Some identities for determinants of structured matrices

A Widom like formula for some Toeplitz plus Hankel determinants

Triangular factorization of general N × N Hermitian matrices requires O(N 3) operations. We present a procedure that factors certain structured matrices in O(N 2) operations. The family of structured matrices considered in this paper includes Toeplitz and Hankel matrices, as well as a variety of ‘Toeplitz-like’ and ‘Hankel-like’ matrices (e.g. inverses, sums and products of Toeplitz and Hankel matrices).

Permutability of Toeplitz and Hankel matrices

CONTENTS Introduction Chapter I. The algebraic structure of the spaces of Toeplitz and Hankel matrices § 1. Basic definitions § 2. Basic notation § 3. Statement of the theorems § 4. Auxiliary lemmas § 5. Proof of the theorems Chapter II. Fast computations in some problems of linear algebra connected with Toeplitz and Hankel matrices § 1. Basic definitions and notation § 2. Statement of the theorems § 3. Auxiliary lemmas § 4. Proof of the theorems Chapter III. The fast prediction of stochastic processes § 1. The fast prediction of stationary stochastic processes § 2. The fast prediction of piecewise constant random functions Chapter IV. The structure of Toeplitz and Hankel integral operators § 1. Definitions § 2. Notation § 3. Statement of the theorems References

Toeplitz and Wiener-Hopf determinants with piecewise continuous symbols

Bounds for the condition number are obtained which indicate the dependence of the conditioning of a symmetric positive definite Toeplitz matrix on the sign distribution of its reflection coefficients. This dependence is illustrated with numerical examples. Similar bounds for symmetric positive definite block Toeplitz matrices are also presented.

On algebras of Toeplitz plus Hankel matrices

Conditions for commuting a Toeplitz matrix and a Hankel matrix were obtained relatively recently (in 2015). The solution to the problem of describing all anti-commuting pairs (T, H), where T is a Toeplitz matrix and H is a Hankel matrix, is sketched below.

This paper is a survey of some works on paracommutators and Hankel operators in China, which are the subjects I have been studying since 1985. The classical Hankel operator is defined on the Hilbert space l 2 by the Hankel matrix Γ = {b n+k } n,k ≥0, and the Toeplitz operator is defined also on l 2 by the toeplitz matrix T = {b n−k } n≥k≥0. By the theory of the Fourier series both of them become the operators on the Hardy space \( {{H}^{2}}(T) = \{ f \in {{L}^{2}}(T):\hat{f}(n) = 0,n < 0\} . \). Let \( \overline H ^2 (T) \) denote the conjugate Hardy space and P,P denote the ortnogonal projections from L 2(T) into H 2(T),H 2 (T) respectively. Then the Toeplitz operator is defined as follows $$ {T_b}(f) = P(\bar bf),for f \in {H^2}(T) $$ (1.1) and the Hankel operator is defined as follows $$ {H_b}(f) = \bar P(\bar bf),for f \in {H^2}(T).$$ (1.2)

Kernel structure of block Hankel and Toeplitz matrices and partial realization

In this work, we consider the parametric data model employed in applications such as line spectral estimation and direction-of-arrival estimation. We focus on the stochastic maximum likelihood estimation (MLE) framework and offer approaches to estimate the parameter of interest in a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">gridless</i> manner, overcoming the model complexities of the past. This progress is enabled by the modern trend of reparameterization of the objective, and exploiting the sparse Bayesian learning (SBL) approach. The latter is shown to be a correlation-aware method, and for the underlying problem it is identified as a grid-based technique for recovering a structured covariance matrix of the measurements. For the case when the structured matrix is expressible as a sampled Toeplitz matrix, such as when measurements are sampled in time or space at regular intervals, additional constraints and reparameterization of the SBL objective leads to the proposed structured matrix recovery technique based on MLE. The proposed optimization problem is non-convex, and we propose a majorization-minimization based iterative procedure to estimate the structured matrix; each iteration solves a semidefinite program. We recover the parameter of interest in a gridless manner by appealing to the Carathéodory-Féjer result on decomposition of positive semidefinite (PSD) Toeplitz matrices. For the general case of irregularly spaced time or spatial samples, we propose an iterative SBL procedure that refines grid points to increase resolution near potential source locations, while maintaining a low per iteration complexity. We provide numerical results to evaluate and compare the performance of the proposed techniques with other gridless techniques, and the Cramér-Rao bound. The proposed correlation-aware approach is more robust to environmental/system effects such as low number of snapshots, correlated sources, small separation between source locations and improves sources identifiability.

Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix T_n(f), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f. A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where mathbf {f} is an stimes s matrix-valued trigonometric polynomial with sge 1, and T_n(mathbf {f}) is the block Toeplitz matrix generated by mathbf {f}, whose size is N(n,s)=sn. The case s=1 corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrix-less. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the mathbb {Q}_p Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.

A Schur-based algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz matrix

In this paper, a new subclass S∗(μ, β, ℘,Un(t)) of univalent functions is defined by using Opoola differential operator involving subordination principle. The upper bounds to the second Hankel determinant denoted by H2(2) for the subclass S∗(μ, β, ℘,Un(t)) is established in connection with Chebyshev polynomials of the second kind and Toeplitz determinants.

This study of Hankel and Hermitian Toeplitz determinants is one of the major areas of interest in Geometric function theory and has wide applications in the areas of signal processing and Applied Mathematics. In our present investigations, we define a new subclass of normalized analytic functions H(λ)(λ≥0), defined using a subordination relation with the sine function K(z)=1+sinz. For the class H(λ), coefficient estimates, upper and lower bounds for the Hermitian Toeplitz determinants of second and third order are found. In addition, estimates are provided for the second and third-order Hankel determinants for the class H(λ).