In this paper, we deal mainly with a class of column upper-plus-lower (CUPL) Toeplitz matrices without Toeplitz structure, which are “close” to the Toeplitz matrices in the sense that their ...(−1,1)-cyclic displacements coincide with cyclic displacement of some Toeplitz matrices. By constructing the corresponding displacement of the matrices, we derive the formulas on representation of the inverses of the CUPL Toeplitz matrices in the form of sums of products of factor (1, 1)-circulants and (−1,−1)-circulants. Furthermore, through the relation between the CUPL Toeplitz matrices and the CUPL Hankel matrices, the inverses of the CUPL Hankel matrices can be obtained as well.
A Helson matrix (also known as a multiplicative Hankel matrix) is an infinite matrix with entries {a(jk)} for j,k≥1. Here the (j,k)'th term depends on the product jk. We study a self-adjoint Helson ...matrix for a particular sequence a(j)=(jlogj(loglogj)α))−1, j≥3, where α>0, and prove that it is compact and that its eigenvalues obey the asymptotics λn∼ϰ(α)/nα as n→∞, with an explicit constant ϰ(α). We also establish some intermediate results (of an independent interest) which give a connection between the spectral properties of a Helson matrix and those of its continuous analogue, which we call the integral Helson operator.
Let zj:=efj with fj∈(-∞,0+i-π,π) be distinct nodes for j=1,…,M. With complex coefficients cj≠0, we consider a nonincreasing exponential sum h(x):=c1ef1x+⋯+cMefMx(x⩾0). Many applications in electrical ...engineering, signal processing, and mathematical physics lead to the following problem: Determine all parameters of h, if 2,N sampled values h(k) (k=0,…,2N-1; N⩾M) are given. This parameter estimation problem is a nonlinear inverse problem. For noiseless sampled data, we describe the close connections between Prony-like methods, namely the classical Prony method, the matrix pencil method, and the ESPRIT method. Further we present a new efficient algorithm of matrix pencil factorization based on QR decomposition of a rectangular Hankel matrix. The algorithms of parameter estimation are also applied to sparse Fourier approximation and nonlinear approximation.
A three-parameter family
of weighted Hankel matrices is introduced with the entries
, supposing
,
,
are positive and
,
,
. The famous Hilbert matrix is included as a particular case. The direct sum
...is shown to commute with a discrete analogue of the dilatation operator. It follows that there exists a three-parameter family of real symmetric Jacobi matrices,
, commuting with
. The orthogonal polynomials associated with
turn out to be the continuous dual Hahn polynomials. Consequently, a unitary mapping
diagonalizing
can be constructed explicitly. At the same time,
diagonalizes
and the spectrum of this matrix operator is shown to be purely absolutely continuous and filling the interval
where
is known explicitly. If the assumption
is relaxed while the remaining inequalities on
,
,
are all supposed to be valid, the spectrum contains also a finite discrete part lying above the threshold
. Again, all eigenvalues and eigenvectors are described explicitly.
Display omitted
•The reconstruction exploits the low rank constraint and the parallel computing technique.•The proposed approach uses a sliding window algorithm to improve the reconstruction ...process.•The proposed method outperforms the typical low rank methods in time performance.•The proposed algorithm is suitable for small computing platforms.
Nuclear Magnetic Resonance (NMR) spectroscopy is one of the most promising analytical chemistry techniques, although it takes a long time to acquire data. Non-uniform sampling (NUS) is an effective way to reduce the sampling time, but faithful reconstruction methods are needed. The low rank Hankel matrix (LRHM) approach uses the low rank constraint to obtain high-quality spectra from NUS signals, but the reconstruction has a considerable time overhead. In this work, we propose a sliding window based low rank Hankel matrix approach to speed up the spectra reconstruction from NUS signals. Using the sliding window to construct a matrix can effectively reduce the size of the Hankel matrix for faster reconstructions. To further decrease the reconstruction time, parallel computation is applied in the proposed approach. The experiments on both synthetic data and realistic data demonstrate that the reconstruction speed of the proposed method is the fastest among compared methods without sacrificing the quality of spectra.
The core of this work is to understand the structural properties of certain coordinate sections of the generic square Hankel matrix over a field k of characteristic zero. In particular, one looks ...closely at the Jacobian ideal of the determinant, its Hessian and the ideal of submaximal minors of the matrix. In contrast to coordinate sections of the fully generic square matrix and the symmetric generic matrix, here the Hessian is nonzero, hence the related polar map is dominant. In the case of the fully generic Hankel matrix one establishes some of the conjectures stated in a previous paper by the second and fourth authors. A major question in this connection is whether the Jacobian ideal is of linear type, a result that would trigger similar results for the coordinate sections considered here. At the moment this is only known for the so-named sub-Hankel coordinate section.
2D magnetic resonance spectroscopy (MRS) is extensively used to analyze the components, structures, and interactions of substances in chemistry and bioengineering. To reduce data acquisition time, ...the spatiotemporally encoded ultrafast (STEU) MRS employs a fast means to acquire data in the hybrid time and frequency (HTF) plane, but relies on non-uniform sampling (NUS) technique. After that, a proper reconstruction method is essential to recover a high quality 2D MRS from undersampled HTF data. In this work, each column and row of 2D MRS are converted into Hankel matrices, by which a simultaneous low rank regularization model is proposed to exploit the bivariate exponential structure of 2D MRS signal. Additionally, a non-convex surrogate function for rank is integrated in the proposed model to more precisely enforce the low rank property of Hankel matrices. To ensure computational accuracy and convergence, an efficient numerical algorithm is further deduced based on alternating direction method of multipliers (ADMM) iteration. The experimental results have shown that the proposed method significantly outperforms existing 2D MRS reconstruction methods, and presents a strong robustness to low sampling rates, various sampling patterns, and measurement noise with a favorable complexity.
This paper presents a robust and efficient method for recovering missing data in subNyquist synthetic aperture radar (SAR) using matrix completion (MC) techniques. Unlike previous MC-based methods ...that have limitations on the required data properties, such as broadside-looking mode, sparse target scenarios, distributed missing, or low missing data ratios (MDR), our method is capable of handling consecutively clustered missing data and high MDR, even in the presence of weakened spectrum sparsity caused by squint-looking mode or dense target scenarios. Our approach involves arranging the incomplete SAR echo data from a single range-time sampling instant into a Hankel structured matrix and recovering it using nuclear-norm based convex relaxation and an augmented Lagrangian multiplier (ALM) method. To address computational complexity, we propose segmenting the SAR echo data into smaller slices and provide guidelines for selecting an appropriate slice size. We demonstrate the effectiveness and accuracy of our method through extensive experiments using simulated data and realistic Radarsat-1 data. The experiments cover various challenging conditions, including squint-looking mode, dense target scenarios, wide-swath coverage, high MDR, and clustered missing. The results highlight the capabilities of our method in handling subNyquist SAR data recovery, even in challenging scenarios. Our proposed method offers a robust and efficient solution for recovering missing SAR data.
PDF
