The frame sets of the B-splines

Abstract

It is not completely clear which elements constitute the frame sets of the B-splines currently, but some considerable results have been obtained. In this paper, firstly, the background of frame set is introduced. Secondly, the main progress of the frame sets of the B-splines in the past more than twenty years are reviewed, and particularly the progress for the frame set of the 2 order B-spline and the frame set of the 3 order B-spline are explained, respectively.

Keywords

B-spline / Gabor frame / frame set

1 Introduction

Suppose that g\in L^2(\mathbb{R}) and (a,b)\in {\mathbb{R}}_{+}^2, where {\mathbb{R}}_{+}^2=\{(x,y)|x>0,y>0\}. Let

(g,a,b)={\left\{{\mathrm{e}}^{\mathrm{i}2\pi mbx}g(x-na)\right\}}_{m,n\in \mathbb{Z}}=\{E_{mb}{T}_{na}g(x){\}}_{m,n\in \mathbb{Z}}.

Then (g,a,b) is called a Gabor system for L^2(\mathbb{R}), where \mathbb{Z} is the integer set, T_a denotes 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 E_b is 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}.

Defintion 1.1　If there exists constants A>0 and such that

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

then (g,a,b) is called a Gabor frame for L^2(\mathbb{R}) with bounds A and B, and (g,a,b) is also called a Weyl-Heisenberg frame, correspondingly, where g is called the window function or the generating function. If only the right inequality holds in (1.1), then (g,a,b) is called a Bessel sequence for L^2(\mathbb{R}); if A=B in (1.1), then the frame is called a tight Gabor frame for L^2(\mathbb{R}).

The existence, construction, and properties of the generating function g and time-frequency parameters (a,b) in Gabor frame (g,a,b) constitute the part of main contents of Gabor analysis [9, 20-22, 26, 37, 41]. The research on Gabor analysis predates wavelet analysis historically, but the research on Gabor frame has gained great attention with the rapid development of wavelet analysis [15, 16, 20-22], this is mainly due to the widespread applications of Gabor frame in fields such as image processing, digital communication and others [20-22, 30, 45, 46]. As a tool, Gabor frame can be used to study quasi-differential operator, complex analysis, non-commutative geometry, etc. [23, 38, 39, 43]. Specifically, Gabor frame is a fundamental research topic in time-frequency analysis [18, 20-22].

Obviously, whether a Gabor system forms a frame depends on the window function and time-frequency parameters. Ideally, the characteristic description of the window function g and time-frequency parameters (a,b) should be established so that the Gabor system (g,a,b) is a frame for L^2(\mathbb{R}), but this is a very challenging problem, which has not been completely solved yet. However, for given g\in L^2(\mathbb{R}), the search for time-frequency parameters (a,b) in the system (g,a,b) forming a frame for L^2(\mathbb{R}) has obtained some results, and the study of this situation is currently one of the basic research contents of Gabor frame. Therefore, the concept of frame set was born.

Definition 1.2　Assume that g\in L^2(\mathbb{R}), the following set

\mathbb{F}(g)=\left\{(a,b)\in {\mathbb{R}}_{+}^2|(g,a,b)\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{G}\mathrm{a}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}{L}^2(\mathbb{R})\right\}

is called the frame set associated with g.

It is easy to know that if (a,b)\in \mathbb{F}(g), then ab⩽1, which comes from the density theorem of Gabor frame as follows (see [9, 15-17, 29, 37]).

Theorem 1.1　 Let g\in L^2(\mathbb{R}) and (a,b)\in {\mathbb{R}}_{+}^2.

(1) If ab>1, then (g,a,b) is not a frame for L^2(\mathbb{R}).

(2) The system (g,a,b) is an orthonormal basis for L^2(\mathbb{R}) if and only if the (g,a,b) is a tight frame for L^2(\mathbb{R}), \Vert g\Vert =1, and ab=1.

In general, for fixed g\in L^2(\mathbb{R}), fully giving its frame set is an unresolved problem. But an important result of the frame set was established in [19].

Theorem 1.2 [19]　 If g\in L^2(\mathbb{R}) and g\in M^1(\mathbb{R}), then \mathbb{F}(g) is a non-empty open set in {\mathbb{R}}_{+}^2, where the modulation space M^1(\mathbb{R}) (see [7]) is defined by

and W{F}_g is the windowed Fourier transform, i.e.,

W{F}_{g}f(x,\xi )={\int }_{\mathbb{R}}f(t)\overline{g(t-x)}{\mathrm{e}}^{-\mathrm{i}2\pi t\xi }\mathrm{d}t.

For any g\in L^2(\mathbb{R}), it is not known which elements \mathbb{F}(g) are composed of, but for the special function g, \mathbb{F}(g) has been fully characterized.

Theorem 1.3　 If g is any of the following functions, then

(1) The Guassian function g(x)={\mathrm{e}}^{-\pi x^2} (see [39, 42, 44]).

(2) The hyperbolic secant function g(x)=\frac{1}{\cosh x} (see [34]).

(3) The one-sided exponential decay function g(x)={\mathrm{e}}^{-x}{\chi }_{[0,\mathrm{\infty })}(x) (see [31]).

(4) The two-sided exponential decay function g(x)={\mathrm{e}}^{-|x|} (see [32]).

(5) The total positive function of finite type g(x), i.e., the Fourier transform of g(x) has the form \hat{g}(\xi )=\prod _{j=1}^K(1+\mathrm{i}2\pi {\beta }_j\xi {)}^{-1}, where K\in {\mathbb{Z}}_{+}, {\mathbb{Z}}_{+} is the positive integer set, and \mathrm{\forall }j\ne k,{\beta }_j\ne {\beta }_k\in \mathbb{R} (see [27]).

(6) The total positive function of Gaussian type g(x), i.e., the Fourier transform of g(x) has the form \hat{g}(\xi )=\prod _{j=1}^K(1+\mathrm{i}2\pi {\beta }_j\xi {)}^{-1}{\mathrm{e}}^{-a{\xi }^2}, where K\in {\mathbb{Z}}_{+} and \mathrm{\forall }j\ne k,{\beta }_j\ne {\beta }_k\in \mathbb{R},a>0 (see [25]).

Remark 1.1　It is easy to see that a two-sided exponential decay function is a total positive function of finite type and the Gaussian function is a total positive function of Gaussian type.

In addition, the structure of \mathbb{F}(g) of g(x)={\chi }_{[0,c]}(x)(c>0) is quite complex, and the \mathbb{F}(g) has been fully described in [13]. In history, earlier relevant results of the \mathbb{F}(g) were established in [28, 33].

The structure of \mathbb{F}(g) of Haar function g(x)=-{\chi }_{[-\frac{1}{2},0)}(x)+{\chi }_{[0,\frac{1}{2})}(x) was completely given in [14]. We have recently seen that the profound results on the frame sets of Hergoltz functions and the rational functions were established in [6].

It is well known that the spline functions are a type of piecewise smooth functions with certain smoothness at the connections of each segment. It is a powerful tool for function approximation and a commonly used method for interpolation and fitting of scattered data. It is not only widely used in data processing, differential equations, numerical calculus and other mathematical branches, but also closely related to functional analysis, computational geometry, variational problems, optimal control, statistics and other directions. At the same time, the spline functions have profound applications in many engineering sciences such as aerospace, computers, and materials. Even the simplest B-spline in such a class of functions, the specific composition of its frame set is not completely clear. In the past twenty years, there have been many results in the study of the B-spline frame sets, but there are still many problems that need to be addressed. The main advances in this area of research will be reviewed in this paper.

The paper is organized as follows: Section 2 reviews the existing results of the B-spline frame set so far. In Sections 3 and 4, the results of the 2 order B-spline and the 3 order B-spline frame sets are described in detail, respectively.

2 Results on the frame sets of the B-splines

The definition of the B-splines is as follows.

g_1(x)={\chi }_{[-\frac{1}{2},\frac{1}{2}]}(x),g_N(x)=(g_1\ast g_{N-1})(x),N⩾2.

For the discontinuous function g_1, the structure of \mathbb{F}(g_1) is very complex, as shown in [13], one of the reason for this situation is that g_1 is the only B-spline that does not belong to M^1(\mathbb{R}). In fact, \mathbb{F}(g_1) is not an open set in {\mathbb{R}}_{+}^2, and (a,b)\in (0,1]\times (0,1]\subset \mathbb{F}(g_1). It was shown in [24] that the window function other than M^1(\mathbb{R}) has poor properties, so its frame set is difficult to determine. Notice that the non-smooth spline does not belong to M^1(\mathbb{R}), hence determining the frame set of the non-smooth spline is more difficult.

For N⩾2, g_N\in M^1(\mathbb{R}), thus by Theorem 1.2, \mathbb{F}(g_N) is an open set in {\mathbb{R}}_{+}^2. In addition, the paper [8] lists the characterization of \mathbb{F}(g_N) as one of the six open questions in frame theory.

In the following, we assume that N⩾2. In [10-12, 35], the necessary and sufficient conditions for the duality of Gabor frames and the properties of Zak transforms are used to discuss in depth the frame sets of different classes of window functions with compact support and certain smoothness, and g_N is a special case of these different types of window functions, so the characterizations of the following \mathbb{F}(g_N) comes from the literatures [10-12, 35]. However, let's first review the number pairs that are not in \mathbb{F}(g_N). Note that supp g_N=[-\frac{N}{2},\frac{N}{2}], hence for \mathrm{\forall }a>N, the support union of T_{na}{g}_N(n\in \mathbb{Z}) cannot cover the real number set, thus \mathrm{\forall }b>0, (g,a,b) cannot be a frame for L^2(\mathbb{R}).

Theorem 2.1　 For \mathrm{\forall }a>N and \mathrm{\forall }b>0, (a,b)\notin \mathbb{F}(g_N).

In addition, using the necessary conditions of Gabor frame (see [37], Theorem 8.2.2), it is easy to prove

Theorem 2.2　 If a>0,b=2,3,\dots , then (a,b)\notin \mathbb{F}(g_N).

For the critical values a and b that satisfy ab=1, the following was obtained by the Zak transform properties of g_N and the Zibulski Zeevi matrix in [40].

Theorem 2.3　 If ab=1, then (a,b)\notin \mathbb{F}(g_N).

Combining Theorem 2.1, Theorem 2.2 and Theorem 2.3, we have

(2.1)

Based on (2.1), Gröchenig [24] proposed the following conjecture.

Conjecture 2.1 (Gröchenig)　 \mathrm{\forall }N⩾2,

Subsequently, it was proved that the following result holds by fully using the Zak transform properties of g_2 and the Zibulski Zeevi matrix in [36, 40].

Theorem 2.4　 Let ab=\frac{5}{6},b\in [\frac{7}{3},\frac{8}{3}], then (a,b)\notin \mathbb{F}(g_2).

Theorem 2.4 negates Gröchenig conjecture, one specific example is that if a=\frac{1}{3} and b=\frac{5}{2}, then (a,b)\notin \mathbb{F}(g_2). In fact, the other values of a and b were provided in [36] such that (a,b)\notin \mathbb{F}(g_N), these values of a and b were included in Theorem 2.4 for N=2. At the same, paper [36] proposed the following conjecture based on Theorem 2.4.

Conjecture 2.2　If a=\frac{1}{2n+1},b=\frac{2l+1}{2},n,l\in {\mathbb{Z}}_{+},l>n and , then (a,b)\notin \mathbb{F}(g_2). Furthermore, if \tilde{a}\tilde{b}=\frac{2l+1}{2(2n+1)},\tilde{b}\in [b-a\frac{l-n}{2},b+a\frac{l-n}{2}], then (\tilde{a},\tilde{b})\notin \mathbb{F}(g_2).

It is obvious to see that when n=1,l=2 in conjecture 2.2, a=\frac{1}{3},b=\frac{5}{2}, \tilde{a}\tilde{b}=\frac{5}{6},\tilde{b}\in [\frac{7}{3},\frac{8}{3}], which is just Theorem 2.4.

In Conjecture 2.2, if a is the reciprocal of a positive even number, then it is believed that this conjecture is also correct by the analysis of number values in [40], that is,

Conjecture 2.3　If , then (a,b)\notin \mathbb{F}(g_2).

To our knowledge, the above two conjectures has been unsolved yet.

Let's discuss the number pairs in the set \mathbb{F}(g_N) below. Based on the results of the literatures [10-12, 35], it can be concluded that

(2.2)(0,N)\times \left(0,\frac{4}{N+3a}\right]\subset \mathbb{F}(g_N),

(2.3)\left[\frac{N}{2},N\right)\times \left(0,\frac{1}{a}\right)\subset \mathbb{F}(g_N).

Indeed, formulas (2.2) and (2.3) come from literatures [12] and [11], respectively. In addition, the following results were provided in [11].

Theorem 2.5　 LetN⩾2,a>0,b>0 and ab⩽1.

(1) If and b⩽N^{-1}, then (a,b)\in \mathbb{F}(g_N).

(2) If there exists a positive integer l such that N^{-1}⩽b⩽N^{-1}2 and , then (a,b)\in \mathbb{F}(g_N).

(3) If b\in \{1,2^{-1},\dots ,(N-1{)}^{-1}\}, then (a,b)\in \mathbb{F}(g_N).

(4) If a=p^{-1}l, where l=1,2,\dots ,N-1 and p is a positive integer, and , then (a,b)\in \mathbb{F}(g_N).

Remark 2.1　It is easy to see that formula (2.2) implies (1) in Theorem 2.5, and in the case l=1, (2) in Theorem 2.5 is just (2.3). Additionally, when l=1, the results (3) and (4) in Theorem 2.5 all stem from [35].

We also noticed that paper [5] used the partitioning technique to obtain

(2.4)\left[\frac{N}{3},\frac{N}{2}\right]\times \left[\frac{4}{N+3a},\frac{2}{N}\right]\subset \mathbb{F}(g_N).

In fact, the frame sets of a class of functions (referred to as generalized B-splines) containing B-spline functions discussed in [12] were considered in [5], and the obtained result is to give a new pair of numbers (a,b)\in \mathbb{F}(g_N) on the basis of the original results. Combining the formulas (2.2), (2.3), and (2.4) immediately yields

(2.5)\begin{array}{l}{\mathbb{E}}_N\subset \mathbb{F}(g_N),\end{array}

where

As far as we know, the set {\mathbb{E}}_N in (2.5) is the largest subset of the frame set \mathbb{F}(g_N) of the B-spline, except for N=2 and N=3. When N=2 and N=3, \mathbb{F}(g_N) contain more values of a and b than in the set {\mathbb{E}}_N, so the following two sections summarize the results of the 2 order B-spline and the 3 order B-spline frame sets, respectively.

3 The case of the 2 order B-spline

The definition of the 2 order B-spline is as follows.

g_2(x)=\{\begin{array}{ll}1+x,& x\in [-1,0],\\ 1-x,& x\in [0,1].\end{array}

Of course, \mathbb{F}(g_2) is an open set in {\mathbb{R}}_{+}^2, and which elements exactly constitute \mathbb{F}(g_2) is still an open question. The proof ideas of existing results all rely on the necessary and sufficient conditions for the dual Gabor frame established by Janssen (see [9, 20, 37]).

Theorem 3.1　 Let g,h\in L^2(\mathbb{R}) and (a,b)\in {\mathbb{R}}_{+}^2. Then two Bessel sequences (g,a,b) and (h,a,b) are dual Gabor frames if and only if

\frac{1}{b}\sum _{l\in \mathbb{Z}}g\left(x-la-\frac{k}{b}\right)\overline{h(x-la)}={\delta }_{k,0},a.e.,x\in [0,a].

For the Bessel sequences (g,a,b) and (h,a,b), if

\mathrm{\forall }f\in L^2(\mathbb{R}),f(x)=\sum _{m,n\in \mathbb{Z}}⟨f,E_{mb}{T}_{na}g⟩E_{mb}{T}_{na}h(x),

then (g,a,b) and (h,a,b) are called the dual Gabor frames, alternatively, h is called a dual of g. In fact, the two are mutually dual.

In order to reflect the historical process of obtaining the elements in \mathbb{F}(g_2) based on Theorem 3.1, we summarize as follows.

Theorem 3.2　 For g_2, the following statements hold.

(1) If and , then (a,b)\in \mathbb{F}(g_2), and there exists a unique dual h\in L^2(\mathbb{R})\bigcap L^{\mathrm{\infty }}(\mathbb{R}) such that supp h⫅[-\frac{a}{2},\frac{a}{2}] (see [12]).

(2) If and , then (a,b)\in \mathbb{F}(g_2), and there is a unique dual h\in L^2(\mathbb{R})\bigcap L^{\mathrm{\infty }}(\mathbb{R}) such that supp h⫅[-\frac{3a}{2},\frac{3a}{2}] (see [36]).

(3) If and , then (a,b)\in \mathbb{F}(g_2), and there is a unique dual h\in L^2(\mathbb{R})\bigcap L^{\mathrm{\infty }}(\mathbb{R}) such that supp h⫅[-\frac{5a}{2},\frac{5a}{2}] (see [3]).

(4) If \frac{1}{2}⩽a⩽\frac{4}{5}, and and b>1, then (a,b)\in \mathbb{F}(g_2), and there exists a unique dual h\in L^2(\mathbb{R})\bigcap L^{\mathrm{\infty }}(\mathbb{R}) such that supp h⫅[-\frac{5a}{2},\frac{5a}{2}] (see [3]).

(5) If \frac{2}{3}⩽a⩽1 and , then (a,b)\in \mathbb{F}(g_2), and there is a unique compactly supported dual h\in L^2(\mathbb{R})\bigcap L^{\mathrm{\infty }}(\mathbb{R}) (see [4]).

Certainly, the formula (2.5) also holds for N=2, that is,

(3.1)\begin{array}{l}{\mathbb{E}}_2\subset \mathbb{F}(g_2).\end{array}

Recently, some new elements of the set \mathbb{F}(g_2) were found in [2] based on the linear algebra method used in [4].

Theorem 3.3　 {\mathrm{\Gamma }}_3\bigcup \mathrm{\Lambda }\subset \mathbb{F}(g_2), where

{\mathrm{\Gamma }}_3=\left\{(a,b)\in {\mathbb{R}}_{+}^2|a\in \left(0,\frac{2}{9}\right]\bigcup \left(\frac{2}{7},\frac{1}{2}\right),b\in \left(\frac{4}{2+3a},\frac{2}{1+a}\right]\right\},

and

\mathrm{\Lambda }=\bigcup _{m=3}^{\mathrm{\infty }}{\mathrm{\Lambda }}_m.

Here

{\mathrm{\Lambda }}_3=\{(a,b)\in {\mathbb{R}}_{+}^2|a\in [\frac{1}{2},\frac{4}{5}],b\in (\frac{4}{2+3a},\frac{6}{2+5a}),b>1\}.

And for m⩾4,

{\mathrm{\Lambda }}_m=\left\{(a,b)\in {\mathbb{R}}_{+}^2|a\in \left[\frac{m-3}{m-2},\frac{2(m-1)}{2m-1}\right],b\in \left(\frac{2(m-1)}{2+(2m-3)a},{\gamma }_{m,a}\right],b>1\right\},

{\gamma }_{m,a}=min\left\{\frac{2m}{2+(2m-1)a},\frac{2}{1+a}\right\}.

The proof idea of Theorem 3.3 is still based on the necessary and sufficient condition of the dual Gabor frames, but this condition is a concretization of the special cases of g and h in Theorem 3.1, that is,

Theorem 3.4　 Let , and h be a bounded real function with supp h\subset [-\frac{2m-1}{2}a,\frac{2m-1}{2}a]. Then Gabor systems (g_N,a,b) and (h,a,b) are dual Gabor frames if and only if

\frac{1}{b}\sum _{l=1-m}^{m-1}{g}_N\left(x-la-\frac{k}{b}\right)h(x-la)={\delta }_{k,0},|k|⩽m-1,a.e.,x\in \left[-\frac{a}{2},\frac{a}{2}\right].

Remark 3.1　For the proof of Theorem 3.4, the cases m=1 and m=2 were given in [36] and [12], respectively, and the cases m⩾3 were established in [4].

After carefully analyzing the proof methods and processes of Theorems 3.2 and 3.3 and formula 3.1, Okoudjou [41] proposed the following unsolved conjecture.

Conjecture 3.1 [41]

4 The case of the 3 order B-spline

The 3 order B-spline is defined by

g_3(x)=\{\begin{array}{rl}& \frac{1}{2}{x}^2+\frac{3}{2}x+\frac{9}{8},x\in \left[-\frac{3}{2},-\frac{1}{2}\right],\\ & -x^2+\frac{3}{4},x\in \left[-\frac{1}{2},\frac{1}{2}\right],\\ & \frac{1}{2}{x}^2-\frac{3}{2}x+\frac{9}{8},x\in \left[\frac{1}{2},\frac{3}{2}\right].\end{array}

Naturely, \mathbb{F}(g_3) is an open set in {\mathbb{R}}_{+}^2, and by (2.5), the following set {\mathbb{E}}_3 is a subset of \mathbb{F}(g_3),

Some new elements of \mathbb{F}(g_3) were founded by making full use of Theorem 3.4 and the invertibility of block matrix in [1], see the following theorem.

Theorem 4.1　 Let

{\mathrm{\Delta }}_3=\{(a,b)\in {\mathbb{R}}_{+}^2|a\in [\frac{3}{4},\frac{6}{5}],b\in (\frac{4}{3+3a},\frac{6}{3+5a}),b>1\}.

For m⩾4,

{\mathrm{\Delta }}_m=\{(a,b)\in {\mathbb{R}}_{+}^2|a\in [\frac{3(m-3)}{(2(m-2)},\frac{3(m-1)}{2m-1}],b\in (\frac{2(m-1)}{3+(2m-3)a},{\theta }_{m,a}],b>\frac{2}{3}\},

{\theta }_{m,a}=min\{\frac{2m}{3+(2m-1)a},\frac{4}{3+2a}\}.

If (a,b)\in {\mathrm{\Delta }}_m, then (a,b)\in \mathbb{F}(g_3), and g_3 exists a unique dual h\in L^2(\mathbb{R}) such that

\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{h}⫅[-\frac{(2\mathrm{m}-1)\mathrm{a}}{2},\frac{(2\mathrm{m}-1)\mathrm{a}}{2}].

Theorem 4.1 shows that

\bigcup _{m=3}^{\mathrm{\infty }}{\mathrm{\Delta }}_m\subset \mathbb{F}(g_3).

References

[1] Atindehou A G D. On the frame set for the 3-spline. Appl Math 2022; 13: 377–400

[2] Atindehou A G D, Frederick C, Kouagou Y B, Okoudjou K A. On the frame set of the second-order B-spline. Appl Comput Harmon Anal 2023; 62: 237–250

[3] AtindehouA G DKouagouY BOkoudjouK A. Frame sets for generalized B-splines. 2018, arXiv: 1804-02450

[4] Atindehou A G D, Kouagou Y B, Okoudjou K A. Frame sets for a class of compactly supported continuous functions. Asian-Eur J Math 2020; 13(5): 2050093

[5] AtindehouA G DYebeniBKouagouY BOkoudjouK. A. Frame sets for generalized B-splines. 2018, arXiv: 1804.02450v2

[6] Belov Y, Kulikov A, Lyubarskii Y. Gabor frames for rational functions. Invent Math 2023; 23(2): 431–466

[7] BényiÁOkoudjouK A. Modulation Spaces: With Applications to Pseudodifferential Operators and Nonlinear Schrödinger Equations. Applied and Numerical Harmonic Analysis. New York: Birkhaüser/Springer, 2020

[8] ChristensenO. Six (seven) problems in frame theory. In: New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis. Cham: Birkhaüser/Springer, 2014, 337–358

[9] ChristensenO. An Introduction to Frames and Riesz Bases, 2nd ed. Applied and Numerical Harmonic Analysis. Cham: Birkhaüser/Springer, 2016

[10] Christensen O, Kim H O, Kim R Y. Gabor windows supported on [−1,1] and compactly supported dual windows. Appl Comput Harmon Anal 2010; 28(1): 89–103

[11] Christensen O, Kim H O, Kim R Y. On Gabor frames generated by sign-changing windows and B-splines. Appl Comput Harmon Anal 2015; 39(3): 534–544

[12] Christensen O, Kim H O, Kim R Y. On Gabor frame set for compactly supported continuous functions. J Inequal Appl 2016; 94

[13] Dai X R, Sun Q Y. The abc-problem for Gabor systems. Mem Amer Math Soc 2016; 244(1152): ix+99 pp

[14] DaiX RZhuM. Frame sets for Gabor systems with Haar window. 2022, arXiv: 2205.06479v1

[15] Daubechies I. The wavelet transform, time-frequency localization and signal analysis. IEEE Trans Inform Theory 1990; 36(5): 961–1005

[16] DaubechiesI. Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol 61. Philadelphia, PA: SIAM, 1992

[17] Daubechies I, Landau H, Landau Z. Gabor time-frequency lattices and the Wexler-Raz identity. J Fourier Anal Appl 1995; 1(4): 437–478

[18] Feichtinger H G, Gröchenig K. Gabor frames and time-frequency analysis of distributions. J Funct Anal 1997; 146(2): 464–495

[19] Feichtinger H G, Kaiblinger N. Varying the time-frequency lattices of Gabor frames. Trans Amer Math Soc 2004; 356(5): 2001–2023

[20] FeichtibgerH GStrohmerT (eds). Gabor Analysis and Algorithms: Theory and Applications. Applied and Numerical Harmonic Analysis. Boston, MA: Birkhaüser Boston, Inc, 1998

[21] FeichtingerH GStrohmerT (eds). Advances in Gabor Analysis. Applied and Nu- merical Harmonic Analysis. Boston, MA: Birkhaüser Boston, Inc, 2003

[22] GröchenigK. Foundations of Time-Frequency Analysis. Applied and Numerical Har- monic Analysis. Boston, MA: Birkhaüser Boston, Inc, 2001

[23] Gröchenig K. Time-frequency analysis of Sjöstrand’s class. Rev Mat Iberoam 2006; 22(2): 703–724

[24] Gröchenig K. The mystery of Gabor frames. J Fourier Anal Appl 2014; 20(4): 865–895

[25] Gröchenig K, Romero J L, Stöckler J. Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions. Invent Math 2018; 211(3): 1119–1148

[26] Gröchenig K, Romero J L, Stöckler J. Sharp results on sampling with derivatives in shift-invariant spaces and multi-window Gabor frames. Constr Approx 2020; 51(1): 1–25

[27] Gröchenig K, Stöckler J. Gabor frames and totally positive functions. Duke Math J 2013; 162(6): 1003–1031

[28] Gu Q, Han D G. When a characteristic function generates a Gabor frame?. Appl Comput Harmon Anal 2008; 24(3): 290–309

[29] Heil C. History and evolution of the density theorem for Gabor frames. J Fourier Anal Appl 2007; 13(2): 113–166

[30] HlawatschFMatzG. Wireless communications over rapidly time-varying channels. Amsterdam: Academic Press, 2011

[31] Janssen A J E M. Some Weyl-Heisenberg frame bound calculations. Indag Math (N S) 1996; 7(2): 165–183

[32] Janssen A J E M. On generating tight Gabor frames at critical density. J Fourier Anal Appl 2003; 9(2): 175–214

[33] JanssenA J E M. Zak transforms with few zeros and the tie. In: Advances in Gabor Analysis. App Numer Harmon Anal. Boston, MA: Birkhaüser Boston, Inc, 2003, 31–70

[34] Janssen A J E M, Strohmer T. Hyperbolic secants yield Gabor frames. Appl Comput Harmon Anal 2002; 12(2): 259–267

[35] Kloos T, Stöckler J. Zak transforms and Gabor frames of totally positive functions and exponential B-splines. J Approx Theory 2014; 184: 209–237

[36] Lemvig J, Nielsen K H. Counterexamples to the B-spline conjecture for Gabor frames. J Fourier Anal Appl 2016; 22(6): 1440–1451

[37] LiD F. Mathematical Theory of Wavelet Analysis. Beijing: Science Press, 2017 (in Chinese)

[38] Luef F. Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces. J Funct Anal 2009; 257(6): 1921–1946

[39] LyubarskiiY I. Frames in the Bargmann space of entire function functions. In: Entire and Subharmonic Functions. Adv Sovit Math, Vol 11. Providence, RI: AMS, 1992, 167–180

[40] NielsenK H. The frame set of Gabor systems with B-spline generators. Master Thesis. København: Technical University of Denmark, 2015

[41] Okoudjou K A. An invitation to Gabor analysis. Notices Amer Math Soc 2019; 66(6): 808–819

[42] Seip K. Density theorems for sampling and interpolation in the Bargmann-Fock space, I. J Reine Angew Math 1992; 429: 91–106

[43] SeipK. Interpolation and sampling in spaces of analytic functions. University Lecture Series, Vol 33. Providence, RI: AMS, 2004

[44] Seip K, Wallstén R. Density theorems for sampling and interpolation in the Bargmann-Fock space, II. J Reine Angew Math 1992; 429: 107–113

[45] Strohmer T. Approximation of dual Gabor frames, window decay, and wireless communications. Appl Comput Harmon Anal 2001; 11(2): 243–262

[46] Strohmer T. Pseudodifferential operators and Banach algebras in mobile communications. Appl Comput Harmon Anal 2006; 20(2): 237–249

RIGHTS & PERMISSIONS

Higher Education Press 2023