aa r X i v : . [ m a t h . F A ] J un A SHORT NOTE ON THE FRAME SET OF ODD FUNCTIONS
MARKUS FAULHUBER
Analysis Group, Department of Mathematical Sciences, NTNU TrondheimSentralbygg 2, Gløshaugen, Trondheim, Norway
Abstract.
In this work we derive a simple argument which shows that Gabor systems con-sisting of odd functions of d variables and symplectic lattices of density 2 d cannot constitutea Gabor frame. In the 1–dimensional, separable case, this is a special case of a result provedby Lyubarskii and Nes, however, we use a different approach in this work exploiting the al-gebraic relation between the ambiguity function and the Wigner distribution as well as theirrelation given by the (symplectic) Fourier transform. Also, we do not need the assumptionthat the lattice is separable and, hence, new restrictions are added to the full frame set ofodd functions. Introduction and Main Result
In this short note we show that the full frame set of any odd function of d variables inFeichtinger’s algebra cannot contain symplectic lattices of density 2 d . In the 1–dimensional,separable case, this is a special case of a more general result derived by Lyubarskii and Nes[16] who could show that no odd window function g ∈ S ( R ) can produce a separable Gaborframe of redundancy n +1 n , n ∈ N by studying the vector–valued Zak transform and Zebulski–Zeevi matrices. For an alternative proof of this result see the survey article by Gr¨ochenig[10].However, our arguments are somewhat simpler and hold for symplectic lattices in arbitrarydimension d , which makes up for the drawback that we do not derive more general results.The key argument is that the Wigner distribution is the symplectic Fourier transform ofthe ambiguity function and that they also fulfill a simple algebraic relation. Moreover, ourarguments show that, after a proper scaling, the cross Wigner distribution of any functionin Feichtinger’s algebra and any even function in Feichtinger’s algebra is an eigenfunction ofthe symplectic Fourier transform with eigenvalue 1 and the pairing with any odd function inFeichtinger’s algebra is an eigenfunction with eigenvalue -1.This work concerns the fine structure of Gabor frames as described in [10], i.e., relationsbetween the properties of a fixed window and its frame set. For a (window) function g ∈ E-mail address : [email protected] .2010 Mathematics Subject Classification.
Key words and phrases.
Ambiguity Function, Feichtinger’s Algebra, Frame Set, Gabor Frame, WignerDistribution.The author was supported by the Erwin–Schr¨odinger program of the Austrian Science Fund (FWF): J4100-N32. The computational results presented have been achieved (in part) using the Vienna Scientific Cluster(VSC). The author wishes to thank Franz Luef for many fruitful discussions on the topic. The author thanksthe anonymous referee for beneficial feedback on the first version of the manuscript. L ( R d ) and an index set Λ ⊂ R d , we denote the resulting Gabor system by G ( g, Λ). The(full) frame set of the window g is given by F full ( g ) = { Λ ⊂ R d , Λ a lattice | G ( g, Λ) is a frame } . Inspired by the work of Lemvig [15], the original intention of this short note was to show upsimple restrictions for the full frame set of odd (1–dimensional) Hermite functions by showingthat certain sums vanish, but the restriction to this very special class of functions turnedout to be unnecessary. Unfortunately, we do not get any new insights into the frame set ofeven (Hermite) functions. Among other counterexamples, Lemvig showed that the squarelattice of density 2 does not generate a Gabor frame for the second Hermite function (theGaussian being indexed as 0–th Hermite function), which was the first known obstruction tothe frame set of the second Hermite function. Numerical inspections suggest that, for thesecond Hermite function, among all separable lattices of density 2 the square lattice is theonly lattice which does not yield a Gabor frame, in particular, in case of the square lattice thelower frame bound is zero and it yields the global minimum of the lower frame bound seen as afunction of the lattice parameters. This example stands in sharp contrast to the results givenin [3], where it is shown that under the same assumptions, but using the Gaussian instead ofthe second Hermite function, the square lattice gives the global maximum of the lower framebound seen as a function of the lattice parameters. The common theme, however, is thatin both cases the highest possible symmetry of the lattice leads to extremal frame bounds.It was proven in [2] that, for a Gabor frame of even redundancy with standard Gaussianwindow, the hexagonal lattice yields the smallest upper frame bound among all lattices. Weconjecture that the hexagonal lattice should also give the largest lower frame bound in thiscase. So, we pose the following question: For the second Hermite function, does the Gaborsystem generated by the hexagonal lattice of density 2 have a positive lower frame bound?The results by Lemvig tempt us to think that this might not the be case, but numericalinspections say that we actually have a Gabor frame with approximate lower frame bound0 . . . . .Our main result, however, concerns odd windows in Feichtinger’s algebra which we denoteby S ( R d ) (another common notation is M ( R d )). Theorem 1.1 (Main Result) . Let g ∈ S ( R d ) be an odd function, i.e., g ( t ) = − g ( − t ) andlet Λ ⊂ R d be a symplectic lattice in the time–frequency plane. If vol(Λ) = 2 − d then G ( g, Λ) cannot be a Gabor frame, or, in shorter notation:If g ∈ S ( R d ) , g ( t ) = − g ( − t ) and vol(Λ) = 2 − d , Λ symplectic = ⇒ Λ / ∈ F full ( g ) . Theorem 1.1 particularly implies that for d = 1 no lattice of density 2 can be contained inthe frame set of an odd function from Feichtinger’s algebra.This work is structured as follows: • In Section 2 we recall the basic properties of Gabor frames for the Hilbert space L ( R d ). After that, we introduce quadratic representations of a function f ∈ L ( R d )with respect to a (fixed) window g ∈ L ( R d ), namely the short–time Fourier transform,the ambiguity function and the Wigner distribution. We show their algebraic relationsas well as their relation under the symplectic Fourier transform and introduce thesymplectic version of Poisson’s summation formula. Also, we will see that Feichtinger’salgebra is a convenient setting for our purposes. SHORT NOTE ON THE FRAME SET OF ODD FUNCTIONS 3 • In Section 3 we show how sharp frame bounds can be calculated, using the resultsestablished by Janssen in the 1990s. These results finally lead to the proof of Theorem1.1.2.
Gabor Frames and Time–Frequency Analysis in a nutshell
We consider Gabor frames for the Hilbert space of square integrable functions in d –dimensional Euclidean space L ( R d ). Concerning the notation we follow mainly the textbookof Gr¨ochenig [9]. A more recent introduction to the topic is the 2 nd edition of Christensen’stextbook [1].As our functions will be defined pointwise and at least continuous in the remainder of thiswork the following notation for the inner product in L ( R d ) is justified; h f, g i = Z R d f ( t ) g ( t ) dt. For two vectors t and t ′ in R d we denote the Euclidean scalar product by t · t ′ .The key elements in time-frequency analysis are the translation operator T x (time shift)and the modulation operator M ω (frequency shift) which are defined as T x f ( t ) = f ( t − x ) and M ω f ( t ) = e πiω · t f ( t ) . For a function in the Schwartz space S ( R d ) we define the Fourier transform by F f ( ω ) = Z R f ( t ) e − πiω · t dt, which extends to a unitary operator on L ( R d ) by the usual density argument. The Fouriertransform has the well–known properties of interchanging translation and modulation, i.e., F ( T x f ) = M − x F f and F ( M ω f ) = T ω F f. The translation (time shift) and modulation (freuqency shift) operator do not commute ingeneral, but they fulfill the following commutation relation(2.1) M ω T x = e πiω · x T x M ω . Hence, the combination of the two operators is called a time–frequency shift and usuallydenoted by π ( λ ) = M ω T x , λ = ( x, ω ) ∈ R , where λ is a point in the time–frequency plane or phase space. The composition of twotime–frequency shifts is given by π ( λ ) π ( λ ′ ) = e − πix · ω ′ π ( λ + λ ′ ) . A Gabor system is a collection of time–frequency shifted copies of a so–called windowfunction g ∈ L ( R ) with respect to an index set Λ ⊂ R and it is denoted by G ( g, Λ) = { π ( λ ) g | λ ∈ Λ } . Throughout this work, Λ will be a lattice, i.e., a discrete subgroup of R d . A lattice canbe represented by an invertible matrix M ∈ GL (2 d, R ) and is then given by Λ = M Z d .The matrix M is not unique since we can choose from countably many possible bases for Z d . For example, if d = 1, then any matrix B with integer entries and determinant 1, MARKUS FAULHUBER i.e.,
B ∈ SL (2 , Z ), satisfies B Z = Z . Although the representing matrix is not unique itsdeterminant is. We define the volume of a lattice Λ = M Z d byvol(Λ) = | det( M ) | . The density of a lattice is given by the reciprocal of the volume, i.e., δ (Λ) = vol(Λ) − .Usually, a lattice is called separable if it can be written as α Z d × β Z d , α, β ∈ R + . Alternativedefinitions of a separable lattice are that the generating matrix is a diagonal matrix, or, inthe most general case, that the lattice separates as M Z d × M Z d with M , M ∈ GL ( d, R ).For d = 1 all definitions coincide.A Gabor system G ( g, Λ) is called a Gabor frame if and only if the frame inequality isfulfilled, i.e.,(2.2) A (cid:13)(cid:13) f (cid:13)(cid:13) ≤ X λ ∈ Λ |h f, π ( λ ) g i| ≤ B (cid:13)(cid:13) f (cid:13)(cid:13) , ∀ f ∈ L ( R d ) , with positive constants 0 < A ≤ B < ∞ called frame bounds. In general, a Gabor frameis a redundant system and the redundancy of a Gabor system is given by the density of theunderlying lattice. If all elements of the Gabor system G ( g, Λ) have unit norm, the redundancyalso reflects itself in the frame bounds. We note that in the case of an orthonormal basis wehave A = B = 1.2.1. Symmetric Time–Frequency Shifts.
It will be advantageous to consider symmetrictime–frequency shifts instead of usual time–frequency shifts. The symmetric time–frequencyshift is given by(2.3) ρ ( λ ) = M ω T x M ω = T x M ω T x = e − πix · ω π ( λ ) . We note that ρ ( λ ) ρ ( λ ′ ) = e − πi ( x · ω ′ − x ′ · ω ) ρ ( λ + λ ′ ) . The Gabor system under consideration is then e G ( g, Λ) = { ρ ( λ ) g | λ ∈ Λ } . This system is a frame if and only if there exist positive constants 0 < A ≤ B < ∞ such that(2.4) A (cid:13)(cid:13) f (cid:13)(cid:13) ≤ X λ ∈ Λ |h f, ρ ( λ ) g i| ≤ B (cid:13)(cid:13) f (cid:13)(cid:13) , ∀ f ∈ L ( R d ) , It follows from (2.3) that the optimal constants
A, B in equations (2.2) and (2.4) are thesame. In particular, G ( g, Λ) is a frame if and only if e G ( g, Λ) is a frame. In the rest of thiswork we will work with the Gabor system e G ( g, Λ) as the phase factors are easier to handle inthis case.2.2.
Phase–Space Methods.
The short–time Fourier transform (STFT) and the ambiguityfunction are often used to measure time frequency concentration. They are defined in similarways and, in fact, they only differ by a phase factor, i.e., a complex exponential of modulus1. We will now introduce the necessary tools to prove Theorem 1.1. For more details we referto the textbooks of Folland [5], de Gosson [6, 7] or Gr¨ochenig [9].
SHORT NOTE ON THE FRAME SET OF ODD FUNCTIONS 5
Definition 2.1 (STFT) . For f ∈ L ( R d ), the short–time Fourier transform with respect tothe window g ∈ L ( R ) is defined as V g f ( x, ω ) = Z R d f ( t ) g ( t − x ) e − πiω · t dt = h f, π ( λ ) g i , λ = ( x, ω ) ∈ R d . Before we continue, we introduce the function space which will be most suitable for ourintentions, namely Feichtinger’s algebra S ( R d ), introduced by Feichtinger in the early 1980s[4]. There are several equivalent definitions of S ( R d ) and we prefer to use the followingdefinition. Definition 2.2 (Feichtinger’s Algebra) . Feichtinger’s algebra S ( R d ) consists of all elements g ∈ L ( R d ) such that (cid:13)(cid:13) V g g (cid:13)(cid:13) L ( R d ) = Z Z R d | V g g ( λ ) | dλ < ∞ , λ = ( x, ω ) ∈ R d . We note the following properties of S ( R d ). It is a Banach space, invariant under theFourier transform and time–frequency shifts. It contains the Schwartz space S ( R d ) and it isdense in L p ( R d ), p ∈ [1 , ∞ [. It is for these properties that it is a quite popular function spacein time–frequency analysis and the literature on the subject is huge. For more details on S we refer to the survey by Jakobsen [11] and the references therein.We turn to another time–frequency representation, which is defined similarly to the STFT. Definition 2.3 (Ambiguity Function) . For f, g ∈ L ( R d ), the (cross) ambiguity function isdefined as A g f ( x, ω ) = Z R d f ( t + x ) g ( t − x ) e − πiω · t dt = h π ( − λ ) f, π ( λ ) g i = h f, ρ ( λ ) g i , λ = ( x, ω ) ∈ R d . Both, V g f and A g f are uniformly continuous on R d . Due to relation (2.3), which is aconsequence of the commutation relation (2.1), we have that A g f ( x, ω ) = e πiω · x V g f ( x, ω ) . In particular this means that |V g f | ≡ | A g f | . We will now introduce a quadratic representationof a function f ∈ L ( R d ) which is usually used in quantum mechanics, the Wigner distribution. Definition 2.4 (Wigner Distribution) . For f, g ∈ L ( R d ), the (cross) Wigner distribution isdefined as W g f ( x, ω ) = Z R f ( x + t ) g ( x − t ) e − πiω · t dt, x, ω ∈ R d . For the rest of this work, we will drop the index in all of the above definitions if f = g .The Wigner distribution is related to the ambiguity function (and, hence, in a similar wayto the STFT) by the symplectic Fourier transform. In order to define the symplectic Fouriertransform, we first equip our phase space with a symplectic structure. In what follows thevectors λ = ( x, ω ) and λ ′ = ( x ′ , ω ′ ) in R d are always seen as column vectors and the scalarproduct of two vectors in the phase space is again denoted by λ · λ ′ . We define the symplecticform σ ( λ, λ ′ ) = x · ω ′ − ω · x ′ = λ · J λ ′ = λ T J λ,
MARKUS FAULHUBER where J = (cid:18) I − I (cid:19) is the standard symplectic matrix and I the d × d identity matrix. Amatrix S is called symplectic if and only if it preserves the symplectic form, i.e., σ ( Sλ, Sλ ′ ) = σ ( λ, λ ′ ) , or, equivalently, S T J S = J. As mentioned, it will turn out to be convenient to use a slightly different version of theFourier transform in phase space, the symplectic Fourier transform.
Definition 2.5 (Symplectic Fourier Transform) . For F ∈ S ( R d ) the symplectic Fouriertransform is given by F σ F ( x, ω ) = Z Z R F ( λ ′ ) e − πi σ ( λ,λ ′ ) dλ ′ , λ = ( x, ω ) , λ ′ = ( x ′ , ω ′ ) ∈ R d . Of course, the symplectic Fourier transform extends to all of L ( R d ) by the usual densityargument (just as the Fourier transform). A tool which is heavily exploited in time-frequencyanalysis is the Poisson summation formula which we will use for 2 d –dimensional lattices.The technical details for the Poisson summation formula to hold pointwise have been workedby Gr¨ochenig in [8]. Since our functions under consideration are in S ( R d ), their Wignerdistributions as well as their ambiguity functions will be elements of Feichtinger’s algebra inphase space, i.e., elements of S ( R d ) (see [11, chap. 5]). This assumption is sufficient forPoisson’s summation formula to hold pointwise. Proposition 2.6 (Poisson Summation Formula) . For F ∈ S ( R d ) and a lattice Λ = M Z d with dual lattice Λ ⊥ = M − T Z d we have X λ ∈ Λ F ( λ + z ) = vol(Λ) − X λ ⊥ ∈ Λ ⊥ F F ( λ ⊥ ) e πiλ ⊥ · z , λ, λ ⊥ , z ∈ R d . Instead of using the 2 d –dimensional Fourier transform we can adjust this result by using thesymplectic Fourier transform and the adjoint lattice instead of the dual lattice. The adjointof a lattice Λ = M Z d is given by Λ ◦ = J M − T Z d . Under the assumptions of Poisson’ssummation formula we get X λ ∈ Λ F ( λ + z ) = vol(Λ) − X λ ◦ ∈ Λ ◦ F σ F ( λ ◦ ) e πi σ ( λ ◦ ,z ) , λ, λ ◦ , z ∈ R d We say that a lattice is symplectic if its generating matrix is a multiple of a symplecticmatrix, i.e., Λ = c S Z d with c > S ∈ Sp ( d ), with Sp ( d ) being the set of all symplectic2 d × d matrices. We note that symplectic matrices actually form a group under matrix mul-tiplication and that any symplectic matrix has determinant 1 and, hence, Sp ( d ) ⊂ SL (2 d, R ).In general Sp ( d ) is a proper subgroup of the special linear group SL (2 d, R ), only for d = 1we have that Sp (1) = SL (2 , R ). In particular, it follows that any 2–dimensional lattice issymplectic. In general, it follows from the definition of a symplectic matrix thatΛ ◦ = vol(Λ) − /d Λ , Λ symplectic , because, by definition, S ∈ Sp ( d ) ⇔ S = J S − T J − and for Λ = c S Z , c > ◦ = c − J S − T J − Z d = c − J S − T Z d , as J − is just another choice of basis for Z d . Hence,for Λ symplectic the adjoint lattice is only a scaled version of the original lattice. SHORT NOTE ON THE FRAME SET OF ODD FUNCTIONS 7
As a last point in this section, we have a closer look at the relation between the ambiguityfunction (and hence the STFT) and the Wigner distribution. We start with their relationgiven by the symplectic Fourier transform.
Proposition 2.7.
For f, g ∈ L ( R d ) , the ambiguity function and the Wigner distribution aresymplectic Fourier transforms of each other, i.e, F σ ( A g f ) ( x, ω ) = W g f ( x, ω ) and F σ ( W g f ) ( x, ω ) = A g f ( x, ω )Also, we have the following algebraic relation between the ambiguity function and theWigner distribution. Proposition 2.8.
For f, g ∈ L ( R d ) , the ambiguity function and the Wigner distributionfulfill W g f ( x, ω ) = 2 d A g ∨ f (2 x, ω ) and A g f ( x, ω ) = 2 − d W g ∨ f (cid:0) x , ω (cid:1) , where g ∨ ( t ) = g ( − t ) denotes the reflection of g . We proceed with some more results regarding the ambiguity function and the Wignerdistribution which we will need in the end to prove our main result. But first, we introducesome notation. For a function F in phase space, the isotropic dilation is given by D α F ( x, ω ) = F ( αx, αω ) , α ∈ R + . The behavior of this operator under the symplectic Fourier transform is given by(2.5) F σ ( D α F )( x, ω ) = α − d D α F σ F ( x, ω ) . Lemma 2.9.
For f, g ∈ L ( R d ) with g ∨ = g we have F σ (cid:16) D √ A g f (cid:17) ( x, ω ) = D √ A g f ( x, ω ) , F σ (cid:18) D √ W g f (cid:19) ( x, ω ) = D √ W g f ( x, ω ) . If − g ∨ = g we have F σ (cid:16) D √ A g f (cid:17) ( x, ω ) = − D √ A g f ( x, ω ) , F σ (cid:18) D √ W g f (cid:19) ( x, ω ) = − D √ W g f ( x, ω ) . Proof.
This is an immediate consequence of Proposition 2.7 and Proposition 2.8. By usingProposition 2.7 and (2.5) we get F σ ( D √ A g f )( x, ω ) = 2 − d D √ W g f ( x, ω ) = D √ (cid:16) − d W g f (cid:0) x , ω (cid:1)(cid:17) . Now, by the algebraic property from Proposition 2.8 we conclude that F σ ( D √ A g f )( x, ω ) = D √ A g ∨ f ( x, ω ) . In a similar manner we derive the analogous statement for W g f . The results follow from thedefinitions of A g f and W g f and the assumptions that ± g ∨ = g . (cid:3) MARKUS FAULHUBER
In [17] it was shown that the (suitably scaled) cross Wigner distributions of two Hermitefunctions as well as tensor products of Hermite functions are eigenfunctions of the planar(2–dimensional) Fourier transform with eigenvalues ±
1, depending on the pairing. In [14]another example of a “nonstandard” eigenfunction of the planar Fourier transform was given,namely the function F ( x, ω ) = √ x + ω xω (integrals have to be understood as Cauchy principalvalues in this case). All these examples are invariant under rotation (also the presentedset of eigenfunctions is countable). Lemma 2.9 gives us an uncountable set of examplesof eigenfunctions of the symplectic Fourier transform which do not necessarily possess anyrotational symmetries.For the next result, we recall that W g f ∈ L ( R d ) if and only if f, g ∈ S ( R d ) (see [7,chap. 7] or [11]). Also, if f, g ∈ S ( R d ), then the Wigner distribution W g f is in S ( R d ) (see[11]), which means that W g f ∈ L ( R d ) ⇐⇒ W g f ∈ S ( R d ) . This statement holds, of course, for the ambiguity function A g f and for the STFT V g f . Also,the assumptions for Poisson’s summation formula to hold pointwise are met and we derivethe following result. Lemma 2.10.
Let f, g ∈ S ( R d ) and let g be an odd function and Λ a symplectic lattice with vol(Λ) − = 2 d . Then X λ ∈ Λ W g f ( λ ) = − X λ ∈ Λ W g f ( λ ) = 0 , X λ ◦ ∈ Λ ◦ A g f ( λ ◦ ) = − X λ ◦ ∈ Λ ◦ A g f ( λ ◦ ) = 0 . Proof.
By the symplectic version of Poisson’s summation formula we have X λ ∈ Λ W g f ( λ ) = vol(Λ) − | {z } =2 d X λ ◦ ∈ Λ ◦ F σ ( W g f ) ( λ ◦ )By Proposition 2.7 we have2 d X λ ◦ ∈ Λ ◦ F σ ( W g f ) ( λ ◦ ) = 2 d X λ ◦ ∈ Λ ◦ A g f ( λ ◦ )and by the algebraic relation in Proposition 2.8 we have2 d X λ ◦ ∈ Λ ◦ A g f ( λ ◦ ) = 2 d X λ ◦ ∈ Λ ◦ − d W g ∨ f (2 − λ ◦ ) = − X λ ∈ Λ W g f ( λ ) , since g ∨ = − g , vol(Λ) − = 2 d and, hence, 2 − Λ ◦ = Λ as Λ is symplectic. Therefore, thestatement about the Wigner distribution follows. The statement for the ambiguity functionfollows analogously. (cid:3) An alternative (but equivalent) proof can be established by using Lemma 2.9: Let f, g ∈ S ( R d ), − g ∨ = g and vol(Λ) − = 1 (note that in this case Λ = Λ ◦ ), then X λ ∈ Λ D √ W g f ( λ ) = X λ ◦ ∈ Λ ◦ F σ (cid:18) D √ W g f (cid:19) ( λ ) = X λ ∈ Λ − D √ W g f ( λ ) . The analogous statement for A g f obviously holds as well. Now, note that dilating the latticeand dilating the Wigner distribution are two equivalent ways to establish the result. SHORT NOTE ON THE FRAME SET OF ODD FUNCTIONS 9 Sharp Frame Bounds
In this section we have a closer look at the frame operator and its spectrum. We will mainlyfollow Janssen’s articles [12, 13]. The main differences are that we formulate the results forsymplectic lattices in 2 d –dimensional phase space rather than for separable lattices in 2–dimensional phase space. Also, we use symmetric time–frequency shifts which only changesthe appearing phase factors. For non–separable lattices, they will be easier to handle lateron with this approach. Building on the results of the previous section, we will finally showthat for odd windows in S ( R d ) and Λ ⊂ R d a lattice in phase space with vol(Λ) − = 2 d , thelower frame bound of the Gabor system e G ( g, Λ) vanishes. By the comments in Section 2.1this is equivalent to the fact that the Gabor system G ( g, Λ) does not generate a frame, whichis our main result.The frame operator associated to the Gabor system e G ( g, Λ) is denoted by e S g, Λ and givenby e S g, Λ f = X λ ∈ Λ h f, ρ ( λ ) g i ρ ( λ ) g, f ∈ L ( R d ) . Another, very useful, representation of the frame operator is due to Janssen [12] and usuallycalled Janssen’s representation of the frame operator e S g, Λ = vol(Λ) − X λ ◦ ∈ Λ ◦ h g, ρ ( λ ◦ ) g i ρ ( λ ◦ ) . The frame operator is the composition of the analysis and the synthesis operator, which areadjoint to each other. The analysis operator maps a function from L ( R d ) to ℓ (Λ), Λ ⊂ R d and is given by e G g, Λ f = ( h f, ρ ( λ ) g i ) λ ∈ Λ . Its adjoint is called the synthesis operator, mapping sequences c = ( c λ ) λ ∈ Λ ∈ ℓ (Λ) to L ( R d ),and is given by e G ∗ g, Λ c = X λ ∈ Λ c λ ρ ( λ ) g. The frame operator can be written as e S g, Λ = e G ∗ g, Λ e G g, Λ . The following result is a straightforward generalization of the main result in [12], whereJanssen showed it for d = 1 and Λ separable. Proposition 3.1.
The following are equivalent:(i) e G ( g, Λ) is a frame with bounds A and B .(ii) A I L ( R d ) ≤ e S g, Λ ≤ B I L ( R d ) .(iii) A I ℓ (Λ ◦ ) ≤ vol(Λ) − e G g, Λ ◦ e G ∗ g, Λ ◦ ≤ B I ℓ (Λ ◦ ) . The most interesting part for this work is that we can compute the frame bounds via theeigenvalues of the bi–infinite matrix, indexed by the adjoint lattice; e G g, Λ ◦ e G ∗ g, Λ ◦ = (cid:0) h ρ ( λ ◦ ) g, ρ ( λ ◦′ ) g i (cid:1) λ ◦ ,λ ◦′ ∈ Λ ◦ . We proceed by calculating the values of the above matrix; h ρ ( λ ◦ ) g, ρ ( λ ◦′ ) g i = h g, ρ ( − λ ◦ ) ρ ( λ ◦′ ) g i = e πi σ ( λ ◦ ,λ ◦′ ) h g, ρ ( λ ◦′ − λ ◦ ) g i , λ ◦ , λ ◦′ ∈ Λ ◦ . Assume that vol(Λ) − /d ∈ N , then the entries in vol(Λ) − e G g, Λ ◦ e G ∗ g, Λ ◦ are constant alongdiagonals, i.e., vol(Λ) − e G g, Λ ◦ e G ∗ g, Λ ◦ has a Laurent structure. For the time–frequency shifts ρ ( λ ◦′ − λ ◦ ) the argument is obvious, the only justification we have to make is that the phasefactor e πi σ ( λ ◦ ,λ ◦′ ) is constant along diagonals. We show that σ ( λ ◦ , λ ◦′ ) is an integer multipleof vol(Λ) − /d . Let Λ = α / d S Z d , then vol(Λ) = α and Λ ◦ = α − / d S Z d . Since our latticeis symplectic by assumption, the symplectic form σ is independent from the matrix S and wehave e πi σ ( λ ◦ ,λ ◦′ ) = e πi σ ( α − / d S ( k,l ) T , α − / d S ( k ′ ,l ′ ) T ) = e vol(Λ) − /d πi ( k · l ′ − k ′ · l ) , k, l, k ′ , l ′ ∈ Z d . In the case that vol(Λ) − /d is even, the phase–factor equals +1 and can be neglected.However, if vol(Λ) − /d is odd, the phase–factor takes the role of an alternating sign, whichis constant along diagonals, i.e., it is either +1 or − λ ◦ − λ ◦′ being constant. For this reason we focus on the case where vol(Λ) − /d is even.It follows from the general theory on Toeplitz (matrices) and Laurent operators that thespectrum of such a (double) bi–infinite matrix can be computed via the essential infimumand supremum of a Fourier series, where the coefficients of the series are derived from theentries in the matrix. Using the above arguments, the following result is a straight forwardgeneralization of the result derived by Janssen in [13] (see Appendix A for Janssen’s result). Proposition 3.2.
For g ∈ L ( R d ) and Λ ⊂ R d with vol(Λ) − /d ∈ N the Gabor system e G ( g, Λ) possesses the optimal frame bounds A = ess inf z ∈ R vol(Λ) − X λ ◦′ − λ ◦ ∈ Λ ◦ e πi σ ( λ ◦ ,λ ◦′ ) A g ( λ ◦′ − λ ◦ ) e πi σ ( λ ◦′ − λ ◦ ,z ) B = ess sup z ∈ R vol(Λ) − X λ ◦′ − λ ◦ ∈ Λ ◦ e πi σ ( λ ◦ ,λ ◦′ ) A g ( λ ◦′ − λ ◦ ) e πi σ ( λ ◦′ − λ ◦ ,z ) . The above series is real–valued, since we sum over a lattice the imaginary parts also appearas complex conjugates an cancel out. Note that the above series need not be convergent. Inthis case the upper bound might not be finite and the Gabor system might not constitute aframe. However, for windows in Feichtinger’s algebra the upper bound is always finite . Asthe frame operator e S g, Λ is self–adjoint and positive semi–definite we have0 ≤ A = ess inf z ∈ R vol(Λ) − X λ ◦′ − λ ◦ ∈ Λ ◦ e πi σ ( λ ◦ ,λ ◦′ ) A g ( λ ◦′ − λ ◦ ) e πi σ ( λ ◦′ − λ ◦ ,z ) , by the theory of Laurent operators. We have now all the tools we need to prove Theorem 1.1. Proof of Theorem 1.1.
In order to prove our main result, we will show that the lowerframe bound vanishes under the assumptions of Theorem 1.1. For vol(Λ) − = 2 d and due tothe fact that λ ◦′ − λ ◦ ∈ Λ ◦ , the series in Proposition 3.2 reduces to φ ( z ) = vol(Λ) − X λ ◦ ∈ Λ ◦ A g ( λ ◦ ) e πi σ ( λ ◦ ,z ) . Now observe that φ (0) = vol(Λ) − X λ ◦ ∈ Λ ◦ A g ( λ ◦ ) , It follows from the results in Tolimieri and Orr [18] that vol(Λ) − P λ ◦ ∈ Λ ◦ |A g ( λ ◦ ) | always is an upperbound, however, usually not the optimal upper bound. For g ∈ S ( R d ) this expression is always finite. SHORT NOTE ON THE FRAME SET OF ODD FUNCTIONS 11 which is, up to the factor vol(Λ) − , just the series from Lemma 2.10. Hence, we concludethat for vol(Λ) − = 2 d and g ∈ S ( R d ), g ∨ = − g we have φ (0) = 0. This is equivalent tothe statement that the lower frame bound of the system e G ( g, Λ) vanishes. The same is truefor the lower frame bound of the Gabor system G ( g, Λ). Hence, the proof of Theorem 1.1 iscomplete.
Appendix A. Janssen’s Proposition
We will shortly state Janssen’s proposition from [13] which he used to compute sharpframe bounds for the L ( R ) case. In his formulation, Janssen used the STFT rather than theambiguity function and separable lattices rather than general lattices. Hence, Janssen’s resultis a special case of Proposition 3.2, but already carries the general idea in it. For g ∈ L ( R )and a lattice Λ ( α,β ) = α Z × β Z , ( αβ ) − ∈ N , the Gabor system G ( g, Λ ( α,β ) ) possesses theoptimal frame bounds A = ess inf ( x,ω ) ∈ R ( αβ ) − X k − k ′ ,l − l ′ ∈ Z V g (cid:16) k − k ′ β , l − l ′ α (cid:17) e πi (( k − k ′ ) x +( l − l ′ ) ω ) B = ess sup ( x,ω ) ∈ R ( αβ ) − X k − k ′ ,l − l ′ ∈ Z V g (cid:16) k − k ′ β , l − l ′ α (cid:17) e πi (( k − k ′ ) x +( l − l ′ ) ω ) We note that the phase factor is now implicitly appearing in the STFT and that the standardPoisson summation formula (and not its symplectic version) was used.
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