Infinite Toeplitz Research Articles

Infinite extensions of Toeplitz matrices

A formula for the smallest of the ranks of infinite Toeplitz extensions of a finite rectangular Toeplitz matrix is derived. Bibliography: 3 titles.

Semi-Infinite Arrays of Isotropic Point Scatterers. A Unified Approach

We solve the two-dimensional problem of acoustic scattering by a semi-infinite periodic array of identical isotropic point scatterers, i.e., objects whose size is negligible compared to the incident wavelength and which are assumed to scatter incident waves uniformly in all directions. This model is appropriate for scatterers on which Dirichlet boundary conditions are applied in the limit as the ratio of wavelength to body size tends to infinity. The problem is also relevant to the scattering of an E-polarized electromagnetic wave by an array of highly conducting wires. The actual geometry of each scatterer is characterized by a single parameter in the equations, related to the single-body scattering problem and determined from a harmonic boundary-value problem. Using a mixture of analytical and numerical techniques, we confirm that a number of phenomena reported for specific geometries are in fact present in the general case (such as the presence of shadow boundaries in the far field and the vanishing of the circular wave scattered by the end of the array in certain specific directions). We show that the semi-infinite array problem is equivalent to that of inverting an infinite Toeplitz matrix, which in turn can be formulated as a discrete Wiener--Hopf problem. Numerical results are presented which compare the amplitude of the wave diffracted by the end of the array for scatterers having different shapes.

Coloring planar Toeplitz graphs and the stable set polytope

Cliques and odd cycles are well known to induce facet-defining inequalities for the stable set polytope. In graph coloring cliques are a class of n-critical graphs whereas odd cycles represent the class of 3-critical graphs. In the first part of this paper we generalize both notions to ( K n ⧹ e)-cycles, a new class of n-critical graphs, and discuss some implications for the class of infinite planar Toeplitz graphs. More precisely, we show that any infinite Toeplitz graph decomposes into a finite number of connected and isomorphic components. Similar to the bipartite case, infinite planar Toeplitz graphs can be characterized by a simple condition on their defining 0–1 sequence. We then address the problem of coloring such graphs. Whereas they can always be 4-colored by a greedy-like algorithm, we are able to fully characterize the subclass of 3-chromatic such graphs. As a corollary, we obtain a König-type characterization of this class by means of ( K 4⧹ e)-cycles. In the second part, we turn to polyhedral theory and show that ( K n ⧹ e)-cycles give rise to a new class of facet-defining inequalities for the stable set polytope. Then we show how Hajós’ construction can be used to further generalize ( K n ⧹ e)-cycles thereby providing a very large class of n-critical graphs which are still facet-inducing for the associated stable set polytope.

The spectra of large Toeplitz band matrices with a randomly perturbed entry

This paper is concerned with the union spΩ(j,k) Tn(a) of all possible spectra that may emerge when perturbing a large n × n Toeplitz band matrix Tn(a) in the (j, k) site by a number randomly chosen from some set Ω. The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of spΩ(j,k) Tn(a) as n → ∞. Also discussed are the cases of small and large sets Ω as well as the discontinuity of the infinite volume case, which means that in general spΩ(j,k) Tn(a) does not converge to something close to spΩ(j,k) Tn(a) as n → ∞ where T(a) is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case.

A Bound for the Inverse of a Lower Triangular Toeplitz Matrix

The expression of a bound of the uniform norm of infinite lower triangular Toeplitz matrices with nonnegative entries is found. All the results are obtained by studying the behavior of the resolvent kernel and of the fundamental matrix of the recurrence relation, which generates the sequence of the entries of the considered matrix.

Asymptotic Spectra of Dense Toeplitz Matrices Are Unstable

The paper is concerned with the limiting set of the eigenvalues of the truncations of an infinite Toeplitz matrix whose symbol is continuous but not rational. This limiting set is shown to be unstable with respect to small perturbations of the symbol in the uniform norm, which reveals that the numerical computation of the asymptotic spectra of dense Toeplitz matrices is a genuine mathematical challenge.

Can spectral value sets of Toeplitz band matrices jump?

The spectral value set sp ε B, C A of a bounded linear operator A on l 2 is the union of the spectra of all operators of the form A+ BKC, where ∥ K∥< ε and B, C are fixed bounded linear operators. It turns out that small changes of ε may cause drastic changes of the set sp ε B, C A. We conjecture that this can never happen if B or C is compact and A is given by an infinite Toeplitz band matrix. In the present paper, this conjecture is proved for certain interesting operators B and C and for several classes of Toeplitz band matrices, including Hessenberg matrices and matrices of small bandwidth. Our approach is based on working with the monodromy group of the Riemann surface associated with the generating function of the matrix.

Infinite Toeplitz and Laurent matrices with localized impurities

This paper is concerned with the change of the spectra of infinite Toeplitz and Laurent matrices under perturbations in a prescribed finite set of sites. The main result says that the spectrum of a Toeplitz matrix with a non-constant rational symbol is not affected by small localized impurities, while such impurities can nevertheless enlarge the spectrum of the corresponding Laurent matrix. We also study the spectra that may emerge when randomly perturbing Toeplitz or Laurent matrices in a randomly chosen single site.

Four short stories about Toeplitz matrix calculations

The stories told in this paper are dealing with the solution of finite, infinite, and biinfinite Toeplitz-type systems. A crucial role plays the off-diagonal decay behavior of Toeplitz matrices and their inverses. Classical results of Gelfand et al. on commutative Banach algebras yield a general characterization of this decay behavior. We then derive estimates for the approximate solution of (bi)infinite Toeplitz systems by the finite section method, showing that the approximation rate depends only on the decay of the entries of the Toeplitz matrix and its condition number. Furthermore, we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems. Finally, some quantitative results on the construction of preconditioners via circulant embedding are derived, which allow us to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems.

Szegö limits for infinite Toeplitz matrices determined by the Taylor series of two rational functions

Explicit expressions for the well-known Szegö limits are obtained for the cases when the upper and lower triangular parts of the Toeplitz matrix are determined separately by the Taylor series of rational matrix functions and the exponential growth of the entries is permissible.

Computations with infinite Toeplitz matrices and polynomials

We relate polynomial computations with operations involving infinite band Toeplitz matrices and show applications to the numerical solution of Markov chains, of nonlinear matrix equations, to spectral factorizations and to the solution of finite Toeplitz systems. In particular two matrix versions of Graeffe's iteration are introduced and their convergence properties are analyzed. Correlations between Graeffe's iteration for matrix polynomials and cyclic reduction for block Toeplitz matrices are pointed out. The paper contains a systematic treatment of known topics and presentation of new results, improvements and extensions.

On the application of numerical methods to Hallen's equation

The so-called Hallen integral equation for the current on a finite linear antenna center-driven by a delta-function generator takes two forms depending on the choice of kernel. The two kernels are usually referred to as the exact and the approximate or reduced kernel. With the approximate kernel, the integral equation has no solution. Nevertheless, the same numerical method is often applied to both forms of the integral equation. In this paper, the behavior of the numerical solutions thus obtained is investigated, and the similarities and differences between the two numerical solutions are discussed. The numerical method is Galerkin's method with pulse functions. We first apply this method to the two corresponding forms of the integral equation for the current on a linear antenna of infinite length. In this case, the method yields an infinite Toeplitz system of algebraic equations in which the width of the pulse basis functions enters as a parameter. The infinite system is solved exactly for nonzero pulse width; the exact solution is then developed asymptotically for the case where the pulse width is small. When the asymptotic expressions for the case of the infinite antenna are used as a guide for the behavior of the solutions of the finite antenna, the latter problem is greatly facilitated. For the approximate kernel, the main results of this paper carry over to a certain numerical method applied to the corresponding equation of the Pocklington type.

Coloring Planar Toeplitz Graphs

Abstract We study structural properties of infinite Toeplitz graphs such as connectivity, planarity and colorability. As to connectivity we show that any infinite Toeplitz graph decomposes into a finite number of connected and isomorphic components. Similar to the bipartite case (cf. [4]), infinite planar Toeplitz graphs can be characterized by a simple condition on their defining 0-1 sequence. We then turn to the problem of coloring such planar graphs. Whereas they can always be 4-colored by a greedy-like algorithm, we are able to fully characterize the subclass of 3-colorable such graphs. As a corollary we obtain a Konig-type characterization of this class by means of ( K 4 \e)-cycles, a family of 4-critical graphs that generalize odd cycles. We also discuss some consequences for general (finite) graph coloring.

Using FFT-based techniques in polynomial and matrix computations: recent advances and applicatons

Computations with univariate polynomials, like the evaluation of product, quotient, remainder, greatest common divisor, etc, are closely related to linear algebra computations performed with structured matrices having the Toeplitz-like or the Hankel-like structures. The discrete Fourier transform, and the FFT algorithms for its computation, constitute a powerful tool for the design and analysis of fast algorithms for solving problems involving polynomials and structured matrices. We recall the main correlations between polynomial and matrix computations and present two recent results in this field: in particular, we show how Fourier methods can speed up the solution of a wide class of problems arising in queueing theory where certain Markov chains, defined by infinite block Toeplitz matrices in generalized Hessenberg form, must be solved. Moreover, we present a new method for the numerical factorization of polynomials based on a matrix generalization of Koenig's theorem. This method, that relies on the evaluation/interpolation technique at the Fourier points, reduces the problem of polynomial factorization to the computation of the LU decomposition of a banded Toeplitz matrix with its rows and columns suitably permuted. Numerical experiments that show the effectiveness of our algorithms are presented

Norms of Large Toeplitz Band Matrices

Let T(bjk) be a finite collection of infinite Toeplitz band matrices, let Tn(bjk) denote their (n+1) x (n+1) truncations, and put \[ A_n=\sum_j\prod_k T_n(b_{jk}),\quad M_p=\lim_{n\to\infty}\|A_n\|_p, \] where $\|\cdot\|_p$ stands for the operator norm associated with the $l^p$-norm $(1\le p<\infty)$ on Cn+1. We establish tight two-sided estimates for the difference Mp-||An||p. It is well known that if T(b) is a Hermitian Toeplitz band matrix and An=Tn(b), then M2-||An||2 goes to zero with polynomial speed. We show that such a slow convergence rate is, in a sense, an exceptional case, and we prove that in the generic case Mp - ||An||p approaches zero with exponential speed. Our results yield good error estimates when computing the norms of certain infinite matrices via their large truncations and, conversely, when determining the norms of certain large matrices by having recourse to known norms of infinite matrices.

Convergence speed estimates for the norms of the inverses of large truncated Toeplitz matrices

Given a continuous function a on the complex unit circle, let T(a) denote the infinite Toeplitz matrix generated by a and let Tn(a) stand for the (n+1)×(n+1) principal section of T(a). We think of T(a) and Tn(a) as operators on l2 spaces. A classical result by Gohberg and Feldman says that if T(a) is invertible, then so is Tn(a) for all sufficiently large n≥n0 and \(\). Only in 1994 did we realize that in fact \(\). In this paper, we provide estimates for the speed with which \(\) converges to \(\). We prove that in the “generic case” the convergence speed can be estimated by the smoothness of a, whereas in some “exceptional cases” (e.g., if T(a) is Hermitian or triangular) it is not the smoothness of $a$ but the orders of certain zeros which determine the convergence speed. Some of the results are extended to operators on lp spaces.

Inversion formulas for infinite generalized Toeplitz matrices

“Inversion formulas are obtained for a certain class of infinite matrices that possess displacement structure similar to that of finite block Toeplitz matrices. Consequences are symmetric inversion formulas for matrix-valued singular integral operators and infinite Toeplitz plus Hankel matrices”.

Normal Toeplitz Matrices

It is well known from the work of Brown and Halmos [J. Reine Angew. Math., 213 (1963/1964), pp. 89–102] that an infinite Toeplitz matrix is normal if and only if it is a rotation and translation of a Hermitian Toeplitz matrix. In the present article we prove that all finite normal Toeplitz matrices are either generalised circulants or are obtained from Hermitian Toeplitz matrices by rotation and translation.

Infinite Toeplitz and Hankel Matrices with Operator-Valued Entries

Infinite Toeplitz matrices with operator-valued entries arise, for example, when interpreting Wiener–Hopf integral operators on $L^2 (0,\infty )$ as matrices acting on the direct sum of countably many copies of $L^2 (0,1)$ . This paper concerns the question of asymptotically inverting such infinite Toeplitz matrices by having recourse to their finite principal sections. As expected from the corresponding theories for the scalar and matrix-valued cases, this problem leads to the investigation of compactness properties of infinite Hankel matrices. By introducing the concept of $Q_n $-compact operators on spaces of square-summable sequences with values in a separable Hilbert space, criteria for the applicability of the finite section method to Toeplitz operators with symbols in $C + H^\infty $, in $PC$, or with locally sectorial symbols are established.

The finite section method for Moore-Penrose inversion of Toeplitz operators

It is shown that for certain classes of infinite block Toeplitz matricesT(a)=[a j-k ] 0 ∞ the Moore-Penrose inverses of the finite sectionT n (a)=[a j-k ] 0 n−1 converge to the Moore-Penrose inverse ofT(a). Furthermore the convergence for modified finite section methods and the finite section method for Wiener-Hopf integral and related operators are studied.