Linear independence of a finite set of time-frequency shifts

Abstract

This paper introduces an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L² function. Firstly, background and motivation for the conjecture are provided. Secondly, the main progress of this linear independence in the past twenty five years is reviewed. Finally, the partial results of the high dimensional case and other cases for the conjecture are briefly presented.

Keywords

Time-frequency shift / Gabor frame / wavelet / linear independence

1 Introduction

In 1996, the following conjecture was raised by Heil et al. [18].

If \mathrm{\Lambda }={\left\{(a_k,b_k)\right\}}_{k=1}^N\subset {\mathbb{R}}^2, g\in L^2(\mathbb{R}) and g\ne 0, then

\mathrm{\Re }(g,\mathrm{\Lambda })={\left\{{\mathrm{e}}^{\mathrm{i}2\pi b_{k}x}g(x-a_k)\right\}}_{k=1}^N

is linearly independent in L^2(\mathbb{R}).

This conjecture is called the linearly independent conjecture with finite time-frequency shifts, and also called the HRT conjecture. 15 years ago, this conjecture was well introduced in [17]. The HRT conjecture seems simple, but it is not. Even if g\in S(\mathbb{R}), the HRT conjecture is also an unsolved problem, where S(\mathbb{R}) denotes the Schwartz function class. Since the HRT conjecture was proposed, it has aroused the close attention of applied and computational harmonic analysis scholars and researchers in other related fields. It had been confirmed that the conjecture is equivalent to that the finite shifts of the square integrable function on the Heisenberg group is linearly independent (see [11]), so it is difficult to solve the conjecture. When some “reasonable” conditions are imposed on the generating function g or time-frequency parameters (a_k,b_k), the conjecture was proved positively. These proofs use C^{\ast } algebra, von Neumann algebra, non-harmonic series, functional analysis and other tools. This fact shows that the conjecture is related to ergodic theory, compact Abel group, finite group, number theory, Heisenberg group, Bargmann Fock space, Segal Bargmann transform, Schrödinger representation etc.

In the past 25 years, much progress has been made in research of the HRT conjecture, but there are still many problems to be explored. This paper will introduce the main research progress in this field, so as to draw the attention of domestic scholars and peers in this field.

This survey is organized as follows: Section 2 introduces the backgrounds of the HRT conjecture, Section 3 summarizes the results of the affirmative conjecture so far, and the last section briefly describes the related results of the high-dimensional case and other cases of the conjecture.

2 Why was the HRT conjecture proposed?

The HRT conjecture follows the following two motivations.

(1) In time-frequency analysis, the application of the Gabor frame operator requires linear independence of finite time-frequency shifts.

Suppose g\in L^2(\mathbb{R}), a,b\in \mathbb{R}. (g,a,b)={\left\{{\mathrm{e}}^{\mathrm{i}2\pi mbx}g(x-na)\right\}}_{m,n\in \mathbb{Z}} is called a Gabor system for L^2(\mathbb{R}), where \mathbb{Z} denotes integer set. Obviously, the Gabor system depends on the shift operator

T_a:L^2(\mathbb{R})\to L^2(\mathbb{R}),(T_{a}f)(x)=f(x-a)(a\in \mathbb{R})

and the modulation operator

E_b:L^2(\mathbb{R})\to L^2(\mathbb{R}),(E_{b}f)(x)={\mathrm{e}}^{\mathrm{i}2\pi bx}f(x)(b\in \mathbb{R}),

that is, (g,a,b)=\{E_{mb}{T}_{na}g(x){\}}_{m,n\in \mathbb{Z}}.

Definition 2.1　If there are constants A>0 and , such that

(2.1)\mathrm{\forall }f\in L^2(\mathbb{R}),A\Vert f|{|}^2\le \sum _{m,n\in \mathbb{Z}}|⟨f,E_{mb}{T}_{na}g⟩{|}^2\le B\Vert f{\Vert }^2,

then (g,a,b) is called a Gabor frame with bounds A and B for L^2(\mathbb{R}), and also called a Weyl-Heisenberg frame. Correspondingly, the function g is called the window function or the generating function. If A=B in (2.1), then the frame is called a tight Gabor frame.

Using the Gabor frame may decompose every function in L^2(\mathbb{R}). Assume g\in L^2(\mathbb{R}), a,b>0. If (g,a,b) forms a frame, then \mathrm{\forall }f\in L^2(\mathbb{R}), the following decomposition formula holds:

(2.2)f(x)=\sum _{m,n\in \mathbb{Z}}⟨f,S^{-1}{E}_{nb}{T}_{ma}g⟩E_{nb}{T}_{ma}g(x),

where S is the frame operator of the frame (g,a,b):

Sf=\sum _{m,n\in \mathbb{Z}}⟨f,E_{nb}{T}_{ma}g⟩E_{nb}{T}_{ma}g.

In particular, if (g,a,b) is a tight frame with bound A for L^2(\mathbb{R}), then the formula (2.2) becomes

f(x)=\frac{1}{A}\sum _{m,n\in \mathbb{Z}}⟨f,E_{nb}{T}_{ma}g⟩E_{nb}{T}_{ma}g(x).

The above formula is similar to the decomposition of functions by the orthonormal basis, but (g,a,b) is a tight frame which is weaker than the orthonormal basis: the tight frame can be linearly dependent. This difference makes the frames more flexible and widely used in specific applications. About the frames and the Gabor frames, one refers [8, 24].

When applying the Gabor frame, the sum in the definition of its frame operator S must be truncated to a finite sum

(2.3)P_{N,M}f=\sum _{n=-N}^N\sum _{m=-M}^M⟨f,E_{mb}{T}_{na}g⟩E_{mb}{T}_{na}g,N,M\ge 1.

So, P_{N,M} is required to be invertible so that the approximation function sequence of f can be obtained from Eq. (2.3). If \{E_{mb}{T}_{na}g{\}}_{|n|\le N,|m|\le M} is linearly independent, then the operator P_{N,M} is naturally invertible. So it is conjectured that the finite time-frequency shift sequence is linearly independent.

(2) A second motivation comes from the comparison with linear dependent properties for the time-scale shift sequence.

In wavelet analysis, the wavelet system depends on the shift operator T_a which was previously defined, and the scale operator:

D_a:L^2(\mathbb{R})\to L^2(\mathbb{R}),(D_{a}f)(x)=|a{|}^{\frac{1}{2}}f(ax)(a\ne 0).

That is, for a given \psi \in L^2(\mathbb{R}), using T_a and D_a can obtain the wavelet system (or called the affine system):

\{{\psi }_{j,k}(x)|j,k\in \mathbb{Z}\}=\{2^{\frac{j}{2}}\psi (2^{j}x-k)|j,k\in \mathbb{Z}\}=\{D_{2^j}{T}_k\psi (x)|j,k\in \mathbb{Z}\}.

Correspondingly, \psi is called the generating function, here we choose a=2 for convenience.

Definition 2.2　If \left\{{\psi }_{j,k}(x)|j,k\in \mathbb{Z}\right\} is an orthonormal basis for L^2(\mathbb{R}), then \left\{{\psi }_{j,k}(x)|j,k\in \mathbb{Z}\right\} is called an orthonormal wavelet basis for L^2(\mathbb{R}), correspondingly, \psi is called a mother wavelet, short for a wavelet.

The research on the existence, construction and properties of the wavelet constitutes the main content of wavelet analysis [12, 24]. In particular, the construction of compactly supported wavelets becomes very important due to application needs. And one of the key steps to construct compactly supported wavelets is to solve the following function refinement equation in L^2(\mathbb{R}):

\varphi (x)=\sum _{n=-N}^M{c}_n{\varphi }_{1,n}(x),

where the above equation solutions \varphi is called the refinement function. Based on this, a multiresolution analysis (affine multiresolution analysis) for L^2(\mathbb{R}) can be obtained, thus the wavelets can be constructed. See [12, 24].

Obviously, the above refined equation can be written as

D_1{T}_0\varphi (x)=\sum _{n=-N}^M{c}_n{D}_2{T}_n\varphi (x).

In other words, the finite time-scale shift system \mathrm{\Im }(\varphi ,\mathrm{\Lambda })=\{T_0{D}_1\varphi (x)\}\cup \{T_n{D}_2\varphi (x){\}}_{n=-N}^M is linearly dependent in L^2(\mathbb{R}), where \mathrm{\Lambda }=\{(0,1)\}\cup \{(n,2){\}}_{n=-N}^M.

It is natural to ask that does time-frequency translates have a “refinement equation” similar to time-scale translates? The answer is no [18]. Therefore there is not any “Heisenberg multiresolution” in L^2. So it is reasonable to guess that the finite time-frequency shift sequence of a non-zero function in L^2 is linearly independent.

3 The positive results of the conjecture

It is clear to see that the HRT conjecture depends on two basic elements: the point set \mathrm{\Lambda } and the function g. Most of the existing positive results are the point set \mathrm{\Lambda } has no restriction and the function g is restricted to some function classes; or g\in L^2 and \mathrm{\Lambda } is restricted to some special point sets.

Firstly, for N\le 3, the following result was proved.

Theorem 3.1 [18]　 If N\le 3, then the HRT conjecture holds.

If the restriction conditions are imposed on \mathrm{\Lambda }, then the following conclusions were stated.

Theorem 3.2　 For any g\in L^2(\mathbb{R}) and g\ne 0, the HRT conjecture holds in each of the following cases:

(1) The points in \mathrm{\Lambda } lie on a line or N-1 points in \mathrm{\Lambda } are collinear and equi-spaced, with the last point located off this line (see [18]).

(2) \mathrm{\Lambda }\subset A({\mathbb{Z}}^2)+z, where A is a full rank matrix and z\in {\mathbb{R}}^2 (see [25]).

(3) N=4, where two of the four points in \mathrm{\Lambda } lie on a line and the remaining two points lie on a second parallel line (see [13, 15]).

Remark 3.1　(1) It is easy to see that (1) in Theorem 3.2 implies Theorem 3.1, and (2) in Theorem 3.2 shows that the HRT conjecture holds when \mathrm{\Lambda } is a finite subset of the lattice point set a\mathbb{Z}\times b\mathbb{Z}.

(2) \mathrm{\Lambda } in (3) of Theorem 3.2 is called a (2,2) configuration (see [28]).

(3) The (2,2) configuration case was proved in [13] and [15], respectively. Although the proof ideas of the two papers are both “conjugate skill”, they are different in essence, and the latter is a further improvement of the former.

If the restriction conditions are imposed on the generating function g, then:

Theorem 3.3　 For all \mathrm{\Lambda }={\left\{(a_k,b_k)\right\}}_{k=1}^N\subset {\mathbb{R}}^2, the HRT conjecture holds in each of the following cases.

(1) g has compactly supported or just supported within [a,\mathrm{\infty }) or (-\mathrm{\infty },a] (see [18]).

(2) g(x)=p(x){\mathrm{e}}^{-\pi |x{|}^2}, where p(x) is a polynomial (see [18]).

(3) g satisfies \underset{x\to \mathrm{\infty }}{lim}|g(x)|{\mathrm{e}}^{c{x}^2}=0 for all c>0 (see [6]).

(4) g satisfies \underset{x\to \mathrm{\infty }}{lim}|g(x)|{\mathrm{e}}^{cx\log x}=0 for all c>0 (see [6]).

Remark 3.2　It is easy to see that (4) in Theorem 3.3 implies (1)−(3). Here listing these results only reflects the historical process of getting them.

The obtaining result in [6] shows that the HRT conjecture holds when g satisfies one-sided decay. Naturally ask:

Question 3.1　When g satisfies exponential decay, that is, \underset{x\to \mathrm{\infty }}{lim}|g(x)|{\mathrm{e}}^{cx}=0 for all c>0, does the HRT conjecture hold?

For every g with exponential decay, if the restriction condition is imposed on \mathrm{\Lambda }, or the weaker monotone condition is imposed on g, then the answer to question 3.1 is yes. In fact, there are the following results.

Theorem 3.4 [6]　 For every g with exponential decay, the HRT conjecture holds in each of the following cases.

(1) \mathrm{\Lambda }\subset \mathbb{R}\times b\mathbb{Z}, b>0.

(2) g(x) is quasi-monotone, i.e.,

The results in Theorem 3.4 support the positive answer of Question 3.1, but as far as we know, the problem is still unsolved. In addition, there are also the following theorems.

Theorem 3.5 [6]　 If g(x)=p(x){\mathrm{e}}^{-|x{|}^{1+ϵ}}, where p(x) is a polynomial and ϵ>0, then the HRT conjecture holds.

If the restriction conditions are imposed on \mathrm{\Lambda } and g, then the following results can be obtained.

Theorem 3.6　 The HRT conjecture holds in each of the following cases.

(1) g\in S(\mathbb{R}) and N=4, where three points in \mathrm{\Lambda } are on a line, and the other point is not on the line (see [13]). Such \mathrm{\Lambda } is called a (1,3) configuration.

(2) g\in S(\mathbb{R}) and g is a real-valued function, and N=4 (see [28]).

(3) g\in L^2(\mathbb{R}) and g is ultimately positive, and \{b_k{\}}_{k=1}^N are linearly independent over the rational number field \mathbb{Q} (see [4]).

(4) g\in L^2(\mathbb{R}) and g is ultimately positive, g(x) and g(-x) are ultimately decreasing, and N=4 (see [4]).

(5) g\in L^2(\mathbb{R}) and g is a real-valued function, and \mathrm{\Lambda } is a (1,3) configuration (see [28]).

Remark 3.3　Two different methods of “extension principle” and “restriction principle” were given to discuss the HRT conjecture in [28]. The positive results of the HRT conjecture for some special (3,2) configuration cases were established by using extension principle in [28], and at the same time, the case of N=4 in detail was analyzed. For N=4, the author of [28] believes that the following questions need to be investigated firstly:

Question 3.2　If 0\ne g\in L^2, \mathrm{\Lambda }=\{(0,0),(0,1),(1,0),(\sqrt{2},\sqrt{2})\}, then the HRT conjecture holds, i.e.,

\left\{g(x),g(x-1),{\mathrm{e}}^{\mathrm{i}2\pi x}g(x),{\mathrm{e}}^{\mathrm{i}2\pi \sqrt{2}x}g(x-\sqrt{2})\right\}\mathrm{i}\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}.

Question 3.3　If 0\ne g\in L^2, \mathrm{\Lambda }=\{(0,0),(0,1),(1,0),(\sqrt{2},\sqrt{3})\}, then the HRT conjecture holds, i.e.,

\left\{g(x),g(x-1),{\mathrm{e}}^{\mathrm{i}2\pi x}g(x),{\mathrm{e}}^{\mathrm{i}2\pi \sqrt{3}x}g(x-\sqrt{2})\right\}\mathrm{i}\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}.

Remark 3.4　(1) Question 3.2 has been solved when g is real-valued (see [28]), and Question 3.3 has been solved when g\in S(\mathbb{R}) (see [28]).

(2) For 0\ne g\in L^2(\mathbb{R}) and the (1,3) configuration \mathrm{\Lambda } with some conditions, it was also proved in [13] that the HRT conjecture holds. But the proof method in [13] is completely different from that of (1) in Theorem 3.2 which is Fourier method (see [18]). Indeed, Fourier method won't work for the general (1,3) configuration \mathrm{\Lambda }.

(3) For 0\ne g\in L^2(\mathbb{R}) and almost every (in Lebesgue measure sense) (1,3) configuration \mathrm{\Lambda }, it was verified in [26] that the HRT conjecture holds.

4 The high dimensional case and other cases

When \mathrm{\Lambda }={\left\{(a_k,b_k)\right\}}_{k=1}^N\subset {\mathbb{R}}^{2d} and g\in L^2({\mathbb{R}}^d) with g\ne 0, the HRT conjecture in this case was definitely proved by using the von Neumann algebra tool in [25]. However it is required that \mathrm{\Lambda }\subset A({\mathbb{Z}}^{2d}), where A is a full rank matrix. Later, an alternative proof for the result in [25] was presented in [29] from an abstract point of view. The HRT conjecture in the high dimensional case was successively discussed in [7, 14, 21, 27]. In these literature, the results of [4] were generalized to the high dimensional case in [7]; in [14], a new proof for the high dimensional case of the results of [25] was provided by the spectral theory based on the random Schrödinger operator; the relation between the uncertainty principle of short-time Fourier transform and the HRT conjecture was discussed in [21, 27]. At the same time, in [21] it was also proved that the HRT conjecture holds when the time coordinates of points in \mathrm{\Lambda }={\left\{(a_k,b_k)\right\}}_{k=1}^N\subset {\mathbb{R}}^{2d} are far apart relative to the decay of g, i.e.,

Theorem 4.1 [21]　 Suppose that g\in C_0({\mathbb{R}}^d), \mathrm{\Lambda }={\left\{(a_k,b_k)\right\}}_{k=1}^N\subset {\mathbb{R}}^{2d}, and there exists R such that . If |a_k-a_l|>R(k\ne l), then the HRT conjecture holds.

Remark 4.1 In \mathbb{R}, the function with the condition of Theorem 4.1 may not satisfy the decay property of the function in the previous Theorem 3.3, for example,

When , the HRT conjecture is completely natural.

Give \mathrm{\Lambda }={\left\{(a_k,b_k)\right\}}_{k=1}^N\subset {\mathbb{R}}^2 and with g\ne 0. If the constants c_k are not all zero such that

\sum _k{c}_k{\mathrm{e}}^{\mathrm{i}2\pi b_{k}x}g(x-a_k)=0,

then g=0.

The case of L^p was firstly discussed and some results were obtained in [1]. Here the linear independence of the finite translates of a function in L^p depends on the values of d and p (see [7, 30]). In addition, the linear independence of finite translates of a function in L^2 was considered in [31], and the l^p linear independence of the function translates in L^2 was studied in [32, 33]. Although the method of proving the results in [25] can work for the finite dimensional case, it doesn’t work for the case of L^p. In [5], the different proofs were given for the one dimensional case of the results of [25] by using the translation invariant subspace theory, but this proof method doesn’t work for the high dimensional case and the case of L^p.

The abstract description of the HRT conjecture under Schrödinger representation was explained in [22], that is, the linear independence of time-frequency shifts on locally compact Abel group. The linear independence of the Gabor system and time-frequency shifts on finite group were discussed in [23], while the connection between the HRT conjecture and the zero factor conjecture on Heisenberg group was showed in [19].

It was shown that the HRT conjecture is not true on l^2(\mathbb{Z}) in [20], and this fact was expounded in detail in [14]. The paper [9] focused on the linear dependence and linear independence of the Gabor system on l^2(\mathbb{Z}). The spectral properties of the algebra of time-frequency shift operators related to the HRT conjecture were given in [2,3]. In [10,16], the estimation of the frame bound of the Gabor system related to the HRT conjecture was obtained. In [34], a connection between the HRT conjecture, Bargmann Fock space and Segal Bargmann transform was investigated. As for the relation between the linear independence of the wavelet system and the linear independence of the Gabor system, it was established by [5].

Finally, the results of time-frequency translation stability are listed as follows.

Theorem 4.2 [18]　 For any \mathrm{\Lambda }={\left\{(a_k,b_k)\right\}}_{k=1}^N\subset {\mathbb{R}}^2 and g\in L^2, if \mathrm{\Re }(g,\mathrm{\Lambda }) is linearly independent, then there exists ϵ>0 such that when \tilde{g}\in L^2 and , \mathrm{\Re }(\tilde{g},\mathrm{\Lambda }) is linearly independent.

Theorem 4.2 shows that for given \mathrm{\Lambda }={\left\{(a_k,b_k)\right\}}_{k=1}^N\subset {\mathbb{R}}^2, the following set is an open set in L^2:

\mathrm{\Omega }=\left\{g|\mathrm{\Re }(g,\mathrm{\Lambda })\mathrm{i}\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\right\}.

Similarly, the following is true.

Theorem 4.3 [18]　 For any \mathrm{\Lambda }={\left\{(a_k,b_k)\right\}}_{k=1}^N\subset {\mathbb{R}}^2 and g\in L^2, if \mathrm{\Re }(g,\mathrm{\Lambda }) is linearly independent, then there exists ϵ>0 such that when

\mathrm{\Re }(g,\tilde{\mathrm{\Lambda }}) is linearly independent, where \tilde{\mathrm{\Lambda }}=\{({\tilde{a}}_k,{\tilde{b}}_k){\}}_{k=1}^N.

Remark 4.2 [18]　The results in [18] can be generalized to the high dimensional case and the L^p case.

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