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Modulation spaces, BMO, and the Balian-Low theorem.

Abstract

The modulation spaces [M.sup.p,q.sub.m] ([R.sup.d]) quantify the time-frequency concentration of functions and distributions. The first main result of this paper proves embeddings of certain modulation spaces into VMO, the space of functions with vanishing mean oscillation. The second main result proves that the Zak transform maps certain modulation spaces on [R.sup.d] into modulation spaces on [R.sup.2d]. These two results allow us to give a Balian-Low-type of uncertainty principle for Gabor systems in the setting of modulation spaces.

Key words and phrases : Balian-Low Theorem, BMO, frames, Gabor systems, modulation spaces, time-frequency analysis, Poincare Inequality, Short-time Fourier transform, VMO, Zak transform

2000 AMS Mathematics Subject Classification--Primary 42C15, 42C25; Secondary 46C15

1 Introduction

Time-frequency analysis interprets the short-time Fourier transform as a measure of simultaneous time and frequency information. To achieve a quantitative analysis, Feichtinger introduced a class of Banach spaces, called the modulation spaces, which measure concentration in terms of a weighted mixed norm on the short-time Fourier transform [8], [11].

The modulation spaces are the "right" spaces for time-frequency analysis and they occur in many problems in the same way that Besov spaces are attached to wavelet theory and issues of smoothness. Many classical spaces are well-described by wavelets, but recently the modulation spaces have been successfully used to address problems not suited to traditional techniques, e.g., the analysis of pseudodifferential operators with nonsmooth symbols [4], [13], or the modeling of narrowband wireless communication channels [18], [19], [20].

For p, q [not equal to] 2, the modulation spaces [M.sup.p,q.sub.m]([R.sup.d]) do not coincide with any of the Besov spaces [B.sup.p,q.sub.s]([R.sup.d]) or the Triebel-Lizorkin spaces [F.sup.p,q.sub.s]([R.sup.d]). The question of which of these classical function spaces embed into the modulation spaces or vice versa is a natural one. The Sobolev embedding theorem [H.sup.s] ([R.sup.d])[subset or equal to][C.sup.k] ([R.sup.d]) for s > k + 1/2d is an example of such a result, as [H.sup.s]([R.sup.d]) is a modulationspace (with p = 2) and [C.sup.k] ([R.sup.d]) is a Besov space. The first systematic study of embeddings of Besov spaces into modulation spaces was by Okoudjou [16], with further results by Tort [21], [22].

One of the important results in time-frequency analysis that is stated in the setting of weighted Sobolev spaces is the Balian-Low theorem. In this paper, we investigate how modulation spaces relate to the Balian-Low theorem, and state a Balian-Low-type theorem for Gabor frames in the setting of modulation spaces. Gautam's results [10] make clear that membership in VMO (the space of functions of vanishing mean oscillation) is central to the Balian-Low Theorem. The embedding of the modulation spaces into BMO or VMO is an important issue in establishing the Balian-Low theorem in terms of modulation spaces.

The first of our main results proves embeddings of certain modulation spaces into VMO. While there are many known embeddings of classical function spaces into BMO or VMO, the question for the modulation spaces has not previously been addressed.

Another important time-frequency tool which plays a key role in the Balian Low theorem is the Zak transform, also known as Weil-Brezin map (first introduced by Gelfand [14]). For a function f [member of] [L.sup.2]([R.sup.d]), its Zak transform Zf is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the series converges unconditionally in [L.sup.2]([R.sup.d]) [11]. Zf provides a useful time-frequency representation of f. The Zak transform's importance in the Balian-Low theorem comes from the fact that the properties of the Zak transform of f can be explicitly tied to whether the associated Gabor system is a frame or not.

In [10], Gautam gives a time-frequency characterization of the Zak transform of a function having certain smoothness and decay properties, quantified in terms of membership in an inhomogeneous Sobolev space (see Lemma 2.3 in [10]). Since the time-frequency content of a function is best quantified by the modulation spaces, this motivated our investigation of the modulation space characterization of Z f. Our second main result proves that the Zak transform maps certain modulation spaces on [R.sup.d] into modulation spaces on [r.sup.2d]. This provides a generalization of Gautam's result, and in fact we obtain his embedding as a corollary. In proving this result, we appeal to Gabor analysis rather than the tools of classical analysis employed in [10]. In particular, we use Janssen's characterization of modulation spaces via the Zak transform [15], which provides a practical approach to our problem and allows us to extend the result to higher dimensions.

Finally, combined with our BMO embeddings, we see that Zak transform maps certain modulation spaces into VMO. As a consequence of these facts, we obtain a Balian-Low-type theorem for modulation spaces that is equivalent to the classical statement of the Balian-Low Theorem.

Our paper is organized as follows. In Section 2 we give some preliminaries on the modulation spaces and the Zak transform. Section 3 contains our results on embeddings of modulation spaces into BMO, and Section 4 gives a modulation space characterization of the Zak transform. Finally, Section 5 discusses the connection to the Balian-Low theorem.

2 Preliminaries: Modulation Spaces and the Zak Transform

2.1 General Notation

S([R.sup.d]) will denote the class of Schwartz functions on [R.sup.d], and S'([R.sup.d]) is its topological dual, the space of tempered distributions. We use a Fourier transform on [L.sup.1]([R.sup.d]) normalized as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This extends to a unitary operator on [L.sup.2]([R.sup.d]), and by duality to an operator mapping S'([R.sup.d]) into itself.

The Fourier coefficients of F [member of] [L.sup.1] [(0, 1).sup.d] are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To avoid dealing with too many intermediate constants, we will write A [??] B to mean that A(x) [less than or equal to] CB(x) for all x where C is a constant independent of x (and possibly other parameters).

Given x [member of] [R.sup.d], we let [x.sup.(r)] denote the [r.sup.th] component of x. Also [absolute value of x] will denote the Euclidean norm when x [member of] [R.sup.d] and absolute value when x is a real or complex number.

2.2 The Modulation Spaces

We introduce the modulation spaces and give some of their properties that are relevant to the results of this paper, referring to Grochenig's text [111 for full details.

The Short-Time Fourier Transform (STFT) of a function f with respect to a window function g is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever this is defined. If f, g lie in dual spaces (typically f [member of] S'([R.sup.d]) and g [member of] S([R.sup.d]), or f, g [member of] [L.sup.2]([R.sup.d])), then [V.sub.g]f is uniformly continuous on [R.sup.2d], and the following fundamental identity of the time-frequency analysis is satisfied:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A weight function is a positive function on [R.sup.n]. We say that a positive function v on [R.sup.2d] is submultiplicative if v([[zeta].sub.1] + [[zeta].sub.2]) [less than or equal to] v([[zeta].sub.1])v([[zeta].sub.2]) for all [[zeta].sub.1], [[zeta].sub.2] [member of] [R.sup.2d]. If v is submultiplicative, then a positive function m on [R.sup.2d] is v-moderate if there exists a constant C > 0 such that re([[zeta].sub.1] + [[zeta].sub.2]) [less than or equal to] Cv([[zeta].sub.1]) m ([[zeta].sub.2]) for all [[zeta].sub.1], [[zeta].sub.2] [member of] [R.sup.2d]. Two weights [m.sub.1], [m.sub.2] are equivalent if there exists a constant C > 0 such that [C.sup.-1] [m.sub.1]([zeta]) [less than or equal to] [m.sub.2]([zeta]) [less than or equal to] [Cm.sub.1]([zeta]) for all z [member of] [R.sup.2d]. The standard class of weights on [R.sup.2d] are weights of polynomial type [v.sub.s]([zeta]) = [(1 + [absulute value of z]).sup.s], s [member of] R, where [zeta] = (X, W) [member of] [R.sup.2d] as usual and s [greater than or equal to] 0. If s [greater than or equal to] 0 then vs is submultiplicative, while if s [less than or equal to] 0 then [v.sub.s] is [v.sub.-s]-moderate. The weights

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

are typical and are used, for example, to define the Bessel potential spaces. The Bessel potential spaces are defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We remark explicitly that vs(z) is equivalent to the weight [m.sub.s]([zeta]).

Let m be a weight function on [R.sup.2d] and fix 1 [less than or equal to] p, q < [infinity]. The weighted mixed-norm space [L.sup.p,q.sub.m]([R.sup.2d]) consists of all (Lebesgue) measurable functions on [R.sup.2d] such that the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is finite, with the usual adjustments if p = [infinity] or q = [infinity]. If m = [m.sub.s] then we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then we write [L.sup.P.sub.m] for [L.sup.p,p.sub.m]

Fix a nonzero window function g [member of] s([R.sup.d]), a v-moderate weight function m on [R.sup.2d], and 1 [less than or equal to] p, q [less than or equal to] [infinity]. Then the modulation space [M.sup.p,q.m]([R.sup.d]) consists of all tempered distributions f [member of] S'([R.sup.d]) such that [V.sub.g]f [member of] [L.sup.p,q.sub.m]([R.sup.2d]). The norm on [M.sup.p,q.sub.m]([R.sup.d]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If p = q then we write [M.sup.p.sub.m]([R.sup.d]) instead of [M.sup.p,p.sub.m]([R.sup.d]). This definition of [M.sup.p,q.sub.n]([R.sup.d]) is independent of the choice of window function g [member of] S([R.sup.d]) in the sense of equivalent norms. Further, [M.sup.p,q.sub.m]([R.sup.d]) is a Banach space, and S([R.sup.d]) is a subspace of [M.sup.p,q.sub.m]([R.sup.d]) if |m([zeta])| [??] [(1 + [absolute value of [zeta]]).sup.N] for some Y. Since |[V.sub.g]f(x,w)| = |[V.sub.[??]][??](w,-x)|, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The following proposition identifies some of the modulation spaces with wellknown function spaces, see [11, Prop. 11.3.1].

Proposition 2.1. Fix g [member of] S([R.sup.d]) \ {0}.

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) If m(x, w) = re(x), then [M.sup.2.sub.m]([R.sup.d]) = [L.sup.2.sub.m]([R.sup.d]), the weighted [L.sup.2] space.

(c) If m(x,w) = m(w), then [M.sup.2.sub.m]([R.sup.d]) = [FL.sup.2.sub.m]([R.sup.d]). In particular, by taking [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where s [member of] R, the space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] coincides with the Bessel potential (or Sobolev) space [H.sup.s]([R.sup.d]).

For a choice of appropriate weights we can identify the space of functions with certain smoothness and decay properties as a modulation space. In light of Proposition 2.1, the following fact can be easily proved.

Proposition 2.2. Fix [s.sub.1], [s.sub.2] > O.

(a) Assume [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then f [member of] [M.sup.2.sub.m] ([R.sup.d]).

(b) Assume [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If [M.sup.2.sub.m] ([R.sup.d]) then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 2.3. F/x [s.sub.1], [s.sub.2] > 0, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Discrete mixed-norm spaces [l.sup.p,q.sub.m]([Z.sup.2d]) are defined as follows. If m is a v-moderate weight, then [l.sup.p,q.sub.m] ([Z.sup.2d]) consists of all sequences [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for which the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is finite, with the usual adjustments if p or q is infinite. If p = q, then we write [l.sup.p.sub.m](Z.sup.2d)

We will need the following version of Young's Inequality for convolution in weighted sequence spaces (see [11] for proof).

Proposition 2.4. Let m be v-moderate weight. Given a = ([a.sub.k,n]) [member of] [l.sup.l.sub.v]([Z.sup.2d]) and b = ([b.sub.k,n]) [member of] [l.sup.p,q.sub.m]([Z.sup.2d]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The modulation spaces possess unconditional bases, and the sequence space associated with the basis for [M.sup.p,q.sub.m]([R.sup.d]) is the mixed-norm space [l.sup.p,q.sub.m]([R.sup.d]). As a consequence, the embeddings within the class of modulation spaces follow the embeddings within the class of the sequence spaces.

Theorem 2.5 (Theorem 12.2.2, [111). If 1 < [P.sub.1] [less than or equal to] [P.sub.2] [less than or equal to] [infinity], 1 [less than or equal to] [q.sub.1] [less than or equal to] [q.sub.2] [less than or equal to] [infinity], and m' [??] m, then [M.sup.p1,q1.sub.m] embeds continuously into [M.sup.p2,q2.sub.m'].

2.3 The Zak Transform

The Zak transform of a function f [member of] [L.sup.2]([R.sup.d]) is the function Zf on [R.sup.2d] defined (almost everywhere) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The series defining Zf converges unconditionally in [L.sup.2]([[0, 1).sup.d]), and the Zak transform is a unitary isomorphism of [L.sup.2]([R.sup.d]) onto [L.sup.2]([[0,1).sup.d]). Zf satisfies the following quasiperiodicity relations:

Z f(x + n, w) = [e.sup.2[pi]iw.n] Z f(x, co), Z f(x, w + n) = Z f(x, w).

A consequence of the quasiperiodicity relations is that if Zf is continuous on [R.sup.2d], then Zf has a zero in every unit square in [R.sup.2d].

The Zak transform maps a function in ,S([R.sup.d]) to an infinitely differentiable function on [R.sup.2d]. A useful property of the Zak Transform is the following equality which describes the Fourier coefficients of Zf x [bar.Zg] as samples of the STFT of f with respect to the window g: Given f, g [member of] [L.sup.2]([R.sup.d]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

This fact allows a characterization of the modulation spaces via the Zak transform. The following result is due to Janssen [15], and we also mention that a different characterization of the modulation spaces was obtained earlier by Walnut [23].

Theorem 2.6. Let m be a v-moderate weight. Let 1 [less than or equal to] p, q [less than or equal to] [infinity] and let [g.sub.r] [member of] [M.sup.1.sub.v]([R.sup.d]) for r = 1, ... , N be such that the functions [Zg.sub.r] have no common zeros. If f [member of] [L.sup.2]([R.sup.d]), then f [member of] [M.sup.p,q.sub.m]([R.sup.d]) if and only if each Zf x [bar.Zgr], has a Fourier series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is necessary to take N [greater than or equal to] 2 in Theorem 2.6 because the functions [Zg.sub.r] will be continuous and therefore will have zeros.

In the sequel, we will use the Zak transform of functions defined on [R.sup.d] and the Zak transform of functions defined on [R.sup.2d]. The intended domain will be clear from context.

3 Embedding of Modulation Spaces into VMO

We recall that BMO([R.sup.d]) is the space of functions (modulo constants) that have bounded mean oscillation on [R.sup.d] :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the supremum is taken over all cubes Q in [R.sup.d], and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

denotes the average of a function over a Lebesgue measurable set E. The space VMO([R.sup.d]) of functions having vanishing mean oscillation on [R.sup.d] consists of those functions f [member of] BMO([R.sup.d]) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

Equivalently, VMO([R.sup.d]) is the closure of the uniformly continuous functions in BMO-norm. We will work mostly with a local version of VMO; given a compact set K we define VMO(K) to be the set of functions satisfying the limit condition in equation (3.1) for cubes Q [subset] K. For details on VMO, we refer to [17].

Now we will obtain an embedding of a certain class of functions defined on [R.sup.2] into BMO([R.sup.2]). Although this embedding (Theorem 3.3) follows from the Sobolev embedding theorem in [5] for the case n : p = 2, for completeness we give a direct proof via Poincare's inequality [7] for functions defined on [R.sup.2].

The Sobolev space [W.sup.1,p](U) consists of locally integrable functions f: U [rigt arrow] R such that each partial derivative of f exists in the weak sense and belongs to [L.sup.P](U).

Theorem 3.1 (Poincare's Inequality). If I [less than or equal to] p [less than or equal to] [infinity] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each ball B(x,r) [subset or equal to] [R.sup.d] and each function u [member of] [W.sup.1,P](B(x,r)).

Corollary 3.2. For all u [member of] [W.sup.1,2]([R.sup.2]) [intersection] [L.sup.1]([R.sup.2]) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. If we take p = 1 and d = 2 in Poincare's Inequality and apply HSlder's Inequality, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we can prove our first main theorem, on the embedding of weighted modulation spaces into VMO([R.sup.2]).

Theorem 3.3. Fix 1 [less than or equal to] p [less than or equal to] 2. Assume that:

(a) v is a submultiplicative weight on [R.sup.4] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some

[N.sub.0] [member of] N,

(b)[??] is a v-moderate weight on [R.sup.4] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then [M.sup.P.sub.[??]]([R.sup.2]) [subset or equal to] VMO([R.sup.2]) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. The Schwartz class is dense in [M.sup.p.sub.[??]]([R.sup.2]), so consider a fixed f [member of] $([R.sup.2]). Setting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and applying Corollary 3.2 and the Plancherel Equality, we compute that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we write re(x,[zeta]) = m(x) then by Proposition 2.1(b) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now let m'(x, w) = [??](-w, x). By assumption we have m [??] m', which by Theorem 2.5 implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

An extension by density argument establishes that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all f [member of] [M.sup.p,[??]]([R.sup.2]). Moreover, since VMO is the closure of the uniformly continuous functions in BMO-norm, this also implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4 Mapping Properties of the Zak Transform on Modulation Spaces

Our second main theorem describes the mapping properties of the Zak transform acting on the modulation spaces.

Theorem 4.1. Fix 1 < p < oc. Assume that:

(a) v is a submultiplicative weight on [R.sup.2d] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some [N.sub.0] [member of] N,

(b) m is a v-moderate weight on [R.sup.2d],

(e) [PSI] [member of] [C.sup.[infinity].sub.c]([R.sup.2d]).

Let m* be any weight function satisfying m*(N, M) [less than or equal to] [C.sub.N] re(M) for all M, N [member of] [Z.sup.2d]. Then for every f [member of] [L.sup.2]([R.sup.d]) [intersection] [M.sup.P.sub.m]([R.sup.d]) we have [PSI] Zf [member of] [M.sup.P.sub.m] x ([R.sup.2d]). Proof. First we note that [PSI]Z f [member of] [L.sup.2] ([R.sup.2d]).

Let g [member of] [C.sup.[infinity].sub.c]([R.sup.d]) be supported on [[0, 1].sup.d] and nonzero on the interior of this cube. Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have [g.sub.j] [member of] [M.sup.1.sub.v] ([R.sup.d]) for each j. Taking the support of [g.sub.j] into consideration, if k [member of] [Z.sup.d] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The zero set of [Zg.sub.j] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [partial derivative]A denotes the boundary of a set A.

Claim 1. The functions [Zg.sub.j] have no common zeros.

To see this, assume that [Zg.sub.j]([x.sub.0], [[omega].sub.0]) = 0 for each j = 1,..., 2d+ 1. Then for each j there exists a [k.sub.j] [member of] [Z.sup.d] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [[theta].sub.j] [partial derivative][[0, 1).sup.d] [intersection] [[0, 1).sup.d] be such that [x.sub.0] [k.sub.j] - (j - 1/2d + 1) [xi] + [[theta].sub.j]. For j [not equal to] j' we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so [k.sub.j] - [k.sub.j,] = (j -j'/2d + 1)[xi] + [[theta].sub.j], - [[theta].sub.j]. Each of [[theta].sub.j] and [[theta].sub.j], must have a zero component. If they both have a zero in the mth coordinate, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This is a contradiction since [k.sub.j.sup.(m)] - [k.sub.j'.sup.(m)] is integer. Hence no [[theta].sub.j] and [[theta].sub.j'] can have a zero in the same component. However, this is impossible as there are 2d + 1 vectors [[theta].sub.j] each with d components. This proves the claim.

Consequently, Theorem 2.6 implies that the Fourier coefficients of Zf [bar.[Zg.sub.j]] belong to [l.sup.p.sub.m]([Z.sup.2d]) for each j = 1,..., 2d + 1.

Now let [phi] [member of] [C.sup.[infinity].sub.c]([R.sup.2d]) be supported in the unit square [[0, 1].sup.2d] and strictly positive on its interior. For j = 1,..., 2d + 1 define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since each [g.sub.j] is Schwartz-class, we have [Zg.sub.j] [member of] [C.sup.[infinity]]([R.sup.2d]). Therefore, for any [N.sub.1] [member of] N and any weight [v.sub.2] satisfying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For the remainder of this proof, let [v.sub.2] be any such weight.

Claim 2. The zero set of [Zg.sub.j] is contained in the zero set of [[phi].sub.j].

This follows from the fact that the zero set of [Zg.sub.j] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

while the zero set of [[phi].sub.j] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [[phi].sub.j] is compactly supported, for (x, w) [member of] (m, n) - (j - 1/2d + 1) ([xi] [xi]) + [[0, 1).sup.2d] we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The zero set of [Zg.sub.j] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Claim 3. The functions [Zg.sub.j] have no common zeros.

To see this, suppose (x, w, p, s) [member of] [R.sup.2d] x [R.sup.2d] is a common zero point for the functions [Zg.sub.j]. Then for each j = 1,..., 2d + 1 there exists ([m.sub.j], [n.sub.j]) [member of] [Z.sup.2d] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that (x, w) = ([m.sub.j], [n.sub.j]) - (j-1/2d + 1) ([xi] [xi]) + [[theta].sub.j]. Each [[theta].sub.j] must have a component that is zero.

If j [not equal to] j' and the sth component of [[theta].sub.j] and [[theta].sub.j'], is both zero then, as in the proof of Claim 1, we obtain that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies j-j'/2d + 1 is an integer, which is a contradiction. Hence no [[theta].sub.j] and [[theta].sub.j], can have a zero in the same component, which is impossible since the number of j's is 2d + 1 while the dimension of [[theta].sub.j] is 2d.

Claim 4. There exists a [v.sub.2]-moderate weight m * : [R.sup.2d] x [R.sup.2d] [right arrow] (0, [infinity]) such that the sequence [c.sub.N,-M]/[ZG.sub.j] belongs to [l.sup.p.sub.m]([Z.sup.2d]), where [C.sub.M,N] are the Fourier coefficients of Z([psi]Zf) * [Zg.sub.j]

To see this, fix j and let K [member of] [Z.sup.2d] and [alpha] [member of] N be such that supp([psi]) [subset or equal to] K + [[0, [alpha]).sup.2d]. Write M = ([m.sub.l], [m.sub.2]), N = ([n.sub.l], [n.sub.2]) [member of] [Z.sup.2d] and X = ([x.sub.l], [x.sub.2]), Y = ([y.sub.l], [y.sub.2]) [member of] [R.sup.2d], etc. Recalling that [G.sub.j] = [[phi].sub.j] [Zg.sub.j], we use equation (2.1) to compute that the Fourier coefficient [c.sub.N,-M] of Z([psi]Zf) * [ZG.sub.j] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we have applied the quasiperiodicity of the Zak transform and the fact that M, N, K belong to [Z.sup.2d].

Note that there are only a finite number of M such that [[phi].sub.j]([alpha]X - M + K) is not identically zero on [[0, 1).sup.2d], and hence a finite number of M such that [c.sub.N,-M] is nonzero. Let B be the set of those M [member of] [Z.sup.2d] such that [c.sub.N,-M] [not equal to] 0. Let us fix an M [member of] B and define the following functions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We note some facts about these functions.

i. [[PHI].sup.(M)] = [U.sup.(M)]W.

ii. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

iii. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

iv. [U.sup.(M)] is a compactly supported infinitely differentiable function whose support lies inside the unit cube [[0, 1].sup.2d]. Consequently, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any polynomial weight function v defined on [R.sup.2d].

v. By Theorem 2.6, the hypothesis f [member of] [M.sup.p.sub.m]([R.sup.2d]) implies that the Fourier coefficients of Zf [bar.[Zg.sub.j]] belong to [l.sup.p.sub.m]([Z.sup.2d]), and therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Fix now any weight function m * : [R.sup.2d] X [R.sup.2d] [right arrow] R such that:

* there exists a C > 0 such that m * (M,N) [less than or equal to] Cm(N) for all N [member of] [Z.sup.2d] and M [member of] B, and

* m * is [v.sub.2]-moderate with respect to some submultiplicative function [v.sub.2].

For example, we can simply take m *(M, N) = m(N).

Define [m.sub.M](N) = m *(M, N). Using Proposition 2.4, we compute that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From above, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

since B is a finite set.

As an application of this theorem, we recover the following result, which is [10, Lemma 2.3].

Corollary 4.2. Assume [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [s.sub.l], [s.sub.2] > 0. Then for any smooth, compactly supported function [phi] [member of] [C.sup.[infinity].sub.c] ([R.sup.2d]) we have [psi]Zf [member of] [FL.sup.2.sub.m]([R.sup.2d]) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. Note that re(x, y) satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence f [member of] [M.sup.2.sub.m]([R.sup.d]) by Proposition 2.2. Now let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then m is a v-moderate weight. By Theorem 4.1, [psi]Zf [member of] [M.sup.2.sub.m*]. ([R.sup.2d]). Taking

m *(x, y, u, v) = m(u, v),

by Proposition 2.1(b) we obtain [psi]Zf [member of] [FL.sup.2.sub.m] ([R.sup.2d]).

Another implication of Theorem 4.1 is that the Zak transform embeds certain modulation spaces into a local VMO space.

Corollary 4.3. Let v be a submultiplicative weight satisfying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some [N.sub.0] [member of] N, and let m be a v-moderate weight such that re(x) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where 1 [less than or equal to] p [less than or equal to] 2, then Zf [member of] VMO(K) for any compact set K [subset] [R.sup.2].

5 Application to the Balian-Low Theorem

We will briefly indicate how the Classical BLT can be recovered from our results. Given g [member of] [L.sup.2](R) and constants [alpha], [beta] > 0, the associated Gabor system is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We are interested in whether 6(f, [alpha], [beta]) is a frame for [L.sup.2](R), which means that there exist A, B > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Frames are a generalization of orthonormal bases, and it is natural to consider under what conditions 6(g, [alpha], [beta]) will generates a frame. The classical Balian--Low Theorem addresses this question for the case [alpha][beta] = 1 (the "critical density"), which by a change of variables can be further reduced to the case [alpha] = [beta] = 1. Other Balian-Low-type theorems can be found in [1], [2], [3], [9], [10], [12].

Theorem 5.1 (Classical BLT). Given g [member of] [L.sup.2](R), if g [member of] [H.sup.1](R) and [??][member of] [H.sup.1](R) then G(g, 1, 1) is not a frame for [L.sup.2](R).

The approach to the proof is based on the Zak transform, which completely determines whether the Gabor system associated with a function g is a frame at the critical density.

Proposition 5.2. Fix g [member of] [L.sup.2](R). Then G(g, 1, 1) is a frame for [L.sup.2](R) if and only if O < [A.sup.1/2] [less than or equal to] |Z f| [less than or equal to] [B.sup.1/2] < infinity a.e.

If a function belongs to the modulation space [M.sup.1](R), which is a subspace of [L.sup.2](R), then the associated Gabor system cannot be a frame for [L.sup.2](R). The reason for this is given in the following proposition, whose proof can be found in [11].

Proposition 5.3. (a) If f [member of] [M.sup.1](R) then Zf is continuous. (b) If f [member of] [L.sup.2](R) and Z f is continuous then Zf must have a zero.

Now we prove a BLT-type result. We word this theorem in terms of modulation spaces, but the strongest conclusion is obtained by taking p = 2 and m(x, w) = [(1 + [|x|.sup.2] + [|w|.sup.2]).sup.1/2], which recovers the Classical BLT in its usual form.

Theorem 5.4. Let v be a submultiplicative weight with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some [N.sub.0] [member of] N, and let m be a v-moderate weight such that m(x,w) [??] [(1 + [|x|.sup.2] + [|w|.sup.2]).sup.1/2] where (x, w) [member of] [R.sup.2.] If f [member of] [L.sup.2](R) [intersection] [M.sup.P.sub.m] (R) where 1 [less than or equal to] p [less than or equal to] 2, then G(f, 1, 1) is not a frame for [L.sup.2](R).

Proof. If G(f, 1, 1) is a frame then Zf [member of] [L.sup.[infinity]] ([R.sup.2]). Corollary 4.3 therefore implies that Zf [member of] VMO(K) for any compact set K. Since Zf is quasiperiodic, it follows from [10, Lemma 4.1] that ess inf |Z f| = O, which is a contradiction. []

ACKNOWLEDGEMENT

The second author was partially supported by NSF Grant DMS-0806532.

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Ramazan Tinaztepe

Department of Mathematics Education, Zirve University,

27260, Gaziantep, Turkey

ramazan.tinaztepe@zirve.edu.tr

Christopher Hell

School of Mathematics, Georgia Institute of Technology,

Atlanta, GA 30332-0160 USA

heil@math.gatech.edu
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Date:Jan 1, 2012
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