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Deguang Han

Publications and source records attributed to Deguang Han.

At least 37 records · Page 2Linked to original sources
arXiv2017

Frame Phase-retrievability and Exact phase-retrievable frames

An exact phase-retrievable frame $\{f_{i}\}_{i}^{N}$ 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 $2n-1\leq N\leq n(n+1)/2$. 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 $|supp(x)| \geq k$ for every nonzero vector $x\in M$. Moreover, if $1\leq k< [(n+1)/2]$, then a $k$-dimensional PR-subspace is maximal if and only if there exists a vector $x\in M$ such that $|supp(x) | = k$.

math.FA↗
arXiv2016

Phase retrieval from multiple-window short-time Fourier measurements

In this paper, we introduce two symmetric directed graphs depending on supports of signals and windows, and we show that the connectivity of those graphs provides either necessary and sufficient conditions to phase retrieval of a signal from magnitude measurements of its multiple-window short-time Fourier transform. Also we propose an algebraic reconstruction algorithm, and provide an error estimate to our algorithm when magnitude measurements are corrupted by deterministic/random noises.

cs.IT↗
arXiv2015

Some Structural Properties of Homomorphism Dilation Systems for Linear Maps

Inspired by some recent development on the theory about projection valued dilations for operator valued measures or more generally bounded homomorphism dilations for bounded linear maps on Banach algebras, we explore a pure algebraic version of the dilation theory for linear systems acting on unital algebras and vector spaces. By introducing two natural dilation structures, namely the canonical and the universal dilation systems, we prove that every linearly minimal dilation is equivalent to a reduced homomorphism dilation of the universal dilation, and all the linearly minimal homomorphism dilations can be classified by the associated reduced subspaces contained in the kernel of synthesis operator for the universal dilation.

math.OA↗
arXiv2014

Dilations of frames, operator valued measures and bounded linear maps

We will give an outline of the main results in our recent AMS Memoir, and include some new results, exposition and open problems. In that memoir we developed a general dilation theory for operator valued measures acting on Banach spaces where operator-valued measures (or maps) are not necessarily completely bounded. The main results state that any operator-valued measure, not necessarily completely bounded, always has a dilation to a projection-valued measure acting on a Banach space, and every bounded linear map, again not necessarily completely bounded, on a Banach algebra has a bounded homomorphism dilation acting on a Banach space. Here the dilation space often needs to be a Banach space even if the underlying space is a Hilbert space, and the projections are idempotents that are not necessarily self-adjoint. These results lead to some new connections between frame theory and operator algebras, and some of them can be considered as part of the investigation about "noncommutative" frame theory.

math.FA↗
arXiv2014

Dilations for Systems of Imprimitivity acting on Banach Spaces

Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in general can not be taken as a Hilbert space. However, it can be taken as a Hilbert space for positive operator valued systems of imprimitivity. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.

math.FA↗
arXiv2013

On common fundamental domains

We find conditions under which two measure preserving actions of two groups on the same space have a common fundamental domain. Our results apply to commuting actions with separate fundamental domains, lattices in groups of polynomial growth, and some semidirect products. We prove that two lattices of equal co-volume in a group of polynomial growth, one acting on the left, the other on the right, have a common fundamental domain.

math.FA↗
arXiv2012

Divergence of mock and scrambled Fourier series

We study divergence properties of Fourier series on Cantor-type fractal measures, also called mock Fourier series. We show that in some cases the $L^1$-norm of the corresponding Dirichlet kernel grows exponentially fast, and therefore the Fourier series are not even pointwise convergent. We apply these results to the Lebesgue measure to show that a certain rearrangement of the exponential functions, which we call scrambled Fourier series, have a corresponding Dirichlet kernel whose $L^1$-norm grows exponentially fast, which is much worse than the known logarithmic bound. The divergence properties are related to the Mahler measure of certain polynomials and to spectral properties of Ruelle operators.

math.FA↗
arXiv2012

Operator-Valued Measures, Dilations, and the Theory of Frames

We develop elements of a general dilation theory for operator-valued measures and bounded linear maps between operator algebras that are not necessarily completely-bounded. We prove our main results by extending and generalizing some known results from the theory of frames and framings.

math.OA↗
arXiv2012

Continuous and Discrete Fourier Frames for Fractal Measures

Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on $\br^d$, as extensions of the notions of Bessel and frame spectra that correspond to bases of exponential functions. Not every finite compactly supported Borel measure admits frame measures. We present a general way of constructing Bessel/frame measures for a given measure. The idea is that if a convolution of two measures admits a Bessel measure then one can use the Fourier transform of one of the measures in the convolution as a weight for the Bessel measure to obtain a Bessel measure for the other measure in the convolution. The same is true for frame measures, but with certain restrictions. We investigate some general properties of frame measures and their Beurling dimensions. In particular we show that the Beurling dimension is invariant under convolution (with a probability measure) and under a certain type of discretization. Moreover, if a measure admits a frame measure then it admits an atomic one, and hence a weighted Fourier frame. We also construct some examples of frame measures for self-similar measures.

math.FA↗
arXiv2012

Iterative approximations of exponential bases on fractal measures

For some fractal measures it is a very difficult problem in general to prove the existence of spectrum (respectively, frame, Riesz and Bessel spectrum). In fact there are examples of extremely sparse sets that are not even Bessel spectra. In this paper we investigate this problem for general fractal measures induced by iterated function systems (IFS). We prove some existence results of spectra associated with Hadamard pairs. We also obtain some characterizations of Bessel spectrum in terms of finite matrices for affine IFS measures, and one sufficient condition of frame spectrum in the case that the affine IFS has no overlap.

math.FA↗
arXiv2010

Bessel sequences of exponentials on fractal measures

Jorgensen and Pedersen have proven that a certain fractal measure $ν$ has no infinite set of complex exponentials which form an orthonormal set in $L^2(ν)$. We prove that any fractal measure $μ$ obtained from an affine iterated function system possesses a sequence of complex exponentials which forms a Riesz basic sequence, or more generally a Bessel sequence, in $L^2(μ)$ such that the frequencies have positive Beurling dimension.

math.FA↗
arXiv2010

On the Beurling dimension of exponential frames

We study Fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems. We prove that, under a mild technical condition, the Beurling dimension of a Fourier frame coincides with the Hausdorff dimension of the fractal.

math.FA↗
arXiv2009

Orthogonal exponentials, translations, and Bohr completions

We are concerned with an harmonic analysis in Hilbert spaces $L^2(μ)$, where $μ$ is a probability measure on $\br^n$. The unifying question is the presence of families of orthogonal (complex) exponentials $e_λ(x) = \exp(2πi λx)$ in $L^2(μ)$. This question in turn is connected to the existence of a natural embedding of $L^2(μ)$ into an $L^2$-space of Bohr almost periodic functions on $\br^n$. In particular we explore when $L^2(μ)$ contains an orthogonal basis of $e_λ$ functions, for $λ$ in a suitable discrete subset in $\br^n$; i.e, when the measure $μ$ is spectral. We give a new characterization of finite spectral sets in terms of the existence of a group of local translation. We also consider measures $μ$ that arise as fixed points (in the sense of Hutchinson) of iterated function systems (IFSs), and we specialize to the case when the function system in the IFS consists of affine and contractive mappings in $\br^n$. We show in this case that if $μ$ is then assumed spectral then its partitions induced by the IFS at hand have zero overlap measured in $μ$. This solves part of the Łaba-Wang conjecture. As an application of the new non-overlap result, we solve the spectral-pair problem for Bernoulli convolutions advancing in this way a theorem of Ka-Sing Lau. In addition we present a new perspective on spectral measures and orthogonal Fourier exponentials via the Bohr compactification.

math.FA↗
arXiv2009

A Duality principle for groups

The duality principle for Gabor frames states that a Gabor sequence obtained by a time-frequency lattice is a frame for $L^{2}(\R^{d})$ if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for $\rm II_{1}$ factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when $G$ is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well known problem about the classification of free group von Neumann algebras.

math.FA↗
arXiv2008

Parseval frames for ICC groups

We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of norm $\frac{1}{\sqrt{N}}$, the Hilbert space is spanned by an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by an Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left regular representation of a free group has this property.

math.FA↗
arXiv2008

On the spectra of a Cantor measure

We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75,1998, pp 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in $\{0, 1, 2, 3\}$) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis.

math.FA↗