Deguang Han | University of Central Florida (original) (raw)

Papers by Deguang Han

Applied and Computational Harmonic Analysis, Sep 1, 2004

Linear Algebra and its Applications, May 1, 2012

Journal of Fourier Analysis and Applications, Nov 1, 2005

arXiv (Cornell University), Mar 7, 2021

Contemporary mathematics, 2004

Acta Applicandae Mathematicae, Apr 4, 2019

Proceedings of SPIE, Dec 4, 2000

Transactions of the American Mathematical Society, Jan 30, 2008

arXiv (Cornell University), Sep 21, 2018

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.

Applied and Computational Harmonic Analysis, 2018

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.

Proceedings of SPIE, Dec 5, 2001

ABSTRACT

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.

Linear Algebra and its Applications, May 1, 2019

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.

Science in China, Dec 1, 2010

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.

arXiv (Cornell University), Jan 31, 2020

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.

arXiv (Cornell University), Jun 23, 2017

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.

arXiv (Cornell University), Aug 25, 2011

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.

Annals of Functional Analysis, Aug 5, 2022

Applied and Computational Harmonic Analysis, Nov 1, 2013

Lemma Let ψ 1 , • • • , ψ m ∈ L 2 (R). Then (ψ 1 , • • • , ψ m) is a Parseval multi-wavelet frame... more Lemma Let ψ 1 , • • • , ψ m ∈ L 2 (R). Then (ψ 1 , • • • , ψ m) is a Parseval multi-wavelet frame for L 2 (R) if and only if (i) m i=1 j∈Z | ψ i (2 j s)| 2 = 1/(2π) a.e. (ii) m i=1 ∞ j=0 ψ i (2 j s) ψ i (2 j (s + 2πq)) = 0 a.e. ∀q ∈ 2Z + 1.