Z ℓ,kX∈ We are interested in the question whether the series deﬁning Janssen’s representation is unconditionally convergent in the strong operator topology (SOT ), as an operator on L2(R). We will construct a function g ∈ H1 for which this fails. INVERTIBILITY OF THE GABOR FRAME OPERATOR ON CERTAIN MODULATION SPACES 13

A.1. Properties of the Zak transform. The construction of the counterexample is based on several properties of the Zak transform that we brieﬂy recall. Given f ∈ L2(R), its Zak 2 R2 transform Zf ∈ Lloc( ) is deﬁned as Zf(x,ω) := f(x − k)e2πikω, Z Xk∈ 2 R2 where the series converges in Lloc( ); this is a consequence of the fact that Z : L2(R) → L2([0, 1]2) is unitary, (A.2) as shown in [10, Theorem 8.2.3] and of the fact that the Zak transform Zf of a function f ∈ L2(R) is always quasi-periodic, meaning that Zf(x + n,ω)= e2πinωZf(x,ω) and Zf(x,ω + n)= Zf(x,ω) (A.3) for (almost) all x,ω ∈ R and all n ∈ Z; see [10, Equations (8.4) and (8.5)]. Another crucial property is the interplay between the Zak transform and the time-frequency shifts TkMn, as expressed by the following formula (found in [10, Equation (8.7)]): 2πinx −2πikω Z[TkMnf](x,ω)= e e Zf(x,ω)= en,−k(x,ω) Zf(x,ω), (A.4) where we used the functions 2πi(nx+kω) en,k(x,ω) := e for n, k ∈ Z and x,ω ∈ R. 2 2 Note that (en,k)n,k∈Z is an orthonormal basis of L ([0, 1] ). Finally, we note the following equivalence, taken from [2, Theorem 3.1]: ∀ g ∈ L2(R) : g is a Bessel vector ⇐⇒ Zg ∈ L∞([0, 1]2). (A.5)

A.2. Properties of H1. A further important property that we will use is the following characterization of the space H1 via the Zak transform, a proof of which is given in [3, Lemma 2.4]. 2 R H1 1,2 R2 ∀ f ∈ L ( ) : f ∈ ⇐⇒ Zf ∈ Wloc ( ). (A.6) It is crucial to observe that the Sobolev space W 1,2(R2) belongs to the “borderline” case of 1,2 R2 ∞ R2 the Sobolev embedding theorem, meaning Wloc ( ) 6֒→ Lloc( ). In fact, it is easy to verify 1 1 T R2 (see e.g. [6, Page 280]) for x0 := ( 2 , 2 ) ∈ that the function 1 u : (0, 1)2 → R, x 7→ ln ln 1+ 0 |x − x | 0 belongs to W 1,2((0, 1)2), but is not essentially bounded. Now, using the chain rule and the product rule for Sobolev functions (see e.g. [19, Exercise 11.51(i)] and [6, Theorem 1 in ∞ 2 Section 5.2.3]), we see that if ϕ ∈ Cc ((0, 1) ) is chosen such that 0 ≤ ϕ ≤ 1 and such that ϕ ≡ 1 on a neighborhood of x0, then the function 2 u : R → [0, ∞), x 7→ ϕ(x) · 1 + sin(u0(x)) (A.7) 1,2 R2 R2 satisﬁes u ∈ W ( ), is continuous and bounded on \ {x0}, but limx→x0 u(x) does not exist; this uses that limx→∞ sin(x) does not exist and that on each small ball Bε(x0), the function u0 attains all values from (M, ∞), for a suitable M = M(ε) > 0. 14 D.G.LEE,F.PHILIPP,ANDF.VOIGTLAENDER

A.3. A connection to Fourier series. In this subsection, we show that for any fixed window g ∈ L2(R) the unconditional convergence of Janssen’s representation in the strong operator topology implies that the partial sums of a certain Fourier series are uniformly bounded in L∞. This connection will be used in the next subsection to disprove the unconditional convergence of Janssen’s representation in the strong operator topology. Precisely, define Q := [0, 1]2. For H ∈ L∞(Q), define the associated multiplication operator as 2 2 MH : L (Q) → L (Q), F 7→ F · H.

It is well-known that kMH kL2→L2 = kHkL∞ . Let us fix any window g ∈ L2(R). Given a finite set I ⊂ Z2, we define

2 2 SI : L (R) → L (R), f 7→ hg, TkMℓgiTkMℓf. (k,ℓX)∈I Using Equation (A.4) and the isometry of the Zak transform, we then see

Z(SIf)= hZg, Z[TkMℓg]iL2(Q)Z[TkMℓf]= Zf · hZg,Zg · eℓ,−kiL2(Q) · eℓ,−k (k,ℓX)∈I (k,ℓX)∈I [2 = Zf · hZg · Zg,eℓ,−kiL2(Q) · eℓ,−k = Zf · |Zg| (ℓ, −k) · eℓ,−k (k,ℓX)∈I (k,ℓX)∈I = M 2 [Zf], FI′ [|Zg| ] where I′ := {(ℓ, −k): (k,ℓ) ∈ I} and Z2 FJ H := H(α) eα with H(α)= hH, eαiL2(Q) for J ⊂ . αX∈J In other words, we have b b −1 S = Z ◦ M 2 ◦ Z. (A.8) I FI′ [|Zg| ] 2 ′ Given J ⊂ Z , deﬁne J∗ := {(−ℓ, k): (k,ℓ) ∈ J} and note (J∗) = J. Now, suppose that (SI )I converges strongly to some (bounded) operator, as I → Z2; this is always the case if Janssen’s representation converges unconditionally (to S or some other operator) in the SOT. Then, Z2 ∞ Z2 given any sequence (Jn)n∈N of ﬁnite subsets Jn ⊂ with Jn ⊂ Jn+1 and n=1 Jn = , 2 we see (Jn)∗ → Z so that the sequence (S )n∈N converges strongly to some bounded (Jn)∗ S 2 2 N operator. By the uniform boundedness principle, this shows kS(Jn)∗ kL →L ≤ C for all n ∈ ′ and some C > 0. By Equations (A.2) and (A.8), and because of ((Jn)∗) = Jn, this implies 2 F [|Zg| ] = M 2 2 2 ≤ C ∀ n ∈ N, Jn L∞(Q) FJn [|Zg| ] L →L 2 2 meaning that the partial Fourier sums F Jn [|Zg| ] of the function |Zg| are uniformly bounded in L∞(Q).

A.4. The counterexample. In this subsection, we prove the following: Proposition A.1. There exists a Bessel vector g ∈ H1(R) such that the series deﬁning Janssen’s representation of the frame operator S = Sg = SZ×Z,g,g associated to g is not unconditionally convergent in the strong operator topology. INVERTIBILITY OF THE GABOR FRAME OPERATOR ON CERTAIN MODULATION SPACES 15

To prove the proposition, we consider the function F := u : (0, 1)2 → [0, ∞) introduced in Equation (A.7). The properties of F that we need are the following: 2 2 1 (1) F has compact support in (0, 1) , say supp F ⊂ (δ, 1 − δ) for some δ ∈ (0, 2 ). 2 (2) F is bounded, but discontinuous at x0 ∈ (0, 1) (even after adjusting F on a set of measure zero). (3) F ∈ W 1,2 (0, 1)2 . We now extend Fby zero to [0, 1)2 and then extend 1-periodically in both coordinates to R2. 1,2 R2 Thanks to the compact support of F , it is easy to see F ∈ Wloc ( ). Furthermore, we consider the function 2 2πi⌊x⌋ω G0 : R → C, (x,ω) 7→ e , where for each x ∈ R, ⌊x⌋ ∈ Z denotes the unique integer such that x ∈ ⌊x⌋ + [0, 1). It is then straightforward to verify that G0 is quasi-periodic (see Equation (A.3)), i.e., 2πimω G0(x + m,ω) = e G0(x,ω) and G0(x,ω + m) = G0(x,ω) for x,ω ∈ R and m ∈ Z. Since F is 1-periodic in both coordinates, it is easy to see that F · G0 is quasi-periodic as well. Finally, we choose a smooth function ψ : R → R satisfying ψ(x)= n for all x ∈ n+[δ, 1−δ] with n ∈ Z, and define G : R2 → C, (x,ω) 7→ e2πiψ(x)ω. Using that F (x,ω) = 0 for n ∈ Z and x ∈ [n,n + 1] \ (n + δ, n + 1 − δ), it is easy to check 1,2 R2 2 R2 F · G0 = F · G, so that H := F · G ∈ Wloc ( ) ⊂ Lloc( ) is quasi-periodic. Since the Zak transform Z : L2(R) → L2((0, 1)2) is unitary, there exists a unique function 2 R g ∈ L ( ) such that (Zg)|(0,1)2 = H|(0,1)2 . Since both Zg and H are quasi-periodic, this 1,2 R2 implies Zg = H almost everywhere. Since H ∈ Wloc ( ) is bounded, Equations (A.5) and (A.6) show that g ∈ H1 is a Bessel vector. Let us assume towards a contradiction that Janssen’s representation of the frame operator associated to g converges unconditionally in the strong operator topology. 2 2 2 2 Note that |Zg| = |H| = F is discontinuous at x0 ∈ (0, 1) (since F is discontinuous there and also non-negative), even after possibly changing |Zg|2 on a null-set. In particular, this im- [2 Z2 1 Z2 plies that the Fourier coefficients cα := |Zg| (α) (for α ∈ ) satisfy c = (cα)α∈Z2 ∈/ ℓ ( ), since otherwise the Fourier series of |Zg|2 would be uniformly convergent. This implies α∈Z2 | Re cα| = ∞ or α∈Z2 | Im cα| = ∞. For simplicity, we assume the first case; the second case can be treated by similar arguments. This implies that there exists an enumeration P Z2 P N (αn)n∈N of such that | n=1 Re cαn |→∞ as N → ∞. Indeed, if α∈Z2 (Re cα)+ < ∞ or α∈Z2 (Re cα)− < ∞, this is trivial (for every enumeration); otherwise, existence of the desired enumeration followsP from the Riemann rearrangement theoremP (see e.g., [22, Theo- remP 3.54]).

Now, deﬁne Jn := {α1,...,αn} for n ∈ N. We have seen in Appendix A.3 that the partial 2 ∞ 2 N Fourier sums FJn [|Zg| ] are uniformly bounded in L , say kFJn [|Zg| ]kL∞ ≤ C for all n ∈ . 2 Since each FJn [|Zg| ] is continuous (in fact, a trigonometric polynomial), this implies

2 [2 2πihα,0i C ≥ FJN |Zg| (0) = |Zg| (α) · e α∈JN X

16 D.G.LEE,F.PHILIPP,ANDF.VOIGTLAENDER

= cα ≥ Re cαn →∞ as N →∞, α∈JN n=1 X X which is the desired contradiction. A.5. Conditional divergence of Janssen’s representation in the operator norm. We showed above that unconditional convergence of Janssen’s representation (A.1) in the strong operator topology fails for some Bessel vector g ∈ H1(R). A similar argument shows that convergence in the operator norm (with respect to any given enumeration) also fails in general: Using Equation (A.8) (or more generally the arguments in Appendix A.3), it is relatively easy 2 to see that if for some enumeration Z = {αn : n ∈ N} and In := {α1,...,αn}, the sequence of partial sums (SIn )n∈N of Janssen’s representation (A.1) converges in operator norm (not 2 ′ N even necessarily to S), then the associated sequence (FIn [|Zg| ])n∈ of partial Fourier sums of |Zg|2 is Cauchy in L∞(Q) and thus converges uniformly on Q to a (necessarily continuous) function H : Q → C. However, since |Zg|2 = |H|2 = F 2 ∈ L∞(Q) ⊂ L2(Q), we know that

′ 2 2 2 2 FIn [|Zg| ] → F with convergence in L (Q). Hence, F = H almost everywhere on Q, where H is continuous.e But we saw above that F 2 is discontinuous on Q, even after (possibly) changing it on a null-set. Thus, we have obtained the desirede contradiction: e Proposition A.2. There are Bessel vectors g ∈ H1 for which Janssen’s representation fails to converge conditionally in the operator norm.

However, we leave it as an open question whether Janssen’s representation converges conditionally in the strong sense for Bessel vectors g ∈ H1.

Acknowledgments D.G. Lee acknowledges support by the DFG Grants PF 450/6-1 and PF 450/9-1. 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 German Science Foundation (DFG) in the context of the Emmy Noether junior research group VO 2594/1-1.

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Author Affiliations

D.G. Lee: KU Eichstatt–Ingolstadt,¨ Mathematisch–Geographische Fakultat,¨ Ostenstra- ße 26, Kollegiengebaude¨ I Bau B, 85072 Eichstatt,¨ Germany Email address: [email protected]

F. Philipp: Technische Universitat¨ Ilmenau, Institute for Mathematics, Weimarer Straße 25, D-98693 Ilmenau, Germany Email address: [email protected]

F. Voigtlaender: Department of Mathematics, Technical University of Munich, 85748 Garch- ing bei Munchen,¨ Germany Email address: [email protected]