114 research outputs found
On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II
AbstractThe main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz–Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved
Convergence and Asymptotic of Multi-Level Hermite-Padé Polynomials
MenciĂłn Internacional en el tĂtulo de doctorPrograma de Doctorado en IngenierĂa Matemática por la Universidad Carlos III de MadridPresidente: Francisco JosĂ© Marcellán Español.- Secretario: Bernardo de la Calle Ysern.- Vocal: Arnoldus Bernardus Jacobus Kuijla
OPUC, CMV matrices and perturbations of measures supported on the unit circle
Let us consider a Hermitian linear functional defined on the linear space of Laurent polynomials with complex coefficients. In the literature, canonical spectral transformations of this functional are studied. The aim of this research is focused on perturbations of Hermitian linear functionals associated with a positive Borel measure supported on the unit circle. Some algebraic properties of the perturbed measure are pointed out in a constructive way. We discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated Verblunsky coefficients. Then, the structure of the Theta matrices of the perturbed linear functionals, which is the main tool for the comparison of their corresponding CMV matrices, is deeply analyzed. From the comparison between different CMV matrices, other families of perturbed Verblunsky coefficients will be considered. We introduce a new matrix, named Fundamental matrix, that is a tridiagonal symmetric unitary matrix, containing basic information about the family of orthogonal polynomials. However, we show that it is connected to another family of orthogonal polynomials through the Takagi decomposition.The authors would like to thank Professor Bernhard Beckermann and Professor RogerA. Horn for valuable and insightful discussions about congruence relations. We also thank the suggestions by the referees which have contributed to improve substantially the presentation of the manuscript. The work of the first author (FM) was partially sup-ported by DirecciĂłn General de PolĂtica CientĂfica y TecnolĂłgica, Ministerio de EconomĂa y Competitividad (MINECO) of Spain, under grant MTM2012-36732-C03-01. The sec-ond author (NS) thanks Alexander von Humboldt Foundation for the support and the Department of Mathematics, Universidad Carlos III de Madrid, for its constant support and friendly atmosphere during the period January–July 2014 when the manuscript was finished
Spectral Density Functions and Their Applications
The Bernstein-Szegő measure moment problem asks when a given finite list of complex numbers form the Fourier coefficients of the spectral density function of a stable polynomial in the one-variable case. Szegő proved that it is possible if and only if the Toeplitz matrix form by these numbers is positive definite. Bernstein later proved a real line analog of the problem. The question remained open in two variables until Geronimo and Woerdeman stated and proved the necessary and sufficient conditions. Unlike the solution in one variable, it does not suffice to write down a single matrix and check whether it is positive definite. A positive definite completion condition is also required. In this thesis, we further pursue the moment problem in two variables and beyond. We first enhance the two-variable results by identifying the eigenstructure of matrices that arise from the theory. We then create a method that allows us to compute the Fourier coefficients in a given infinite region by using a finite portion of the coefficients. Use is made of determinantal representations of stable polynomials. In addition, we compute the asymptotics for the Fourier coefficients and later generalize the result to higher dimensions. In the final chapter, we draw a connection between offset words and a particular type of spectral density functions and compute the asymptotics of the number of offset words as different parameter changes.Ph.D., Mathematics -- Drexel University, 201
Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems
of Toda-like integrable systems are connected us-ing the Gauss-Borel factorization of two, left and a right, Cantero-Morales-Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. Ablock Gauss-Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix SzegĹ‘ polynomials, which can be expressed in terms of Schur complements of bordered trun-cations of the block moment matrix. The corresponding block extension of the Christoffel-Darboux theory is derived. De-formations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov-Shabat equations, bilinear equa-tions and discrete flows-connected with Darboux transformations. We generalize the integrable flows of the Cafasso’s matrix extension of the Toeplitz lattice for the Verblunsky coefficients of SzegĹ‘ polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel-Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomialsM.M. thanks economical support from the Spanish “Ministerio de EconomĂa y Competitividad” research project MTM2012-36732-C03-01, Ortogonalidad y aproximaciĂłn; teorĂa y aplicaciones
Local properties of Hilbert spaces of Dirichlet series
We show that the asymptotic behavior of the partial sums of a sequence of
positive numbers determine the local behavior of the Hilbert space of Dirichlet
series defined using these as weights. This extends results recently obtained
describing the local behavior of Dirichlet series with square summable
coefficients in terms of local integrability, boundary behavior, Carleson
measures and interpolating sequences. As these spaces can be identified with
functions spaces on the infinite-dimensional polydisk, this gives new results
on the Dirichlet and Bergman spaces on the infinite dimensional polydisk, as
well as the scale of Besov-Sobolev spaces containing the Drury-Arveson space on
the infinite dimensional unit ball. We use both techniques from the theory of
sampling in Paley-Wiener spaces, and classical results from analytic number
theory.Comment: 27 pages, 1 figur
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