The theme of this thesis is combinatorics, complex analysis and algebraic geometry. The thesis consists of six articles divided into four parts. Part A: Spectral properties of the Schrödinger equation This part consists of Papers I-II, where we study a univariate Schrödinger equation with a complex polynomial potential. We prove that the set of polynomial potentials that admits solutions to the...

In this article various existence results for the LDU-factorization of semi-innnite and bi-innnite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonor-malization of splines is indicated. Both banded and non-banded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Results on the a...

This paper is concerned with the influence of frequency modulation on the semiFredholm properties of Toeplitz operators with oscillating matrix symbols. The main results give conditions on an orientation-preserving homeomorphism α of the real line that ensure the following: if b belongs to a certain class of oscillating matrix functions (periodic, almost periodic, or semi-almost periodic matrix...

ABSTllACT-In this article various existence results for the LDU-factorization of semi-infinite and bi-infinite scalar and block Toeplitz matrices and numerical methods for computing them are reviewed. Moreover, their application to the orthonormal-ization of splines is indicated. Both banded and non-banded Toeplitz matrices are considered. Extensive use is made of matrix polynomial theory. Resu...

The Fredholm properties of Toeplitz operators on the Bergman space A have been well-known for continuous symbols since the 1970s. We investigate the case p = 1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on A that ari...

This paper is concerned with the construction of circulant preconditioners for Toeplitz systems arising from a piecewise continuous generating function with sign changes. If the generating function is given, we prove that for any " > 0, only O(log N) eigenvalues of our preconditioned Toeplitz systems of size N N are not contained in ?1?"; ?1+"] 1?"; 1+"]. The result can be modiied for trigonome...

The paper is devoted to the study of Toeplitz operators with piecewise continuous symbols. We clarify the geometric regularities of the behaviour of the essential spectrum of Toeplitz operators in dependence on their crucial data: the angles between jump curves of symbols at a boundary point of discontinuity and on the limit values reached by a symbol at that boundary point. We show then that t...

The EM algorithm is a commonly cited solution in the literature for the problem of maximum likelihood estimation of covariance matrices under a Toeplitz constraint. In this paper, the solution is extended to the case of twodimensional signals, where spatial stationarity enforces a Toeplitz-block-Toeplitz structure on the covariance matrix. A further generalisation which is presented involves th...

The inverse problem of constructing a symmetric Toeplitz matrix with prescribed eigenvalues has been a challenge both theoretically and computationally in the literature. It is now known in theory that symmetric Toeplitz matrices can have arbitrary real spectra. This paper addresses a similar problem — Can the three largest eigenvalues of symmetric pentadiagonal Toeplitz matrices be arbitrary? ...

In this paper, we consider solutions of Toeplitz systems A n u = b where the Toeplitz matrices A n are generated by nonnegative functions with zeros. Since the matrices A n are ill-conditioned, the convergence factor of classical iterative methods, such as the damped Jacobi method, will approach 1 as the size n of the matrices becomes large. Here we propose to solve the systems by the multigrid...