Orientadores: Dimitar Kolev Dimitrov, Roberto Andreani / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-23T19:23:17Z (GMT). No. of bitstreams: 1 Yen_ChiLun_D.pdf: 3950162 bytes, checksum: 1812f3dd736abbe2d4ff070c7877fdff (MD5) Previous issue date: 2013 / Resumo: O objetivo deste trabalho é apresentar uma nova formulação do Teorema de comparação de Sturm e suas aplicações na teoria dos zeros de polinômios ortogonais, que são: monotonicidade dos zeros dos polinômios ortogonais X1-Jacobi, desigualdades de Gautschi sobre os zeros dos polinômios ortogonais de Jacobi e o comportamento assintótico dos zeros dos polinômios ultrasféricos / Abstract: In this thesis we state a new formulation of the Sturm comparison Theorem and its applications to the zeros of orthogonal polynomials. Specifically, these applications deal with the monotonicity of zeros of X1-Jacobi orthogonal polynomials, Gautschi's conjectures about inequalities of zeros of Jacobi polynomials and the asymptotic of zeros of ultrasphricals polynomials / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

In this thesis we present some fundamental results regarding orthogonal polynomials and biorthogonal polynomials, the latter defined as in the article "Cauchy Biorthogonal Polynomials", authored by Bertola, Gekhtman, and Szmigielski. We show that total positivity of the kernel can be weakened and how this implies that interlacement for biorthogonal polynomials holds in general. A counterexample is provided showing that in general there does not exist a four-term recurrence relation such as the one found for the Cauchy kernel. As a direct consequence we show that biorthogonal polynomial sequences cannot be considered orthogonal polynomial sequences by an appropriate choice of orthogonality measure. Furthermore, we motivate a conjecture stating that the more general form of interlacement that exists for orthogonal polynomials also exists for biorthogonal polynomials. We end with suggesting some further work that could be of interest.

The integrable Schrödinger operators often have a singularity on the real line, which creates problems for their spectral analysis. In several particular cases we show that all closed gaps lie on the infinite spectral arc. In the second part we develop a theory of complex exceptional orthogonal polynomials corresponding to integrable rational and trigonometric Schrödinger operators, which may have a singularity on the real line. In particular, we study the properties of the corresponding complex exceptional Hermite polynomials related to Darboux transformations of the harmonic oscillator, and exceptional Laurent orthogonal polynomials related to trigonometric monodromy-free operators.

One of the major tasks in electronic circuit design is the ability to predict the performance of general circuits in the presence of uncertainty in key design parameters. In the mathematical literature, such a task is referred to as uncertainty quantification. Uncertainty about the key design parameters arises mainly from the difficulty of controlling the physical or geometrical features of the underlying design, especially at the nanometer level. With the constant trend to scale down the process feature size, uncertainty quantification becomes crucial in shortening the design time. To achieve the uncertainty quantification, this thesis presents a new approach based on the concept of generalized Polynomial Chaos (gPC) to perform variability analysis of general nonlinear circuits. The proposed approach is built upon a decoupling formulation of the Galerkin projection (GP) technique, where the large matrix is transformed into a block-diagonal whose diagonal blocks can be factorized independently. The proposed methodology provides a general framework for decoupling the GP formulation based on a general system of orthogonal polynomials. Moreover, it provides a new insight into the error level that is caused by the decoupling procedure, enabling an assessment of the performance of a wide variety of orthogonal polynomials. For example, it is shown that, for the same order, the Chebyshev polynomials outperforms other commonly used gPC polynomials.

La compression des signaux ECG trouve encore plus d’importance avec le développement de la télémédecine. En effet, la compression permet de réduire considérablement les coûts de la transmission des informations médicales à travers les canaux de télécommunication. Notre objectif dans ce travail de thèse est d’élaborer des nouvelles méthodes de compression des signaux ECG à base des polynômes orthogonaux. Pour commencer, nous avons étudié les caractéristiques des signaux ECG, ainsi que différentes opérations de traitements souvent appliquées à ce signal. Nous avons aussi décrit de façon exhaustive et comparative, les algorithmes existants de compression des signaux ECG, en insistant sur ceux à base des approximations et interpolations polynomiales. Nous avons abordé par la suite, les fondements théoriques des polynômes orthogonaux, en étudiant successivement leur nature mathématique, les nombreuses et intéressantes propriétés qu’ils disposent et aussi les caractéristiques de quelques uns de ces polynômes. La modélisation polynomiale du signal ECG consiste d’abord à segmenter ce signal en cycles cardiaques après détection des complexes QRS, ensuite, on devra décomposer dans des bases polynomiales, les fenêtres de signaux obtenues après la segmentation. Les coefficients produits par la décomposition sont utilisés pour synthétiser les segments de signaux dans la phase de reconstruction. La compression revient à utiliser un petit nombre de coefficients pour représenter un segment de signal constitué d’un grand nombre d’échantillons. Nos expérimentations ont établi que les polynômes de Laguerre et les polynômes d’Hermite ne conduisaient pas à une bonne reconstruction du signal ECG. Par contre, les polynômes de Legendre et les polynômes de Tchebychev ont donné des résultats intéressants. En conséquence, nous concevons notre premier algorithme de compression de l’ECG en utilisant les polynômes de Jacobi. Lorsqu’on optimise cet algorithme en supprimant les effets de bords, il dévient universel et n’est plus dédié à la compression des seuls signaux ECG. Bien qu’individuellement, ni les polynômes de Laguerre, ni les fonctions d’Hermite ne permettent une bonne modélisation des segments du signal ECG, nous avons imaginé l’association des deux systèmes de fonctions pour représenter un cycle cardiaque. Le segment de l’ECG correspondant à un cycle cardiaque est scindé en deux parties dans ce cas: la ligne isoélectrique qu’on décompose en séries de polynômes de Laguerre et les ondes P-QRS-T modélisées par les fonctions d’Hermite. On obtient un second algorithme de compression des signaux ECG robuste et performant. / Developing new ECG data compression methods has become more important with the implementation of telemedicine. In fact, compression schemes could considerably reduce the cost of medical data transmission through modern telecommunication networks. Our aim in this thesis is to elaborate compression algorithms for ECG data, using orthogonal polynomials. To start, we studied ECG physiological origin, analysed this signal patterns, including characteristic waves and some signal processing procedures generally applied ECG. We also made an exhaustive review of ECG data compression algorithms, putting special emphasis on methods based on polynomial approximations or polynomials interpolations. We next dealt with the theory of orthogonal polynomials. We tackled on the mathematical construction and studied various and interesting properties of orthogonal polynomials. The modelling of ECG signals with orthogonal polynomials includes two stages: Firstly, ECG signal should be divided into blocks after QRS detection. These blocks must match with cardiac cycles. The second stage is the decomposition of blocks into polynomial bases. Decomposition let to coefficients which will be used to synthesize reconstructed signal. Compression is the fact of using a small number of coefficients to represent a block made of large number of signal samples. We realised ECG signals decompositions into some orthogonal polynomials bases: Laguerre polynomials and Hermite polynomials did not bring out good signal reconstruction. Interesting results were recorded with Legendre polynomials and Tchebychev polynomials. Consequently, our first algorithm for ECG data compression was designed using Jacobi polynomials. This algorithm could be optimized by suppression of boundary effects, it then becomes universal and could be used to compress other types of signal such as audio and image signals. Although Laguerre polynomials and Hermite functions could not individually let to good signal reconstruction, we imagined an association of both systems of functions to realize ECG compression. For that matter, every block of ECG signal that matches with a cardiac cycle is split in two parts. The first part consisting of the baseline section of ECG is decomposed in a series of Laguerre polynomials. The second part made of P-QRS-T waves is modelled with Hermite functions. This second algorithm for ECG data compression is robust and very competitive.

No description available.

We consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.

Pólya and Schur completely characterized all real-rootedness preserving linear operators acting on the standard monomial basis in their famous work from 1914. The corresponding eigenvalues are from then on known as multiplier sequences. In 2009 Borcea and Br\"and\'en gave a complete characterization for general linear operators preserving real-rootedness (and stability) via the symbol. Relying heavily on these results, in this thesis, we are able to completely characterize multiplier sequences for generalized Laguerre bases. We also apply our methods to reprove the characterization of Hermite multiplier sequences achieved by Piotrowski in 2007.

In 1991, A. Máté precisely calculated the first-order asymptotic behavior of the sequence of Christoffel functions associated with Szego measures on the unit circle. Our principal goal is the abstraction of his result in two directions: We compute the translated asymptotics, limn λn(µ, x + a/n), and obtain, as a corollary, a universality limit for the fairly broad class of Szego weights. Finally, we prove Máté’s result for measures supported on smooth curves in the plane. Our proof of the latter derives, in part, from a precise estimate of certain weighted means of the Faber polynomials associated with the support of the measure. Finally, we investigate a variety of applications, including two novel applications to ill-posed problems in Hilbert space and the mean ergodic theorem.

Indiana University-Purdue University Indianapolis (IUPUI) / We consider the behavior of zeros of random polynomials of the from \begin{equation*} P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z) \end{equation*} as \( n\to\infty \), where \( m \) is a non-negative integer (most of the work deal with the case \( m =0 \) ), \( \{\eta_n\}_{n=0}^\infty \) is a sequence of i.i.d. Gaussian random variables, and \( \{\varphi_n(z)\}_{n=0}^\infty \) is a sequence of orthonormal polynomials on the unit circle \( \mathbb T \) for some Borel measure \( \mu \) on \( \mathbb T \) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.