Let L and K be two full rank lattices in R d . We prove that if v(L) = v(K), i.e. they have the s... more Let L and K be two full rank lattices in R d . We prove that if v(L) = v(K), i.e. they have the same volume, then there exists a measurable set Ω such that it tiles R d by both L and K. A counterexample shows that the above tiling result is false for three or more lattices. Furthermore, we prove that if v(L) ≤ v(K) then there exists a measurable set Ω such that it tiles by L and packs by K. Using these tiling results we answer a well known question on the density property of Weyl-Heisenberg frames.
Applied and Computational Harmonic Analysis, Nov 1, 2021
While frequency-resolved optical gating (FROG) is widely used in characterizing the ultrafast pul... more While frequency-resolved optical gating (FROG) is widely used in characterizing the ultrafast pulse in optics, analytic signals are often considered in timefrequency analysis and signal processing, especially when extracting instantaneous features of events. In this paper we examine the phase retrieval (PR) problem of analytic signals in C N by their FROG measurements. After establishing the ambiguity of the FROG-PR of analytic signals, we found that the FROG-PR of analytic signals of even lengths is different from that of analytic signals of odd lengths, and it is also different from the case of B-bandlimited signals with B ≤ N/2. The existing approach to bandlimited signals can be applied to analytic signals of odd lengths, but it does not apply to the even length case. With the help of two relaxed FROG-PR problems and a translation technique, we develop an approach to FROG-PR for the analytic signals of even lengths, and prove that in this case the generic analytic signals can be uniquely (up to the ambiguity) determined by their (3N/2 + 1) FROG measurements.
Applied and Computational Harmonic Analysis, Sep 1, 2004
From the Weyl-Heisenberg (WH) density theorem, it follows that a WH-frame (g mα,nβ ) m,n∈Z for L ... more From the Weyl-Heisenberg (WH) density theorem, it follows that a WH-frame (g mα,nβ ) m,n∈Z for L 2 (R) has a unique WH-dual if and only if αβ = 1. However, the same argument does not apply to the subspace WH-frame case and it is not clear how to use standard methods of Fourier analysis to deal with this situation. In this paper, we apply operator algebra theory to obtain a very simple necessary and sufficient condition for a given frame (induced by a projective unitary representation of a discrete group) to admit a unique dual (induced by the same system). As a special case, we obtain a characterization for the subspace WH-frames that have unique WH-duals (within the subspace). Using this characterization and the Zak transform, we are able to prove that if (g mα,nβ ) m,n∈Z is a WH-frame for a subspace M of L 2 (R), then, (i) (g mα,nβ ) m,n∈Z has a unique WH-dual in M when αβ is an integer; (ii) if αβ is irrational, then (g mα,nβ ) m,n∈Z has a unique WH-dual in M if and only if (g mα,nβ ) m,n∈Z is a Riesz sequence; (iii) if αβ < 1, then the WH-dual for (g mα,nβ ) m,n∈Z in M is not unique.
We define a new numerical range of an n × n complex matrix in terms of correlation matrices and d... more We define a new numerical range of an n × n complex matrix in terms of correlation matrices and develop some of its properties. We also define a related numerical range that arises from Alain Connes' famous embedding problem.
Journal of Fourier Analysis and Applications, Nov 1, 2005
A wavelet frame is called decomposable whenever it is equivalent to a superwavelet frame of lengt... more A wavelet frame is called decomposable whenever it is equivalent to a superwavelet frame of length greater than one. Decomposable wavelet frames are closely related to some problems on super-wavelets. In this article we first obtain some necessary or sufficient conditions for decomposable Parseval wavelet frames. As an application of these conditions, we prove that for each n > 1 there exists a Parseval wavelet frame which is m-decomposable for any 1 < m ≤ n, but not k-decomposable for any k > n. Moreover, there exists a super-wavelet whose components are non-decomposable. Similarly we also prove that for each n > 1, there exists a Parseval wavelet frame that can be extended to a super-wavelet of length m for any 1 < m ≤ n, but can not be extended to any super-wavelet of length k with k > n. The connection between decomposable Parseval wavelet frames and super-wavelets is investigated, and some necessary or sufficient conditions for extendable Parseval wavelet frames are given. k 2 ψ 2 k t -: k, ∈ Z is an orthonormal basis for L 2 (R). For notational convenience, let T and D be the translation and dilation unitary operators on L 2 (R), respectively, such that Tf (t) = f (T -1), Df (t) = Math Subject Classifications. 42C15, 46C05.
While frequency-resolved optical gating (FROG) is widely used in characterizing the ultrafast pul... more While frequency-resolved optical gating (FROG) is widely used in characterizing the ultrafast pulse in optics, analytic signals are often considered in timefrequency analysis and signal processing, especially when extracting instantaneous features of events. In this paper we examine the phase retrieval (PR) problem of analytic signals in C N by their FROG measurements. After establishing the ambiguity of the FROG-PR of analytic signals, we found that the FROG-PR of analytic signals of even lengths is different from that of analytic signals of odd lengths, and it is also different from the case of B-bandlimited signals with B ≤ N/2. The existing approach to bandlimited signals can be applied to analytic signals of odd lengths, but it does not apply to the even length case. With the help of two relaxed FROG-PR problems and a translation technique, we develop an approach to FROG-PR for the analytic signals of even lengths, and prove that in this case the generic analytic signals can be uniquely (up to the ambiguity) determined by their (3N/2 + 1) FROG measurements.
For an arbitrary full rank lattice Λ in R 2d and a function g ∈ L 2 (R d ) the Gabor (or Weyl-Hei... more For an arbitrary full rank lattice Λ in R 2d and a function g ∈ L 2 (R d ) the Gabor (or Weyl-Heisenberg) system is G(Λ, g) := {e 2πi ,x g(xκ) ˛(κ, ) ∈ Λ}. It is well-known that a necessary condition for G(Λ, g) to be an orthonormal basis for L 2 (R d ) is that the density of Λ has D(Λ) = 1. However, except for symplectic lattices it remains an unsolved question whether D(Λ) = 1 is sufficient for the existence of a g ∈ L 2 (R d ) such that G(Λ, g) is an orthonormal basis. We investigate this problem and prove that this is true for some of the important cases. In particular we show that this is true for Λ = M Z d where M is either a block triangular matrix or any rational matrix with | det M | = 1. Moreover, if M is rational we prove that there exists a compactly supported g such that G(Λ, g) is an orthonormal basis. We also obtain similar results for Gabor frames when D(Λ) ≥ 1. j |c j | 2
Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) tha... more Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure F : Ω → B(H ) has an integral representation of the form for some weakly measurable maps G k (1 ≤ k ≤ m) from a measurable space Ω to a Hilbert space H and some positive measure μ on Ω. Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.
We discuss three applications of operator algebra techniques in Gabor analysis: the parametrizati... more We discuss three applications of operator algebra techniques in Gabor analysis: the parametrizations of Gabor frames, the incompleteness property, and the unique Gabor dual problem for subspace Gabor frames.
Transactions of the American Mathematical Society, Jan 30, 2008
Let {x n } be a frame for a Hilbert space H. We investigate the conditions under which there exis... more Let {x n } be a frame for a Hilbert space H. We investigate the conditions under which there exists a dual frame for {x n } which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether {x n } can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is obtained in terms of the frame excess. For a frame {π(g)ξ : g ∈ G} induced by a projective unitary representation π of a group G, it is possible that {π(g)ξ : g ∈ G} can have a Parseval dual, but does not have a Parseval dual of the same type. The primary aim of this paper is to present a complete characterization for all the projective unitary representations π such that every frame {π(g)ξ : g ∈ G} (with a necessary lower frame bound condition) has a Parseval dual of the same type. As an application of this characterization together with a result about lattice tiling, we prove that every Gabor frame G(g, L, K) (again with the same necessary lower frame bound condition) has a Parseval dual of the same type if and only if the volume of the fundamental domain of L × K is less than or equal to 1 2 .
Motivated by the research on sampling problems for a union of subspaces (UoS), we investigate in ... more Motivated by the research on sampling problems for a union of subspaces (UoS), we investigate in this paper the phase-retrieval problem for the signals that are residing in a union of (finitely generated) cones (UoC for short) in R n. We propose a two-step PR-scheme: PR = detection + recovery. We first establish a sufficient and necessary condition for the detectability of a UoC, and then design a detection algorithm that allows us to determine the cone where the target signal is residing. The phase-retrieval will be then performed within the detected cone, which can be achieved by using at most Γ-number of measurements and with very low complexity, where Γ(≤ n) is the maximum of the ranks of the generators for the UoC. Numerical experiments are provided to demonstrate the efficiency of our approach, and to exhibit comparisons with some existing phase-retrieval methods.
In this paper, we study the feasibility and stability of recovering signals in finite-dimensional... more In this paper, we study the feasibility and stability of recovering signals in finite-dimensional spaces from unordered partial frame coefficients. We prove that with an almost self-located robust frame, any signal except from a Lebesgue measure zero subset can be recovered from its unordered partial frame coefficients. However, the recovery is not necessarily stable with almost self-located robust frames. We propose a new class of frames, namely self-located robust frames, that ensures stable recovery for any input signal with unordered partial frame coefficients. In particular, the recovery is exact whenever the received unordered partial frame coefficients are noise-free. We also present some characterizations and constructions for (almost) selflocated robust frames. Based on these characterizations and construction algorithms, we prove that any randomly generated frame is almost surely self-located robust. Moreover, frames generated with cube roots of different prime numbers are also self-located robust.
Journal of Mathematical Analysis and Applications, Jun 1, 2018
It is known that the Naimark complementary frames for a given frame are not necessarily unique up... more It is known that the Naimark complementary frames for a given frame are not necessarily unique up to the similarity. In this paper we introduce the concept of joint complementary frame pairs for a given dual frame pair, and prove that they are unique up to the joint similarity. As an application, we give a necessary and sufficient condition under which two Naimark complementary frames are similar. For different pairs of dual frames, we present an operator parameterization for their joint complementary frame pairs.
Proceedings of the American Mathematical Society, Sep 29, 2004
We introduce the concept of the modular function for a shiftinvariant subspace that can be repres... more We introduce the concept of the modular function for a shiftinvariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix A and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.
Abstract In this paper, we study ( G , α ) -frames for finite dimensional Hilbert spaces, where G... more Abstract In this paper, we study ( G , α ) -frames for finite dimensional Hilbert spaces, where G is a finite group and α is a unitary Schur multiplier of G. We apply the characterizations for tight ( G , α ) -frames and central ( G , α ) -frames to investigation of ( G , α ) -frames that have the maximal spanning property. We apply the theory of twisted group frames to obtain a criterion for maximal spanning vectors that generalizes the main result of [16, Theorem 1.7] . Moreover, we prove that a projective representation does not admit any maximal spanning vector if it has an at least 2-dimensional subrepresentation that is equivalent to a central projection induced subrepresentation of the α-regular representation.
For a time-frequency lattice Λ = AZ d × BZ d , it is known that an orthonormal super Gabor frame ... more For a time-frequency lattice Λ = AZ d × BZ d , it is known that an orthonormal super Gabor frame of length L exists with respect to this lattice if and only if |det(AB)| = 1 L. The proof of this result involves various techniques from multi-lattice tiling and operator algebra theory, and it is far from constructive. In this paper we present a very general scheme for constructing super Gabor frames for the rational lattice case. Our method is based on partitioning an arbitrary fundamental domain of the lattice in the frequency domain such that each subset in the partition tiles R d by the lattice in the time domain. This approach not only provides us a simple algorithm of constructing various kinds of orthonormal super Gabor frames with flexible length and multiplicity, but also allows us to construct super Gabor (non-Riesz) frames with high order smoothness and regularity. Several examples are also presented.
The Sobolev space H ς (R d), where ς > d/2, is an important function space that has many applicat... more The Sobolev space H ς (R d), where ς > d/2, is an important function space that has many applications in various areas of research. Attributed to the inertia of a measurement instrument, it is desirable in sampling theory to recover a function by its nonuniform sampling. In the present paper, based on dual framelet systems for the Sobolev space pair (H s (R d), H −s (R d)), where d/2 < s < ς, we investigate the problem of constructing the approximations to all the functions in H ς (R d) by nonuniform sampling. We first establish the convergence rate of the framelet series in (H s (R d), H −s (R d)), and then construct the framelet approximation operator that acts on the entire space H ς (R d). We examine the stability property for the framelet approximation operator with respect to the perturbations of shift parameters, and obtain an estimate bound for the perturbation error. Our result shows that under the condition d/2 < s < ς, the approximation operator is robust to shift perturbations. Motivated by some recent work on nonuniform sampling and approximation in Sobolev space (e.g., [20]), we don't require the perturbation sequence to be in ℓ α (Z d). Our results allow us to establish the approximation for every function in H ς (R d) by nonuniform sampling. In particular, the approximation error is robust to the jittering of the samples.
An exact phase-retrievable frame {fi} N i for an n-dimensional Hilbert space is a phase-retrievab... more An exact phase-retrievable frame {fi} N i for an n-dimensional Hilbert space is a phase-retrievable frame that fails to be phase-retrievable if any one element is removed from the frame. Such a frame could have different lengths. We shall prove that for the real Hilbert space case, exact phase-retrievable frame of length N exists for every 2n − 1 ≤ N ≤ n(n+1)/2. For arbitrary frames we introduce the concept of redundancy with respect to its phase-retrievability and the concept of frames with exact PR-redundancy. We investigate the phase-retrievability by studying its maximal phase-retrievable subspaces with respect to a given frame which is not necessarily phase-retrievable. These maximal PR-subspaces could have different dimensions. We are able to identify the one with the largest dimension, which can be considered as a generalization of the characterization for phase-retrievable frames. In the basis case, we prove that if M is a k-dimensional PR-subspace, then |supp(x)| ≥ k for every nonzero vector x ∈ M. Moreover, if 1 ≤ k < [(n + 1)/2], then a k-dimensional PR-subspace is maximal if and only if there exists a vector x ∈ M such that |supp(x)| = k.
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