The solution of many problems in physics and engineering reduces ultimately to the solution of linear equations of the form Ra = m, where JR and m are given N x N and N x 1 matrices and a is to be determined. Here our concern is with the fact that it generally takes 0(N) computations (one computation being the multiplication of two real numbers) to do this, and this might be a substantial burde...

One of the common strategies in the study of Toeplitz operators consists in selecting of various special symbol classes S ⊂ L∞ so that the properties of both the individual Toeplitz operators Ta, with a ∈ S, and of the algebra generated by such Toeplitz operators can be characterized. A motivation to study an algebra generated by Toeplitz operators (rather than just Toeplitz operators themselve...

The technique known as multiple signal classification (MUSIC) is a semi-empirical way to obtain pseudo-spectra that highlight the spectral-energy distribution of a time series. It is based on a certain canonical decomposition of a Toeplitz matrix formed out of an estimated autocorrelation sequence. The purpose of this paper is to present an analogous canonical decomposition of the state-covaria...

and Applied Analysis 3 For commuting problem, in 1963, Brown and Halmos 2 showed that two bounded Toeplitz operators Tφ and Tψ on the classical Hardy space commute if and only if i both φ and ψ are analytic, ii both φ and ψ are analytic, or iii one is a linear function of the other. On the Bergman space of the unit disk, some similar results were obtained for Toeplitz operators with bounded har...

A weighted Toeplitz operator on H(β) is defined as Tφf = P (φf) where P is the projection from L(β) onto H(β) and the symbol φ ∈ L(β) for a given sequence β = 〈βn〉n∈Z of positive numbers. In this paper, a matrix characterization of a weighted multiplication operator on L(β) is given and it is used to deduce the same for a weighted Toeplitz operator. The eigenvalues of some weighted Toeplitz ope...

2 Two-Toda lattice and reductions (Hänkel and Toeplitz) 17 2.1 Two-Toda on Moment Matrices and Identities for τ -Functions 18 2.2 Reduction to Hänkel matrices: the standard Toda lattice and a Virasoro algebra of constraints . . . . . . . . . . . . . . . . . 26 2.3 Reduction to Toeplitz matrices: two-Toda Lattice and an SL(2,Z)algebra of constraints . . . . . . . . . . . . . . . . . . . . . . 29...

We study Toeplitz operators on the Hardy spaces of connected compact abelian groups and of tube-type bounded symmetric domains. A representation theorem for these operators and for classes of abstract Toeplitz elements in C*-algebras is proved. This is used to give a unified treatment to index theory in this setting, and a variety of new index theorems are proved that generalize the Gohberg–Kre...

The preconditioned conjugate gradient method is employed to solve Toeplitz systems T n x = b where the generating functions of the n-by-n Toeplitz matrices T n are functions with zeros. In this case, circulant preconditioners are known to give poor convergence, whereas band-Toeplitz preconditioners only ooer linear convergence and can only handle real-valued functions with zeros of even orders....