Dual Szegö pairs of sequences of rational matrix-valued functions

We study certain sequences of rational matrix-valued functions with poles outside the unit circle. These sequences are recursively constructed based on a sequence of complex numbers with norm less than one and a sequence of strictly contractive matrices. We present some basic facts on the rational matrix-valued functions belonging to such kind of sequences and we will see that the validity of some Christoffel-Darboux formulae is an essential property. Furthermore, we point out that the considered dual pairs consist of orthogonal systems. In fact, we get similar results as in the classical theory of Szegö's orthogonal polynomials on the unit circle of the first and second kind

Toeplitz Operators on Semi-Simple Lie Groups

Let $G/K$ be a Hermitian symmetric space of non-compact type. We consider for the so-called minimal Olshanskii semigroup $\Gamma\subset G^C$, the C$^*$-algebra $T$ generated by all Toeplitz operators $T_f$ on the Hardy space $H^2(\Gamma)\subset L^2(G)$. We describe the construction of ideals of $T$ associated to boundary strata of the domain $\Gamma$

Toeplitz Operators on Semi-Simple Lie Groups

Let $G/K$ be a Hermitian symmetric space of non-compact type. We consider for the so-called minimal Olshanskii semigroup $\Gamma\subset G^C$, the C$^*$-algebra $T$ generated by all Toeplitz operators $T_f$ on the Hardy space $H^2(\Gamma)\subset L^2(G)$. We describe the construction of ideals of $T$ associated to boundary strata of the domain $\Gamma$

Rational Covariance Extension, Multivariate Spectral Estimation, and Related Moment Problems: Further Results and Applications

This dissertation concerns the problem of spectral estimation subject to moment constraints. Its scalar counterpart is well-known under the name of rational covariance extension which has been extensively studied in past decades. The classical covariance extension problem can be reformulated as a truncated trigonometric moment problem, which in general admits infinitely many solutions. In order to achieve positivity and rationality, optimization with entropy-like functionals has been exploited in the literature to select one solution with a fixed zero structure. Thus spectral zeros serve as an additional degree of freedom and in this way a complete parametrization of rational solutions with bounded degree can be obtained. New theoretical and numerical results are provided in this problem area of systems and control and are summarized in the following. First, a new algorithm for the scalar covariance extension problem formulated in terms of periodic ARMA models is given and its local convergence is demonstrated. The algorithm is formally extended for vector processes and applied to finite-interval model approximation and smoothing problems. Secondly, a general existence result is established for a multivariate spectral estimation problem formulated in a parametric fashion. Efforts are also made to attack the difficult uniqueness question and some preliminary results are obtained. Moreover, well-posedness in a special case is studied throughly, based on which a numerical continuation solver is developed with a provable convergence property. In addition, it is shown that solution to the spectral estimation problem is generally not unique in another parametric family of rational spectra that is advocated in the literature. Thirdly, the problem of image deblurring is formulated and solved in the framework of the multidimensional moment theory with a quadratic penalty as regularization

Multivariate moment problems with applications to spectral estimation and physical layer security in wireless communications

This thesis focuses on generalized moment problems and their applications in the framework of information engineering. Its contribution is twofold. The first part of this dissertation proposes two new techniques for tackling multivariate spectral estimation, which is a key topic in system identification: Relative entropy rate estimation and multivariate circulant rational covariance extension. The former provides a very natural multivariate extension of a state-of-the-art approach for scalar parametric spectral estimation with a complexity bound, known as THREE (Tunable High-Resolution Estimator). It allows to take into account available a priori information on the spectral density. It exhibits high resolution features and it is robust in case of short data records. As for multivariate circulant rational covariance extension, it is a new convex optimization approach to spectral estimation for periodic multivariate processes, in which the computation of the solution can be tackled efficiently by means of Fast Fourier Transform. Numerical examples show that this procedure also provides an efficient tool for approximating regular covariance extension for multivariate processes. The second part of this dissertation considers the problem of deriving a universal performance bound for a message source authentication scheme based on channel estimates in a wireless fading scenario, where an attacker may have correlated observations available and possibly unbounded computational power. Under the assumption that the channels are represented by multivariate complex Gaussian variables, it is proved that the tightest bound corresponds to a forging strategy that produces a zero mean signal that is jointly Gaussian with the attacker observations. A characterization of their joint covariance matrix is derived through the solution of a system of two nonlinear matrix equations. Based upon this characterization, the thesis proposes an efficient iterative algorithm for its computation: The solution to the matricial system appears as fixed point of the iteration. Numerical examples suggest that this procedure is effective in assessing worst case channel authentication performance

Wavelet Analysis on the Sphere

The goal of this monograph is to develop the theory of wavelet harmonic analysis on the sphere. By starting with orthogonal polynomials and functional Hilbert spaces on the sphere, the foundations are laid for the study of spherical harmonics such as zonal functions. The book also discusses the construction of wavelet bases using special functions, especially Bessel, Hermite, Tchebychev, and Gegenbauer polynomials