We point out a link between the theorem of Balian and Low on the
non-existence of well-localized Gabor-Riesz bases and a constant curvature
connection on projective modules over noncommutative tori.Comment: 7 pages, to appear in Expositiones Mathematica

Luef, Franz

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Franz Luef

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Expositiones Mathematicae

The Balian–Low theorem and noncommutative tori

We point out a link between the theorem of Balian and Low on the non-existence of well-localized Gabor–Riesz bases and a constant curvature connection on projective modules over noncommutative tori.acceptedVersion© 2018. This is the authors’ accepted and refereed manuscript to the article. Locked until 16.3.2020 due to copyright restrictions. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

NTNU Open (Norwegian University of Science and Technology)

THE BALIAN-LOW THEOREM AND NONCOMMUTATIVE TORIFRANZ LUEFAbstract. We point out a link between the theorem of Balian and Low on the non-existence of well-localized Gabor-Riesz bases and a constant curvature connection onprojective modules over noncommutative tori.1. IntroductionThe theorem of Balian-Low on the non-existence of well-localized Gabor-Riesz basesfor L2(R) is one of the cornerstones of time-frequency analysis [2, 16]. For the formula-tion we first introduce the multiplication operator and differentiation operator, denotedby (∇1g)(t) = 2πi t g(t) and (∇2g)(t) = g′(t) , respectively. Let π(z)g(t) = e2πiωtg(t−x)be the time-frequency shift of a function g by z = (x, ω) in phase space. Gabor studiedin [9] systems of the form G(g, θZ×Z) = {π(θk, l)g : k, l ∈ Z}, so-called Gabor systemswith Gabor atom g. The density theorem for Gabor frames says that if G(g, θZ× Z) θis a frame, then θ ∈ (0, 1].A natural question about Gabor systems G(g, θZ × Z) for a fixed Gabor atom g is tostudy for which θ the system G(g, θZ × Z) is a frame. If g is well-localized in timeand frequency, then θ = 1 will be excluded as for the Gaussian. It turns out that forsome Gabor atoms g, such as the Gaussian or any totally positive function of finitetype G(g, θZ× Z) is a Gabor frame for any θ in (0, 1), [13, 19, 25]. On the other handthe answer for the indicator function of an interval [0, c] is much more intricate andthe values θ and c for which one gets a Gabor frame are known as Janssen’s tie [7, 15].The theorem of Balian-Low provides an explanation for these facts.Theorem 1.1 (Balian-Low). Suppose the Gabor system G(g,Z2) is an orthonormalbasis for L2(R). Then(∫R|(∇1g)(t)|2 dt)(∫R|(∇2g)(t)|2 dt)=∞.In particular, the theorem of Balian-Low implies that if G(g,Z2) is an orthonormalbasis for L2(R), then g is not well-localized in time and frequency, e.g. g cannot bein the Schwartz class S (R) or in Feichtinger’s algebra S0(R). The definition of Gaborsystems does not indicate a link to regularity properties of the Gabor atom. Hence,the very reason for the incompatibility between orthonormal Gabor bases of the formG(g,Z2) and good time-frequency localization is not well understood despite the vast1991 Mathematics Subject Classification. Primary 42C15, 58B34; Secondary 46L08, 22D25.Key words and phrases. Gabor frames, noncommutative tori, gauge connections .2 F. Luefliterature on the Balian-Low theorem [1, 2, 3, 4, 10, 11, 14, 20, 21]. Note that someauthors refer to statements of the form Theorem (1.1) as weak Balian-Low theorems.The main aim of this investigation is to present an approach to Gabor frames thatprovides an explanation of the link between regularity properties of Gabor atoms andtheir behavior at the critical density. We are building on the correspondence betweenGabor frames and projective modules over noncommutative tori [17, 18]. The standardargument to demonstrate Theorem 1.1 is due to Battle [3]. We are demonstrating thatBattle’s argument is best understood in terms of noncommutative geometry.2. Noncommutative toriIn noncommutative geometry one attempts to define geometric objects and no-tions for general C∗-algebras. For our purpose we need the noncommutative torusAθ equipped with its structure as a noncommutative manifold. We briefly recall theconstruction of vector bundles over noncommutative tori, which are finitely generatedprojective modules over Aθ, the differential structure on Aθ is given by derivations andon the vector bundles by a connection, and the notion of curvature of a connection [5].We denote the operators π(0, 1) and π(θ, 0) as M1 and Tθ, respectively. Note that wehave M1Tθ = e2πiθTθM1 and hence the norm closure of {π(kθ, l) : k, l ∈ Z} defines thenoncommutative torus Aθ, [22]. The smooth noncommutative torus is the subalgebraA∞θ of Aθ consisting of operators(1) π(a) =∑k,l∈Zaklπ(θk, l), for a = (akl) ∈ S (Z2).The standard derivations on Aθ are given by∂1(a) = 2πi∑k,lkaklπ(θk, l) and ∂2(a) = 2πi∑k,llaklπ(θk, l) .The Schwartz space S (R) turns out to be vector bundle over A∞θ , [5, 17, 23].Proposition 2.1 (Connes). The Schwartz space S (R) is a finitely generated projectivemodule over the smooth noncommutative torus A∞θ with respect to the following A∞θ -valued inner product and left action:•〈f, g〉 =∑k,l∈Z〈f, π(θk, l)g〉π(θk, l) for f, g ∈ S (R),a · g =∑k,l∈Zaklπ(θk, l)g for a ∈ S (Z2), g ∈ S (R).Connes defined also a constant curvature connection on S (R), see [5], given by covari-ant derivatives ∇1, ∇2:(2) (∇1g)(t) = 2πi θ−1 t g(t) and (∇2g)(t) = g′(t) .The covariant derivatives satisfy the left Leibniz rule(3) ∇i(a · g) = (∂ia) · g + a · (∇ig), i = 1, 2.The Balian-Low theorem and noncommutative tori 3For example, we have for a = π(θk, l) that ∇1(π(θk, l)g) = 2πikπ(θk, l)g+π(θk, l)∇1g.The covariant derivatives are compatible with the hermitian structure of the A∞θ -module S (R):∂i(•〈f, g〉) = •〈∇if, g〉+ •〈f,∇ig〉 , i = 1, 2.Finally we observe that the connection has constant curvature:(4) F1,2 := [∇1,∇2] = −2πi θ−1 IS (R) ,which acts on the left on the A∞θ -module S (R), see [5, 6]. In the case of the Moyalplane we have a similar setting and the constant curvature connection provides a gauge-theoretic description of the canonical commutation relations [6], which also sheds someadditional light on Battle’s proof in the next section.3. Balian-Low TheoremThe theorem of Connes that S (R) is a finitely generated projective module over Aθimplies that there exist generators g1, ..., gn in S (R) such thatf = •〈f, g1〉 g1 + · · ·+ •〈f, gn〉 gn for all f ∈ S (R).In this investigation we restrict our interest to the case, when the A∞θ -module S (R)has one generator g. In [17] it was shown that this is equivalent to the Gabor systemG(g, θZ × Z) to be a frame for L2(R), i.e. there exist constants A and B, so calledframe constants, such thatA‖f‖22 ≤∑k,l|〈f, π(θk, l)g〉|2 ≤ B‖f‖22 for all f ∈ L2(R),where A is the maximal and B the minimal constant with these properties. Equiv-alently, we have that the frame operator Sf := •〈f, g〉 g has spectrum contained in[A,B].If G(g, θZ× Z) is a Gabor frame, then one has expansions of the formf =∑k,l〈f, π(θk, l)g〉π(θk, l)S−1g =∑k,l〈f, π(θk, l)S−1g〉π(θk, l)gfor all f ∈ L2(R). The atom S−1g in the above expansions is known as the canonicaldual Gabor atom. We showed in [18] that there is a correspondence between tightGabor frames, i.e. Gabor frames G(g, θZ× Z) with A = B and atoms g in S (R), andprojections p = •〈g, g〉 in noncommutative tori A∞θ .The Balian-Low theorem is a statement on the finer properties of the generators forthe A∞θ -module S (R) in the case θ = 1. Note that A1 is the commutative C∗-algebraC(T2) of continuous functions over the torus T2 and that the smooth subalgebra A∞1is the space of smooth functions over the torus C∞(T2).Our main contribution is to place Battle’s proof of the theorem of the (weak) Balian-Low Theorem into the framework of noncommutative geometry. Namely, Battle uses4 F. Luefthe left Leibniz rule for the covariant derivations ∇1 and ∇2 for the time-frequencyshifts π(k, l) and that the connection defined by ∇1,∇2 has constant curvature −2πi.We present now Battle’s argument for a more general formulation of the theorem ofBalian-Low [11].Theorem 3.1 (Weak-Balian-Low). Suppose G(g, θZ × Z) for a g ∈ L2(R) is a Rieszbasis for L2(R) and we denote by h the canonical dual Gabor atom S−1g. Then either( ∫R |(∇1g)(t)|2 dt)( ∫R |(∇2g)(t)|2 dt)=∞ or( ∫R |(∇1h)(t)|2 dt)( ∫R |(∇2h)(t)|2 dt)=∞.Proof. Suppose ∇ig and ∇ih are in L2(R) for i = 1, 2. Then the left Leibniz rule for∇1 applied to π(k, l) implies that∇1(π(k, l)g) = 2πikπ(k, l)g + π(k, l)∇1g.Now, the biorthogonality of g and h yields that the second term on the right hand sidevanishes〈∇1g, π(k, l)h〉 = 〈g,∇1π(k, l)h〉 = 2πik〈g, π(k, l)h〉+ 〈g, π(k, l)∇1h〉and consequently〈∇1g, π(k, l)h〉 = 〈π(−k,−l)g,∇1h〉and in a similar manner one shows that 〈π(k, l)g,∇2h〉 = 〈∇2g, π(−k,−l)h〉. Byassumption G(g, θZ × Z) is a Riesz basis for L2(R) and h = S−1g is the unique dualGabor atom. Hence we have∇1g =∑k,l〈∇1g, π(k, l)h〉π(k, l)g and ∇2h =∑k,l〈∇2h, π(k, l)g〉π(k, l)h.We use these relations to derive at a contradiction to the fact that ∇1,∇2 is a constantcurvature connection on S (R).〈∇1g,∇2h〉 =〈∑k,l〈∇1g, π(k, l)h〉π(k, l)g,∇2h〉=∑k,l〈π(−k,−l)g,∇1h〉〈∇2g, π(−k,−l)h〉=∑m,n〈π(m,n)g,∇1h〉〈∇2g, π(m,n)h〉=∑m,n〈∇2g, π(m,n)h〉〈π(m,n)g,∇1h〉=〈∇2g,∑m,n〈∇1h, π(m,n)h〉π(m,n)g〉= 〈∇2g,∇1h〉(5)yields that〈(∇1∇2 −∇2∇1)g, h〉 = −2πiθ−1〈g, h〉.The Balian-Low theorem and noncommutative tori 5Hence the curvature F1,2 = ∇1∇2 − ∇2∇1 has a kernel as an operator on L2(R) thatvanishes which is a contradiction to F1,2 = −2πiθ−1I. Hence we have arrived at acontradiction to the assumption ∇ig and ∇ih are in L2(R) for i = 1, 2. In particular, the Balian-Low theorem implies that a g ∈ S (R) cannot generate aRiesz basis of the form G(g,Z2). For θ ∈ (0, 1) there exist Schwartz functions, e.g.the Gaussian g(t) = e−πt2, that generate Gabor frames for θZ × Z. These Gaborframes may be used to construct non-trivial projections in Aθ with trace θ, see [18]. Incontrast to the commutative case, where Aθ=1 only has the trivial projections p = 0and p = I. Recall that A∞1 is Morita equivalent to C∞(T2). Hence one may concludethat the generators of line bundles over noncommutative tori A∞θ for θ ∈ (0, 1) aremore well-behaved, then the line bundles over C∞(T2).Theorem 3.2 (Weak-Balian-Low-Revisited). Let g be a generator of the A∞1 = C∞(T2)-module S (R). Suppose the generator yields a Riesz basis for S (R) and h = S−1g itscanonical dual atom. Then either ∇1g or ∇2h are not S (R).Finally, there is an extension of the Balian-Low theorem to the case of Gabor-Rieszbases generated by atoms in Feichtinger’s algebra S0(R), see [1, 12]. Consequently, thepreceding statement also holds for singly-generated projective S0(R) modules over thesubalgebra A1θ = {a =∑k,l∈Z aklπ(θk, l) : a ∈ `1(Z2)} of Aθ. Note that A11 is theWiener algebra A(T2) of absolutely convergent Fourier series.Theorem 3.3. Let g be a generator of the A11 = A(T2)-module S0(R). Then g is notin S0(R).There is an argument based on results from operator algebras. We know from [17, 18]that generators of A1θ-modules are given by Gabor frames with generators in S0(R).Proof. Suppose g ∈ S0(R) generates a Gabor frame G(g,Z2). Then its canonical dualatom h is also in S0(R) andp =∑k,l∈Z2〈g, π(k, l)h〉π(k, l)defines a projection in A11. Hence S0(R) is an A11-module and this projection representsS0(R) in K0(A11), the K0 group of the algebra of absolutely convergent Fourier series.We know that K0(A11) is isomorphic to Z2 and has two generators. One generatorcorresponds to the trivial element 1 and the other to the Bott element. The projectivemodule defined by S0(R) above corresponds to the Bott element, as can be seen by acomputation of the Connes-Chern character [23]. However, since the C∗-completion ofA11 is isomorphic to C(T2) we know that there cannot be any non-trivial projections inC(T2) since T2 is connected. Thus we have a contradiction, as the projection p abovewould be non-trivial. This argument extends to higher dimensional symplectic lattices [11], as well in muchthe same way, since time-frequency shifts from symplectic lattices generate commuta-tive algebras, which leads naturally to topological obstructions and the results in [24]might be useful in this context.6 F. LuefNote how the argument fails when we have a multi-window Gabor frame for Z2 gen-erated by g1, ..., gn in S0(R). In that case, we can define a projection P = (pij)ni,j=1 inthe matrix algebra Mn(C(T2)) over C(T2):pij =∑k,l∈Z〈gi, π(k, l)hj〉π(k, l),where hj denotes the canonical dual Gabor atom to the jth Gabor atom gj. An ele-mentary computation shows that P is a projection in Mn(C(T2)), which does not yielda contradiction, because there are non-trivial projections in Mn(C(T2)) for n ≥ 2.References[1] G. Ascensi, H. G. Feichtinger, and N. Kaiblinger. Dilation of the Weyl symbol and Balian-Lowtheorem. Trans. Amer. Math. Soc., 366(7):3865–3880, 2014.[2] R. Balian. Un principe d’incertitude fort en théorie du signal ou en mécanique quantique. C. R.Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 292(20):1357–1362, 1981.[3] G. Battle. Heisenberg proof of the Balian-Low theorem. Lett. Math. Phys., 15(2):175–177, 1988.[4] J. J. Benedetto, C. Heil, and D. F. Walnut. Differentiation and the Balian-Low theorem. J.Fourier Anal. Appl., 1(4):355–402, 1995.[5] A. Connes. C∗-algébres et géométrie différentielle. C. R. Acad. Sci. Paris Sér. A-B, 290(13):A599–A604, 1980.[6] L. Dabrowski, G. Landi and F. Luef. Sigma-model solitons on noncommutative spaces. Lett. 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Janssen. Zak transforms with few zeros and the tie. In H. G. Feichtinger andT. Strohmer, editors, Advances in Gabor Analysis, Applied and Numerical Harmonic Analysis,pages 31–70. Birkhäuser, Basel, 2003.[16] F. Low. Complete sets of wave packets. In C. DeTar, J. Finkelstein, and C. Tan, editors, A Passionfor Physics - Essays in Honor of Geoffrey Chew, pages 17–22. World Scientific, Singapore, 1985.[17] F. Luef. Projective modules over noncommutative tori are multi-window Gabor frames for mod-ulation spaces. J. Funct. Anal., 257(6):1921–1946, 2009.[18] F. Luef. Projections in noncommutative tori and Gabor frames. Proc. Amer. Math. Soc.,139(2):571–582, 2011.[19] Y. I. Lyubarskii. Frames in the Bargmann space of entire functions. In Entire and SubharmonicFunctions, volume 11 of Adv. Sov. Math., pages 167–180. American Mathematical Society (AMS),Providence, RI, 1992.The Balian-Low theorem and noncommutative tori 7[20] S. Nitzan and J.-F. Olsen. A Quantitative Balian-Low Theorem. J. Fourier Anal. Appl.,19(5):1078–1092, 2013.[21] S. Nitzan and J.-F. Olsen. Balian-Low type theorems in finite dimensions. arXiv:1707.06449,2017.[22] M. A. Rieffel. C*-algebras associated with irrational rotations. Pacific J. Math., 93:415–429, 1981.[23] M. A. Rieffel. Projective modules over higher-dimensional noncommutative tori. Can. J. Math.,40(2):257–338, 1988.[24] J. Rosenberg. Noncommutative variations on Laplace’s equation. Anal. PDE, 1(1):95–114, 2008.[25] K. Seip. Density theorems for sampling and interpolation in the Bargmann-Fock space. I. J. ReineAngew. Math., 429:91–106, 1992.Department of Mathematical Sciences, NTNU Trondheim, 7041 Trondheim, NorwayE-mail address: franz.luef@math.ntnu.no

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The Balian-Low theorem and noncommutative tori

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