An Introduction to Frames and Riesz Bases
Abstract
This revised and expanded monograph presents the general theory for frames and Riesz bases in Hilbert spaces as well as its concrete realizations within Gabor analysis, wavelet analysis, and generalized shift-invariant systems. Compared with the first edition, more emphasis is put on explicit constructions with attractive properties. Based on the exiting development of frame theory over the last decade, this second edition now includes new sections on the rapidly growing fields of LCA groups, generalized shift-invariant systems, duality theory for as well Gabor frames as wavelet frames, and open problems in the field.
Key features include:
*Elementary introduction to frame theory in finite-dimensional spaces
* Basic results presented in an accessible way for both pure and applied mathematicians
* Extensive exercises make the work suitable as a textbook for use in graduate courses
* Full proofs includ
ed in introductory chapters; only basic knowledge of functional analysis required
* Explicit constructions of frames and dual pairs of frames, with applications and connections to time-frequency analysis, wavelets, and generalized shift-invariant systems
* Discussion of frames on LCA groups and the concrete realizations in terms of Gabor systems on the elementary groups; connections to sampling theory
* Selected research topics presented with recommendations for more advanced topics and further readin
g
* Open problems to stimulate further research
An Introduction to Frames and Riesz Bases will be of interest to graduate students and researchers working in pure and applied mathematics, mathematical physics, and engineering. Professionals working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find this book a useful self-study reference.
Review of the first edition:
"Ole Christensen’s An Introduction to Frames and Riesz Bases is a first-rate introduction to the field … . The book provides an excellent exposition of these topics. The material is broad enough to pique the interest of many readers, the included exercises supply some interesting challenges, and the coverage provides enough background for those new to the subject to begin conducting original research."
— Eric S. Weber, American Mathematical Monthly, Vol. 112, February, 2005
Chapters (24)
In the study of vector spaces, one of the most important concepts is that of a basis. A basis provides us with an expansion of all vectors in terms of “elementary building blocks” and hereby helps us by reducing many questions concerning general vectors to similar questions concerning only the basis elements. However, the conditions to a basis are very restrictive – no linear dependence between the elements is possible, and sometimes we even want the elements to be orthogonal with respect to an inner product. This makes it hard or even impossible to find bases satisfying extra conditions, and this is the reason that one might look for a more flexible tool.
After the introduction to frames in finite-dimensional vector spaces in Chapter 1, the rest of the book will deal with expansions in infinite-dimensional vector spaces. Here great care is needed: we need to replace finite sequences \(\{f_{k}\}_{k=1}^{n}\) by infinite sequences \(\{f_{k}\}_{k=1}^{\infty }\), and suddenly the question of convergence properties becomes a central issue. The vector space itself might also cause problems, e.g., in the sense that Cauchy sequences might not be convergent. We expect the reader to have a basic knowledge about these problems and the way to circumvent them, but for completeness we repeat the central definitions and results concerning Banach spaces and operators hereon in Sections 2.1–2.2. In Sections 2.3–2.4 we specialize to Hilbert spaces and their operators. Section 2.5 deals with pseudo-inverse operators; this subject is not expected to be known and is treated in more detail. Section 2.6 introduces the so-called moment problems in Hilbert spaces. In Sections 2.7–2.9, we discuss the Hilbert space \(L^{2}(\mathbb{R})\) consisting of the square integrable functions on \(\mathbb{R}\) and three classes of operators hereon, as well as the Fourier transform. The material in those sections is not needed for the study of frames and bases on abstract Hilbert spaces, but it forms the basis for all the constructions in Chapters 9–20
Bases play a prominent role in the analysis of vector spaces, as well in the finite-dimensional as in the infinite-dimensional case. The idea is the same in both cases, namely, to consider a family of elements such that all vectors in the considered space can be expressed in a unique way as superpositions of these elements. In the infinite-dimensional case, the situation is complicated: we are forced to work with infinite series, and different concepts of a basis are possible, depending on how we want the series to converge. For example, are we asking for the series to converge with respect to a fixed order of the elements (conditional convergence) or do we want it to converge regardless of how the elements are ordered (unconditional convergence)? We define the relevant types of bases in general Banach spaces in Section 3.1; the case of a basis in a Hilbert space is considered in Section 3.3. Sequences satisfying the Bessel inequality are considered in Section 3.2 and characterized in terms of an associated operator, the synthesis operator. In Section 3.4 we discuss the most important properties of orthonormal bases in Hilbert spaces; we expect the reader to have some basic knowledge about this subject. Section 3.5 deals with the Gram matrix and its relationship with Bessel sequences. In Section 3.6, one of the key subjects of the current book, namely, Riesz bases, is introduced and treated in detail; a subspace version of these is discussed in Section 3.7. Several characterizations of Riesz bases and Riesz sequences are provided. Orthonormal bases and Riesz bases both satisfy the Bessel inequality, which is the key to the observation that they deliver unconditionally convergent expansions and can be ordered in an arbitrary way.
The next chapters will deal with generalizations of the basis concept, so it is natural to ask why they are needed. Bases exist in all separable Hilbert spaces and in practically all Banach spaces of interest, so why do we have to search for generalizations?
The main feature of a basis \(\{f_{k}\}_{k=1}^{\infty }\) in a Hilbert space \(\mathcal{H}\) is that every \(f \in \mathcal{H}\) can be represented as a superposition of the elements f
k
in the basis: $$\displaystyle\begin{array}{rcl} f =\sum _{ k=1}^{\infty }c_{ k}(f)f_{k}.& &{}\end{array}$$ (5.1) The coefficients c
k
(f) are unique. We now introduce the concept of frames. A frame is also a sequence of elements \(\{f_{k}\}_{k=1}^{\infty }\) in \(\mathcal{H}\), which allows every \(f \in \mathcal{H}\) to be written as in ( 5.1). However, the corresponding coefficients are not necessarily unique. Thus a frame might not be a basis; arguments for generalizing the basis concept were given in Chapter 4
We have already highlighted the frame decomposition, which shows that a frame \(\{f_{k}\}_{k=1}^{\infty }\) for a Hilbert space \(\mathcal{H}\) leads to the decomposition $$\displaystyle\begin{array}{rcl} f =\sum _{ k=1}^{\infty }\langle f,S^{-1}f_{ k}\rangle f_{k},\ \ \forall f \in \mathcal{H};& &{}\end{array}$$ (6.1) here \(S: \mathcal{H}\rightarrow \mathcal{H}\) denotes the frame operator. In practice, it is difficult to apply the general frame decomposition, due to the fact that we need to invert the frame operator. We have mentioned two ways to circumvent the problem. The first one is to restrict our attention to tight frames: as we have seen in Corollary 5. 1. 7, for a tight frame \(\{f_{k}\}_{k=1}^{\infty }\) with frame bound A, the frame decomposition takes the much simpler form $$\displaystyle\begin{array}{rcl} f = \frac{1} {A}\sum _{k=1}^{\infty }\langle f,f_{ k}\rangle f_{k},\ \forall f \in \mathcal{H}.& &{}\end{array}$$ (6.2)
We have already seen that Riesz bases are frames. In this chapter we exploit the relationship between these two concepts further. In particular, we give a number of equivalent conditions for a frame to be a Riesz basis.
We have often spoken about a frame in an intuitive sense as some kind of “overcomplete basis.” It turns out that, in the technical sense, one has to be careful with such statements. In fact, we will prove the existence of a frame which has no relation to a basis: no subfamily of the frame forms a basis. On the other hand, sufficient conditions for a frame to contain a Riesz basis as a subfamily are also given.
The content in Chapters 5–7 forms the fundament for frame theory, but much more is known. We will not be able to describe all the interesting directions of frame theory, but the purpose of this chapter is to give short presentations of certain topics that appear repeatedly in the literature. Section 8.1 deals with the theory for g-frames as developed by Sun; it “lifts” frame theory from a condition dealing with vectors in a Hilbert space to a condition dealing with operators on the Hilbert space and hereby provides more general ways of obtaining “frame-like” decompositions. Section 8.2 discusses localized frames; they were introduced by Gröchenig with the purpose to identify frames that not only lead to series expansions in the underlying Hilbert space but also in a class of associated Banach spaces. By now, localized frames appear in many papers where frames with “particularly good properties” are needed. Section 8.3 deals with the so-called R-dual \(\{\omega _{j}\}_{j=1}^{\infty }\) of a sequence \(\{f_{k}\}_{k=1}^{\infty }\) in a Hilbert space, originally introduced by Casazza, Kutyniok, and Lammers. The R-dual provides a way of checking that \(\{f_{k}\}_{k=1}^{\infty }\) is a frame, by checking the (conceptually more accessible) condition that \(\{\omega _{j}\}_{j=1}^{\infty }\) forms a Riesz sequence. The construction mimics the result in Proposition 1. 4. 3 and is strongly related to the duality principle in Gabor analysis, to which we return in Section 13. 1 In Section 8.4 we consider a generalization of frame theory to the case where the upper frame condition is violated. In this case, the synthesis operator is unbounded, but under certain conditions it is still possible to develop a frame-like theory. Finally, Section 8.5 relates frame theory to signal processing and signal transmission. Much more can be said about this, but this should be done by the people who are deeply involved in these applications.
The previous chapters have concentrated on general frame theory. We have only seen a few concrete frames, and most of them were constructed via manipulations on an orthonormal basis for an arbitrary separable Hilbert space. An advantage of this approach is that we obtain universal constructions, valid in all Hilbert spaces.
Chapter 9 dealt with systems of functions generated by integer-translates of a single function in \(L^{2}(\mathbb{R})\). We will now generalize this setup and consider translates of a given countable family of functions rather than just one function. Such systems of functions are called shift-invariant systems. Our goal is to characterize various frame properties for shift-invariant systems, a subject that was treated first in the paper [559] by Ron and Shen. The presentation is inspired by the approach by Janssen in [430]. The derived results will play an important role in the analysis of Gabor systems in Chapter 11
The mathematical theory for Gabor analysis in \(L^{2}(\mathbb{R})\) is based on two classes of operators on \(L^{2}(\mathbb{R})\), namely, $$\displaystyle\begin{array}{rcl} \mbox{ Translation by}\ a \in \mathbb{R},\ & T_{a}:& L^{2}(\mathbb{R}) \rightarrow L^{2}(\mathbb{R}),(T_{ a}f)(x) = f(x - a), {}\\ \mbox{ Modulation by}\ b \in \mathbb{R},\ & E_{b}:& L^{2}(\mathbb{R}) \rightarrow L^{2}(\mathbb{R}),(E_{ b}f)(x) = e^{2\pi ibx}f(x). {}\\ \end{array}$$ Gabor
analysis aims at representing functions \(f \in L^{2}(\mathbb{R})\) as superpositions of translated and modulated versions of a fixed function \(g \in L^{2}(\mathbb{R})\). There are two ways one can think about this. The first is to ask for integral representations involving all possible translations and modulations, i.e., representations like
$$\displaystyle\begin{array}{rcl} f(x) =\int _{ -\infty }^{\infty }\int _{ -\infty }^{\infty }c_{ f}(a,b)e^{2\pi ibx}g(x - a)dbda;& &{}\end{array}$$ (11.1) here we have to search for a function c
f
of two variables making this true. Note that we also have to specify in which sense we want (11.1) to be valid, i.e., how the integral shall be interpreted. The second approach is to restrict the translation and modulation parameters to a discrete subset \(\varLambda \subset \mathbb{R}^{2}\) and ask for series representations of f in terms of the functions $$\displaystyle\begin{array}{rcl} \{e^{2\pi ibx}g(x - a)\}_{ (a,b)\in \varLambda }.& &{}\end{array}$$ (11.2)
The main issue in Chapter 11 was to state necessary and/or sufficient conditions for a Gabor system \(\{E_{mb}T_{na}g\}_{m,n\in \mathbb{Z}}\) in \(L^{2}(\mathbb{R})\) to form a frame. We will now take the next step and consider Gabor frames that are convenient to apply in practice. From the general frame theory in Chapter 5 and Chapter 6 we know that frames are particularly useful when the frame decomposition takes a simple form, which is the case if either the frame is tight or we have access to a convenient dual frame.
Gabor analysis has now been a very active research field for about 30 years, and even a description of its connection to frame theory would cover an entire book. Based on the core material in Chapters 11–12, we will now present selected topics and tools. All of them are of general importance, in the sense that they find applications within several areas of time–frequency analysis. The sections are to a large extent independent of each other.
In concrete applications, a model involving a Gabor frame \(\{E_{mb}T_{na}g\}_{m,n\in \mathbb{Z}}\) for \(L^{2}(\mathbb{R})\) will ultimately have to be transferred into a model involving a finite number of vectors in a finite-dimensional space. In this chapter, we show that it indeed is possible to construct Gabor-type frames in \(\mathbb{C}^{L}\) for \(L \in \mathbb{N},\) based on certain Gabor frames for \(L^{2}(\mathbb{R}).\) As intermediate steps, we will construct frames for \(\ell^{2}(\mathbb{Z})\) based on sampling of the frame \(\{E_{mb}T_{na}g\}_{m,n\in \mathbb{Z}},\) as well as frames for the space \(L^{2}(0,L)\) based on periodization. Each of the mentioned steps keeps the frame bounds; furthermore, dual pairs of Gabor frames in one space are turned into dual pairs in the other spaces as well.
A fundamental question in wavelet analysis is what conditions we have to impose on a function ψ such that a given signal \( f \in L^{2}(\mathbb{R}) \) can be expanded via translated and scaled versions of ψ, i.e., via functions
In this chapter we consider dyadic wavelet systems,
i.e., wavelet systems for \(L^{2}(\mathbb{R})\) with scaling parameter a = 2 and translation parameter b = 1. We will usually denote the resulting wavelet systems \(\{2^{j/2}\psi (2^{j}x - k)\}_{j,k\in \mathbb{Z}}\) by \(\{D^{j}T_{k}\psi \}_{j,k\in \mathbb{Z}}\) or \(\{\psi _{j,k}\}_{j,k\in \mathbb{Z}}.\) Recall that bases of this type were considered already in Section 3. 9
The introduction of multiresolution analysis by Mallat and Meyer was the beginning of a new era; the short descriptions in Section 3. 9 and Section 4. 3 only give a glimpse of the research activity based on this new tool, aiming at construction of orthonormal bases \( \{\psi _{j,k}\}_{j,k\in \mathbb{Z}}. \)
Frame multiresolution analysis is just one way to construct wavelet frames via multiscale techniques. We already mentioned in Section 17. 3 that the conditions can be weakened further, and the purpose of this chapter is to show how one can still construct frames.
Continuing the style from the chapters on Gabor frames, we will now present a few selected topics concerning wavelet frames. The sections deal with issues that appear in several places in the wavelet literature, and they can be read independently of each other. We begin in Section 19.1 with a discussion of irregular wavelet frames. Section 19.2 states a few results about oversampling of wavelet frames. We analyze the relationship between two wavelet systems with the same scaling parameter, but different translation parameters. In particular, we consider the case where one translation parameter is an integer multiple of the other; surprisingly, it turns out to play an important role whether this integer is even or odd. Section 19.3 returns to the extension problem considered for general frames in Section 6. 4 and for Gabor frame in Section 12. 7; while these sections contain complete and satisfying answers, the corresponding wavelet question is open and challenging. Section 19.4 gives a short description of wavelet theory from the signal processing perspective.
We have already seen that a Gabor system \(\{E_{mb}T_{na}g\}_{m,n\in \mathbb{Z}}\) in \(L^{2}(\mathbb{R})\) is a special case of a shift-invariant system. In contrast, a wavelet system is not a shift-invariant system. Indeed, looking for example at a dyadic wavelet system \(\{D^{j}T_{k}\psi \}_{j,k\in \mathbb{Z}},\) we can use the commutator relations for the operators D and T
k
to rewrite the system as $$\displaystyle\begin{array}{rcl} \{D^{j}T_{ k}\psi \}_{j,k\in \mathbb{Z}} =\{ T_{k2^{-j}}D^{j}\psi \}_{ j,k\in \mathbb{Z}};& & {}\\ \end{array}$$ thus, the system is in fact a collection of shifts of the functions \(D^{j}\psi,\,j \in \mathbb{Z},\) but the translation parameters depend on \(j \in \mathbb{Z}.\) Therefore the system does not fall into the framework of shift-invariant systems in Section 10.On the other hand, some of the results we have derived for Gabor systems and wavelet systems are very similar, with similar proofs. This is most evident from the sufficient conditions for such systems to form frames, derived in Theorem 11. 4. 2 and Theorem 15. 2. 3: except for the fact that the Gabor result is derived in the time domain and the wavelet result in the frequency domain, the results are clearly parallel.
In this chapter we will consider frame theory from a broader viewpoint than before, namely, as a part of general harmonic analysis. A central part of harmonic analysis deals with functions on groups and ways to decompose such functions in terms of either series representations or integral representations of certain “basic functions.” One of the strengths of harmonic analysis is that it allows very general results that cover several cases at once; for example, instead of developing parallel theories for various groups, we might obtain all of them as special manifestations of a single theory.
The question of stability
plays an important role in connection with bases. That is, if \(\{f_{k}\}_{k=1}^{\infty }\) is a basis and \(\left \{g_{k}\right \}_{k=1}^{\infty }\) is in some sense “close” to \(\{f_{k}\}_{k=1}^{\infty }\), does it follow that \(\left \{g_{k}\right \}_{k=1}^{\infty }\) is also a basis? A classical result states that if \(\{f_{k}\}_{k=1}^{\infty }\) is a basis for a Banach space X, then a sequence \(\left \{g_{k}\right \}_{k=1}^{\infty }\) in X is also a basis if there exists a constant \(\lambda \in ]0,1[\) such that for all finite sequences of scalars \(\{c_{k}\}_{k=1}^{\infty }\). The result is usually attributed to Paley and Wiener [533], but it can be traced back to Neumann [524]: in fact, it is an almost immediate consequence of Theorem 2. 2. 3 with Uf
k
: = g
k
.
Consider a frame \(\{f_{k}\}_{k=1}^{\infty }\) for a Hilbert space \(\mathcal{H}\) and the associated frame operator, $$\displaystyle\begin{array}{rcl} S: \mathcal{H}\rightarrow \mathcal{H},\ \ Sf =\sum _{ k=1}^{\infty }\langle f,f_{ k}\rangle f_{k}.& & {}\\ \end{array}$$ One of the main results in frame theory, the frame decomposition (5. 7), states that each \(f \in \mathcal{H}\) has the representation $$\displaystyle\begin{array}{rcl} f =\sum _{ k=1}^{\infty }\langle f,S^{-1}f_{ k}\rangle f_{k}.& &{}\end{array}$$ (23.1)
The material presented in this book naturally splits in two parts: a functional analytic treatment of frames in general Hilbert spaces, and a more direct approach to structured frames like Gabor frames and wavelet frames. For the second part the most general results were presented in Chapter 21, in the setting of generalized shift-invariant systems on an LCA group.The current chapter is in a certain sense a natural continuation of both tracks. We consider connections between frame theory and abstract harmonic analysis and show how we can construct frames in Hilbert spaces via the theory for group representations. In special cases the general approach will bring us back to the Gabor systems and wavelet systems. The abstract framework adds another new aspect to the theory: we will not only obtain expansions in Hilbert spaces but also in a class of Banach spaces.
... Indeed, S : H → H is bounded, positive, and invertible. See [7] for more details on frames. ...
... Proof. It is well-known (see [7], Lemma 5.4.2) that for a fixed f ∈ C n , the frame coefficients { f , S −1 f j } N i=1 has the least l 2 -norm energy among any other sequence representing f . That is to say ...
... has the least l 2 -norm energy among any other sequences representing f (see [7], Lemma 5.4.2). We obtain an analogous result for probabilistic frames, which shows that among all reconstructions, the canonical dual frame representation has the least L 2 (µ, R n ) energy. ...
This paper studies probabilistic dual frames and associated dual frame potentials from the optimal mass transport perspective. The main contribution in this work shows that given a probabilistic frame, its dual frame potential is minimized if and only if the probabilistic frame is tight and the probabilistic dual frame is the canonical dual. In particular, the tightness condition can be dropped if the probabilistic dual frame potential is minimized only among probabilistic dual frames of pushforward type.
... The problem we study in this paper is the recovery of an unknown signal g 0 ∈ H from partial measurements of the form ( g 0 , ψ l H ) l , under a sparsity assumption on g 0 with respect to a suitable family of vectors {ϕ j } j . The main assumption of this paper is the following: these families of vectors are required to be frames of H [28,30,31]. Hypothesis 1. Let L and J be two index sets 1 . ...
... Given the generality of our setting, we need to consider the dual frames of {ψ l } l and {ϕ j } j . By classical frame theory (see [31,Lemma 5.1.5]), the frame operators U * U and D * D are invertible, and we can consider the dual frames ...
... The frames {ψ l } l and {φ j } j are the canonical dual frames, but in general many other choices are possible. These are in correspondence with all possible bounded left inverses of U and D, and it is possible to characterize all dual frames [31,Section 6.3]. ...
We consider a compressed sensing problem in which both the measurement and the sparsifying systems are assumed to be frames (not necessarily tight) of the underlying Hilbert space of signals, which may be finite or infinite dimensional. The main result gives explicit bounds on the number of measurements in order to achieve stable recovery, which depends on the mutual coherence of the two systems. As a simple corollary, we prove the efficiency of nonuniform sampling strategies in cases when the two systems are not incoherent, but only asymptotically incoherent, as with the recovery of wavelet coefficients from Fourier samples. This general framework finds applications to inverse problems in partial differential equations, where the standard assumptions of compressed sensing are often not satisfied. Several examples are discussed, with a special focus on electrical impedance tomography.
... It follows from the definition of topology onĜ that Λ ⊥ is also a lattice inĜ. Further, a lattice in G can be used to obtain a splitting of groups G andĜ into disjoint cosets (see [3], Chapter 21): ...
... For more details, we refer [3]. Proof of this lemma may be deduced using various results given in [3]. ...
... For more details, we refer [3]. Proof of this lemma may be deduced using various results given in [3]. ...
We have explored the concept of Riesz multiresolution analysis on a locally compact Abelian group G, and have extensively studied the methods of construction of a Riesz wavelet from the given Riesz MRA. We have proved that, if δ α is the order of dilation, then precisely δ α −1 functions are required to construct a Riesz wavelet basis for L 2 (G). An example, supporting our theory and illustrating the methods developed, has also been discussed in detail.
... Thanks to some of their nice properties, frames have already been applied to many research fields (see [3][4][5][6]). We refer also to [7] for more details about frame theory. ...
... See [7] for more information. Suppose that ( ) ∈ K has closed range and that { } ∈ Λ i i is a K -g-frame for with g-frame bounds C Λ and D Λ . ...
In this work, several bilateral inequalities for K K - g g -frames in subspaces are established, drawing support from two kinds of operators induced, respectively, by the K K - g g -frame itself and the K K -dual pair, which, compared with previous ones concerning this topic, possess novel structures. It is indicated that new types of inequalities for some other generalized frames can be naturally presented following our approaches.
... which is reminiscent of the well-known refinement equation [36], [37]. This can be interpreted in terms of a convolution in the space domain. ...
... An equation of the form (25) is called a refinement equation [30] and a function ϕ that satisfies (25) is called a refinable function. The function h is called a refinement mask [30] or two-scale symbol [37]. It is assumed to be 1-periodic. ...
Resolving the details of an object from coarse-scale measurements is a classical problem in applied mathematics. This problem is usually formulated as extrapolating the Fourier transform of the object from a bounded region to the entire space, that is, in terms of performing extrapolation in frequency. This problem is ill-posed unless one assumes that the object has some additional structure. When the object is compactly supported, then it is well-known that its Fourier transform can be extended to the entire space. However, it is also well-known that this problem is severely ill-conditioned. In this work, we assume that the object is known to belong to a collection of compactly supported functions and, instead performing extrapolation in frequency to the entire space, we study the problem of extrapolating to a larger bounded set using dilations in frequency and a single Fourier multiplier. This is reminiscent of the refinement equation in multiresolution analysis. Under suitable conditions, we prove the existence of a worst-case optimal multiplier over the entire collection, and we show that all such multipliers share the same canonical structure. When the collection is finite, we show that any worst-case optimal multiplier can be represented in terms of an Hermitian matrix. This allows us to introduce a fixed-point iteration to find the optimal multiplier. This leads us to introduce a family of multipliers, which we call -multipliers, that can be used to perform extrapolation in frequency. We establish connections between -multipliers and multiresolution analysis. We conclude with some numerical experiments illustrating the practical consequences of our results.
... Concerning the notation we follow mainly the textbook of Gröchenig [9]. A more recent introduction to the topic is the 2 nd edition of Christensen's textbook [1]. ...
... In order to prove our main result, we will show that the lower frame bound vanishes under the assumptions of Theorem 1.1. For vol(Λ) −1 = 2 d and due to the fact that λ •′ − λ • ∈ Λ • , the series in Proposition 3.2 reduces to 1 It follows from the results in Tolimieri and Orr [18] that vol(Λ) −1 λ • ∈Λ • |Ag(λ • )| always is an upper bound, however, usually not the optimal upper bound. For g ∈ S0(R d ) this expression is always finite. ...
In this work we derive a simple argument which shows that Gabor systems consisting of odd functions of d variables and symplectic lattices of density cannot constitute a Gabor frame. In the 1--dimensional, separable case, this is a special case of a result proved by Lyubarskii and Nes, however, we use a different approach in this work exploiting the algebraic relation between the ambiguity function and the Wigner distribution as well as their relation given by the (symplectic) Fourier transform. Also, we do not need the assumption that the lattice is separable and, hence, new restrictions are added to the full frame set of odd functions.
... For music signals it is often more appropriate to use good time resolution for the onset of attack transients and good frequency resolution for the sinusoidal components. We will consider the task of time stretching in the framework of Gabor theory [13], [14]. Applying nonstationary Gabor frames (NSGFs) [15], [16] we extend the theory of the PV to incorporate TF representations with the above-mentioned adaptive TF resolution. ...
... If {g m,n } m,n spans C L , then it is called a Gabor frame. The associated frame operator S : C L → C L , defined by g m,n g m,n , ∀f ∈ C L , is invertible if and only if {g m,n } m,n is a Gabor frame[13]. If S is invertible, then we have the expansions g m,n g m,n , ∀f ∈ C L , ...
We propose a new algorithm for time stretching music signals based on the theory of nonstationary Gabor frames (NSGFs). The algorithm extends the techniques of the classical phase vocoder (PV) by incorporating adaptive time-frequency (TF) representations and adaptive phase locking. The adaptive TF representations imply good time resolution for the onsets of attack transients and good frequency resolution for the sinusoidal components. We estimate the phase values only at peak channels and the remaining phases are then locked to the values of the peaks in an adaptive manner. During attack transients we keep the stretch factor equal to one and we propose a new strategy for determining which channels are relevant for reinitializing the corresponding phase values. In contrast to previously published algorithms we use a non-uniform NSGF to obtain a low redundancy of the corresponding TF representation. We show that with just three times as many TF coefficients as signal samples, artifacts such as phasiness and transient smearing can be greatly reduced compared to the classical PV. The proposed algorithm is tested on both synthetic and real world signals and compared with state of the art algorithms in a reproducible manner.
... One of these is the frame theoretic characterization of a closed span of translations. This is our bailiwick here, and it has become a topic with great generalization, applicability, intricacy, and abstraction, and with a large number of contributors, see, e.g., [8], [13], [17], [16], [12], [11], [1], [2], [22] and the references therein. ...
... We refer to [14], [4], [13] for the theory of frames. ...
Frames of translates of f in L^2(G) are characterized in terms of the zero-set of the so-called spectral symbol of f in the setting of a locally compact abelian group G having a compact open subgroup H. We refer to such a G as a number theoretic group. This characterization was first proved in 1992 by Shidong Li and one of the authors for L^2(R^d) with the same formal statement of the characterization. For number theoretic groups, and these include local fields, the strategy of proof is necessarily entirely different; and it requires a new notion of translation that reduces to the usual definition in R^d.
... In the regular case many necessary and sufficient conditions on g, a and b are known (see e.g. Christensen (2003) and the references cited there). An early article by Gröchenig Gröchenig (1993) provided some partial sufficient conditions for the existence of irregular Gabor frames. ...
... We refer the reader to the the excellent textbooks Heil (2011) and Young (2001) for the definitions of frame and Riesz bases and preliminary results. See also Christensen (2003), Gröchenig (2001). ...
We prove stability results for a class of Gabor frames in . We consider window functions in the Sobolev spaces and B-splines of order . Our results can be used to describe the effect of the timing jitters in the p-order hold models of signal reconstruction.
... To prove this result, we use a result from Bakic and Beric about frame excess [3]. For more about frame excess and dual frames, see [8]. ...
... where F is finite and {g n } n∈F c is a Riesz basis. For details see Theorem 5.4.7 of [8]. Let S be the frame operator of {g n } n∈F c , and choose k so that since {S −1 g n } n∈F c is biorthogonal to {g n } n∈F c . ...
Frames in a Hilbert space that are generated by operator orbits are vastly studied because of the applications in dynamic sampling and signal recovery. We demonstrate in this paper a representation theory for frames generated by operator orbits that provides explicit constructions of the frame and the operator. It is known that the Kaczmarz algorithm for stationary sequences in Hilbert spaces generates a frame that arises from an operator orbit. In this paper, we show that every frame generated by operator orbits in any Hilbert space arises from the Kaczmarz algorithm. Furthermore, we show that the operators generating these frames are similar to rank one perturbations of unitary operators. After this, we describe a large class of operator orbit frames that arise from Fourier expansions for singular measures. Moreover, we classify all measures that possess frame-like Fourier expansions arising from two-sided operator orbit frames. Finally, we show that measures that possess frame-like Fourier expansions arising from two-sided operator orbits are weighted Lebesgue measure with weight satisfying a weak condition, even in the non-frame case. We also use these results to classify measures with other types of frame-like Fourier expansions.
... The window function of the dual frame is usually found either by expanding in a Neumann series (see [7], Ch. 3, § 3.2, and [10]) or by finite-dimensional reduction (see [11] and [12]). A detailed account of Gabor frames can be found in [13] and [14]. In the case when ...
... Definition 3 (see [7], Ch. 3, and [14], Ch. 1). Functions g k,m (x) ∈ L 2 (R), k, m ∈ Z, form a frame if there exist positive constants A F and B F such that, for all f ∈ L 2 (R), ...
Gabor frames generated by the Gaussian function are considered. The localization of the window functions of dual frames is estimated in terms of the uncertainty constants, it its dependence on the relation between the parameters of the time-frequency window and the degree of overcompleteness. It is shown that localization worsens rapidly with the increasing disproportion in the parameters of the window. On the other hand, the higher the system of functions forming the frame is overdetermined, the better the window function of the dual frame is localized. For a tight frame the localization of the window function with the same set of parameters is much better than that for the dual frame. This problem is closely related to the problem of interpolation by we have uniform shifts of the Gaussian function. Both the nodal interpolation function and the window function of the dual frame are constructed from the same coefficients. These coefficients play an important role also in the derivation of formulae for the uncertainty constants. This is why their properties related to sign alternation and the monotonicity of decrease of the absolute value are considered in the paper. Bibliography: 38 titles.
... This theorem provides us with simply formulated three sufficient conditions for an unbounded operator A with point spectrum to generate the Riesz basis of A-invariant subspaces. Riesz bases are main blocks of spectral approach in an infinite-dimensional linear and nonlinear systems theory and frequently appear in problems of mathematical physics as well as in modern signal processing, see, e.g., [1,3,4,5,8,9,12,17,22,23,24] and the references therein. ...
... In [2] N. Bari proved the second important step, that a basis, quadratically close to Riesz basis, is again a Riesz basis. For more concerning stability of bases we refer, e.g., to [1,2,3,4,6,9,10,11,12,14,15,23] and the references therein. ...
... As an example, a collection of vectors {x i } k i=1 satisfying the reconstruction formula x = k i=1 x, x i x i is called a Parserval frame. In a smilar fashion, given a frame {x i } k i=1 in a Hilbert space F n , a sequence {y i } k i=1 in F n is called a dual frame for {x i } k i=1 if it satisfies the reconstruction formula x = k i=1 x, y i x i for all x in F n , [1,2,3,9]. The problem of signal recovery from a partially erased data set of analysis coefficients was studied in [8], where the authors reconstruct an analyzed signal f via a procedure they called bridging. ...
In this paper we present the construction of an exact dual frame under specific structural assumptions posed on the dual frame. When given a frame F for a finite dimensional Hilbert space, and a set of vectors H that is assumed to be a subset of a dual frame of F, we answer the following question: Which dual frame G for F - if it exists - completes the given set H? Solutions are explored through a direct and an indirect approach, as well as via the singular value decomposition of the synthesis operator of F.
... Let be a frame in a Hilbert space À with frame bounds and . Then the frame operator Ë À é À is defined as Ë It is well-known that the operator Ë is bounded, self-adjoint, positive, and invertible (See [27]). Moreover, Ë ¶1 ...
In this paper, we investigate the problem of sampling and reconstruction in principal shift‐invariant spaces generated by Hilbert space‐valued functions. Given any signal f f and data point xk∈ℝ, the sample f(xk) is stored along a sequence of directions {νkm}m∈ℤ. Specifically, the inner products ⟨f(xk),νkm⟩ are stored. First, we define what we mean by a stable set of sampling and provide equivalent conditions for proving that a given set is a stable set of sampling. We then present a reconstruction formula for f f from its integer samples ⟨f(k),νkm⟩. Finally, we address the cases of perturbed and irregular sampling, examining their impact on the reconstruction process.
... See e.g. [3] or [13]. ...
Let be a set with positive and finite Lebesgue measure. Let be a lattice in with density dens. It is well-known that if M is a diagonal block matrix with diagonal matrices A and B, then is an orthonormal basis for if and only if K tiles both by and . However, there has not been any intensive study when M is not a diagonal matrix. We investigate this problem for a large class of important cases of M. In particular, if M is any lower block triangular matrix with diagonal matrices A and B, we prove that if is an orthonormal basis, then K can be written as a finite union of fundamental domains of and at the same time, as a finite union of fundamental domains of . If is an integer matrix, then there is only one common fundamental domain, which means K tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than one fundamental domains is also possible. We also provide a constructive way for forming a Gabor window functions for a given upper triangular lattice. Our study is related to a Fuglede's type problem in Gabor setting and we give a partial answer to this problem in the case of lattices.
... 1. Let φ ∈ W (L 1 )(R n ) be such that for some positive constants A 1 and B 1 , [7,Theorem 9.2.5]. In this case, we have F γ = φ(· − γ), γ ∈ Γ = Z n . ...
The paper is devoted to studying the stability of random sampling in a localized reproducing kernel space. We show that if the sampling set on (compact) discretizes the integral norm of simple functions up to a given error, then the sampling set is stable for the set of functions concentrated on . Moreover, we prove with an overwhelming probability that many random points uniformly distributed over yield a stable set of sampling for functions concentrated on .
... If B 1 = B 2 = 1, then we say that {ϕ n } n∈N is a Parseval frame. For the theory of frames and some of their applications, see e.g., [7], [6], [5] and the papers cited there. ...
We consider conditions on a given system of vectors in Hilbert space , forming a frame, which turn into a reproducing kernel Hilbert space. It is assumed that the vectors in are functions on some set . We then identify conditions on these functions which automatically give the structure of a reproducing kernel Hilbert space of functions on . We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.
... Let us first collect some of the known results concerning frame properties for continuous compactly supported functions; (i) is classical, and we refer to [2] for a proof. Proposition 1.1 Let N > 0, and assume that g : R → C is a continuous function with supp g ⊆ [− N 2 , N 2 ]. ...
We identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the known obstructions. Easy verifiable sufficient conditions for a function to belong to the class are derived, and it is shown that the B-splines as well as certain "continuous and truncated" versions of several classical functions (e.g., the Gaussian and the two-sided exponential function) belong to the class. The sufficient conditions for the frame property guarantees the existence of a dual window with a prescribed size of the support.
... Frames were first introduced in 1952 by Duffin and Schaeffer [34] and have become the subject of intense study since the 1980s. e.g., see [29], [16], [5], [25], [22]. (In fact, Paley and Wiener gave the technical definition of a frame in [67], but they only developed the completeness properties.) ...
Vector-valued discrete Fourier transforms (DFTs) and ambiguity functions are defined. The motivation for the definitions is to provide realistic modeling of multi-sensor environments in which a useful time-frequency analysis is essential. The definition of the DFT requires associated uncertainty principle inequalities. The definition of the ambiguity function requires a component that leads to formulating a mathematical theory in which two essential algebraic operations can be made compatible in a natural way. The theory is referred to as frame multiplication theory. These definitions, inequalities, and theory are interdependent, and they are the content of the paper with the centerpiece being frame multiplication theory. The technology underlying frame multiplication theory is the theory of frames, short time Fourier transforms (STFTs), and the representation theory of finite groups. The main results have the following form: frame multiplication exists if and only if the finite frames that arise in the theory are of a certain type, e.g., harmonic frames, or, more generally, group frames. In light of the complexities and the importance of the modeling of time-varying and dynamical systems in the context of effectively analyzing vector-valued multi-sensor environments, the theory of vector-valued DFTs and ambiguity functions must not only be mathematically meaningful, but it must have constructive implementable algorithms, and be computationally viable. This paper presents our vision for resolving these issues, in terms of a significant mathematical theory, and based on the goal of formulating and developing a useful vector-valued theory.
... [11], [12]). В [13], [14] приводится большой набор фактов о фреймах Габора. В серии статей А. Янссена [15]- [18] в случае ...
Рассматриваются фреймы Габора, порожденные функцией Гаусса.
С помощью констант неопределенности оценивается локализация функ-
ций двойственных фреймов в зависимости от соотношения параметров
частотно-временного окна и степени переполненности. Общий вывод та-
ков: при увеличении диспропорции окна локализация быстро ухудшается.
С другой стороны, чем более переопределена исходная система функций,
тем лучше локализованы функции двойственного фрейма. Для жестко-
го фрейма локализация при одном и том же наборе параметров сущест-
венно лучше, чем для двойственного фрейма. Рассматриваемая задача
тесно связана с задачей интерполяции по равномерным сдвигам функции
Гаусса. Построение узловой функции при интерполяции и функции окна
двойственного фрейма осуществляется с помощью одних и тех же коэффи-
циентов. Эти коэффициенты играют важную роль и при выводе формул
для констант неопределенности. Поэтому в работе изучаются их свойства,
связанные со знакочередуемостью и монотонностью убывания по модулю.
Библиография: 38 названий.
... y, z) ∶= exp − i <x,y,z> B (A.5)that due to Stokes Formula satisfies the following equality:Ω B (x, y, z) = Λ A (x, y) Λ A (y, z) Λ A (z, x). (A.6)We also recall the following theorem in[9] (Theorem 5.5.1.)Theorem B.3. ...
We consider periodic (pseudo)differential {elliptic operators of Schr\"odinger type} perturbed by weak magnetic fields not vanishing at infinity, and extend our previous analysis in \cite{CIP,CHP-2,CHP-4} to the case {of a semimetal having a finite family of Bloch eigenvalues whose range may overlap with the other Bloch bands but remains isolated at each fixed quasi-momentum.} We do not make any assumption of triviality for the associated Bloch bundle. In this setting, we formulate a general form of the Peierls-Onsager substitution {via strongly localized tight-frames and magnetic matrices. We also} prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control.
... Lemma 2.2. [15] Let H 1 , H 2 be two Hilbert spaces and T 1 ∈ B(H 1 , H 2 ), where R(T 1 ) is closed. Then, there exists T + 1 : H 2 → H 1 , the pseudo-inverse of T 1 , such that T 1 T + 1 x = x, ∀x ∈ R(T 1 ). ...
K K n K n K n K n $-Hilbert spaces is developed, and examples are given. By virtue of auxiliary operators, such as the preframe operator, analysis operator, and frame operator, some new properties and characterizations of these frames are presented, and several new methods for their construction are given. Stability and perturbation results are discussed and new inequalities are established as applications.
... Large body of work on frames for Hilbert spaces (see [14,22,27,28]) lead to the well developed theory of frames (known as Banach frames and X d -frames) for Banach spaces (see [7,8,18,24]) lead to the beginning of frames for metric spaces (known as metric frames) [35]. ...
... Such frames are called Gabor frames. For an introduction to the theory of frames, we refer to [Chr03]. ...
A fundamental result in pseudodifferential theory is the Calder\'on-Vaillancourt theorem, which states that a pseudodifferential operator defined from a H\"ormander symbol of order 0 defines a bounded operator on . In this work we prove an analog for pseudodifferential \emph{super} operator, \ie operators acting on other operators, in the presence of magnetic fields. More precisely, we show that magnetic pseudodifferential super operators of order 0 define bounded operators on the space of Hilbert-Schmidt operators . Our proof is inspired by the recent work of Cornean, Helffer and Purice and rests on a characterization of magnetic pseudodifferential super operators in terms of their "matrix element" computed with respect to a Parseval frame.
... Some references for frame theory are [8] and [10]. Since the theory in the L 2 -case is enough complex, we carefully analyze the range function J s acting on spaces V s = S s (A I,s ). ...
This paper has the characteristics of a review paper in which results of shift-invariant subspaces of Sobolev type are summarized without proofs. The structure of shift-invariant spaces , , generated by at most countable family of generators, which are subspaces of Sobolev spaces , are announced in \cite{aap} and Bessel sequences, frames and Riesz families of such spaces are characterized. With the Fourier multiplier , we are able to extend notions and theorems in \cite{MB} to spaces of the Sobolev type.
... Frames introduced by Duffin and Shaeffer [7], reintroduced in 1986 by Daubechies, Grossmann, and Meyer [6], have recently received great attention owing to their wide range of applications in both pure and applied mathematics, specially that it has been extensively used in many fields such as filter bank theory, signal and image processing, coding and communication [9] and other areas. We refer to [3][4][5]8] for an introduction to frame theory and its applications. ...
In this paper, we give a simple characterization of the eigenvalues and eigenvectors for the continous G-frame operator for {ΛωP∈B(H,Kω), ω∈Ω}, where HN is an N-dimensional Hilbert space and P is a rank k orthogonal projection on HN. Using this, we derive several results.
Using a contraction between the affine group and the Weyl-Heisenberg group, we derive several new sufficient conditions, with explicit frame bounds, for perturbation of Gabor frames and wavelet frames. Firstly, we give perturbation of Gabor frames with respect to the translation parameter, modulation parameter, and window functions. An equivalent criteria for frame conditions in terms of a given frame and its perturbed sequence is given. We give necessary and sufficient conditions for the existence of wavelet frames from the extended affine group. Sufficient conditions for perturbation of frames from the extended affine group are derived. Finally, we give an interplay between perturbation of frames from the Weyl-Heisenberg group and extended affine group.
Рассмотрены системы элементов гильбертова пространства, которые позволяют восстановить вектор по модулям его скалярных произведений с выбранными элементами. Обсуждена взаимосвязь задачи восстановления со свойствами альтернативной полноты и переполненности системы, а также возможность построения гильбертовых и банаховых фреймов с названными свойствами на основе воспроизводящего ядра пространства Харди.
In this work, a characterization in terms of weak observability inequalities is built up for output feedback stabilizability of linear periodic sampled-data control systems with a given decay rate. Based on this characterization, we give a verifiable condition to ensure the output feedback stabilizability of a class of periodic sampled-data control systems coupled by constant matrices. Finally, we provide several valuable examples as applications of our main result.
Based on finite-dimensional time-frequency analysis, we study the properties of time-frequency shift equivariant maps that are generally nonlinear. We first establish a one-to-one correspondence between Λ-equivariant maps and certain phase-homogeneous functions and also provide a reconstruction formula that expresses Λ-equivariant maps in terms of these phase-homogeneous functions, leading to a deeper understanding of the class of Λ-equivariant maps. Next, we consider the approximation of Λ-equivariant maps by neural networks. In the case where Λ is a cyclic subgroup of order N in ZN×ZN, we prove that every Λ-equivariant map can be approximated by a shallow neural network whose affine linear maps are simply linear combinations of time-frequency shifts by Λ. This aligns well with the proven suitability of convolutional neural networks (CNNs) in tasks requiring translation equivariance, particularly in image and signal processing applications.
We consider the problem of reconstructing a function given phase-less samples of its Gabor transform, which is defined by
More precisely, given sampling positions the task is to reconstruct f (up to global phase) from measurements . This non-linear inverse problem is known to suffer from severe ill-posedness. As for any other phase retrieval problem, constructive recovery is a notoriously delicate affair due to the lack of convexity. One of the fundamental insights in this line of research is that the connectivity of the measurements is both necessary and sufficient for reconstruction of phase information to be theoretically possible. In this article we propose a reconstruction algorithm which is based on solving two convex problems and, as such, amenable to numerical analysis. We show, empirically as well as analytically, that the scheme accurately reconstructs from noisy data within the connected regime. Moreover, to emphasize the practicability of the algorithm we argue that both convex problems can actually be reformulated as semi-definite programs for which efficient solvers are readily available. The approach is based on ideas from complex analysis, Gabor frame theory as well as matrix completion. As a byproduct, we also obtain improved truncation error for Gabor expensions with Gaussian generators.
A Zak transform is a effective method to investigate the theory of Gabor frames. Due to being not a group with usual addition, admits no usual Gabor frames and a usual Zak transform is not suitable for the study of -Gabor frames. However, is a group with “” addition. This paper addresses a class of multi-window Gabor systems associated with “” addition. Using a Zak transform associated with “” addition, we characterize “”-based Gabor frames (Riesz bases, orthonormal bases) and their (weak) Gabor dual frames. Also some examples are provided.
Quaternionic Hilbert spaces have important applications in quantum physics and the frame theory in quaternionic Hilbert spaces can handle many practical problems in physics. In this paper, we investigate oblique dual and g-dual frames in separable quaternionic Hilbert spaces. We provide a sufficient condition for commutativity of the oblique dual frames in quaternion Hilbert spaces. We present a parametric and algebraic formula for all oblique duals of any given frame in a closed subspace by utilizing a new (right) pre-frame operator. We prove that the canonical oblique dual has a minimum norm representation of elements in . We provide a method for constructing an oblique dual frame from two Bessel sequences. We show that g-duality relation is symmetric and get a necessary condition for g-duality of the sum of two g-dual frames. We also present a parametric expression of all g-dual frames of any given frame and show that approximate dual frame pairs form g-dual frames. We study perturbation-stability of g-dual frames.
In this article, we study the construction of norm retrievable frames that have a dynamical sampling structure. For a closed subspace W of , we show that when the collection of subspaces is norm retrievable in for a unitary or Jordan operator A, then there always exists a collection of norm retrievable frame vectors that have a dynamical sampling structure in .
Thegeneralizedtranslationinvariant(GTI)systemsunifythediscreteframetheory of generalized shift-invariant systems with its continuous version, such as wavelets, shearlets, Gabor transforms, and others. This article provides sufficient conditions to construct pairwise orthogonal Parseval GTI frames in L2(G) satisfying the local integrability condition (LIC) and having the Calder ́on sum one, where G is a second countable locally compact abelian group. The pairwise orthogonality plays a crucial role in multiple access communications, hiding data, synthesizing superframes and frames, etc. Further, we propose algorithms for constructing N numbers of GTI Parseval frames, which are pairwise orthogonal. Consequently, we obtain an explicit construction of pairwise orthogonal Parseval frames in L2(R) and L2(G), using B-splines as a generating function. In the end, the results are particularly discussed for wavelet systems.
In this paper, we introduce a new concept of K-biframes for Hilbert spaces. We then examine several characterizations with the assistance of a biframe operator. Moreover , we investigate their properties from the perspective of operator theory by establishing various relationships and properties.
We have developed new methods for constructing exponential Riesz bases by combining existing ones. These methods involve taking unions of frequency sets and domains respectively, offering easier construction compared to known techniques. Along with examples illustrating our methods, we also provide several examples that highlight the intricate nature of exponential Riesz bases.
In this article, fractional Gabor frames in L2ℝ are defined and studied. Also, a finite linear combination of fractional Gabor frames for L2ℝ is discussed. It is proved that under certain conditions, a finite linear combination of fractional Gabor frames is a fractional Gabor frame for L2ℝ. Finally, the stability of fractional Gabor frames is discussed, and some results for the stability of fractional Gabor frames are proved.
A characterization is obtained for the system of generalized twisted translates , which is associated with a certain matrix pair X=(A,B), to be a Parseval frame. This characterization is also extended to a dual frame pair consisting of generalized twisted translates. Further, it is proved that if is a Bessel sequence, then there exists a countable collection of generalized twisted translates such that is a tight frame for .
In this paper, we consider a perturbation of operator-valued frames (OPV-frames) and obtain conditions for their stability in terms of operators associated with the OPV-frames. Also, some duality relations of OPV-frames are discussed. Finally, some properties of the duals of OPV-frames are proven.
G?vruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely ?-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator?. For a locally compact abelian groupGand a positive integer n, westudy frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space L2(G,Cn?n) , where a bounded linear operator ? on L2(G,Cn?n) controls not only lower but also the upper frame condition. We term such frames matrix-valued (?,?*)-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued (?,?*)- Gabor frames in terms of hyponormal operators. It is shown that if ? is adjointable hyponormal operator, then L2(G,Cn?n) admits a ?-tight (?,?*)-Gabor frame for every positive real number ?. A characterization of matrix-valued (?,?*)-Gabor frames is given. Finally, we show that matrix-valued (?,?*)-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.
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