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Dissertations / Theses on the topic 'Orthogonal polynomials'
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Griffin, James Christopher. "Topics in orthogonal polynomials." Thesis, Imperial College London, 2004. http://hdl.handle.net/10044/1/7620.
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Thomas, William Howard Fischer Ismor. "Introduction to real orthogonal polynomials /." Monterey, Calif. : Springfield, Va. : Naval Postgraduate School; Available from the National Technical Information Service, 1992. http://handle.dtic.mil/100.2/ADA256448.
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Topkara, Mustafa. "Orthogonal Polynomials And Moment Problem." Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/1109164/index.pdf.
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The generalized moment of order k of a mass distribution sigma for a natural number k is given by integral of lambda to the power k with respect to mass distribution sigma and variable lambda. In extended moment problem, given
a sequence of real numbers, it is required to find a mass distribution whose generalized moment of order k is k'th term of the sequence. The conditions of existence and uniqueness of the solution obtained by Hamburger are studied in this thesis by the use of orthogonal polynomials determined by a measure on real line. A chapter on the study of asymptotic behaviour of orthogonal functions on compact subsets of complex numbers is also included.
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Larsson-Cohn, Lars. "Gaussian structures and orthogonal polynomials." Doctoral thesis, Uppsala : Matematiska institutionen, Univ. [distributör], 2002. http://publications.uu.se/theses/91-506-1535-1/.
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Thomas, William Howard II. "Introduction to real orthogonal polynomials." Thesis, Monterey, California. Naval Postgraduate School, 1992. http://hdl.handle.net/10945/23932.
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Approved for public release; distribution is unlimitedThe fundamental concept of orthogonality of mathematical objects occurs in a wide variety of physical and engineering disciplines. The theory of orthogonal functions, for example, is central to the development of Fourier series and wavelets, essential for signal processing. In particular, various families of classical orthogonal polynomials have traditionally been applied to fields such as electrostatics, numerical analysis, and many others. This thesis develops the main ideas necessary for understanding the classical theory of orthogonal polynomials. Special emphasis is given to the Jacobi polynomials and to certain important subclasses and generalizations, some recently discovered. Using the theory of hypergeometric power series and their q -extensions, various structural properties and relations between these classes are systematically investigated. Recently, these classes have found significant applications in coding theory and the study of angular momentum, and hold much promise for future applications.
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Graneland, Elsa. "Orthogonal polynomials and special functions." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-418820.
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Smith, James. "Painleve equations and orthogonal polynomials." Thesis, University of Kent, 2016. https://kar.kent.ac.uk/54758/.
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In this thesis we classify all of the special function solutions to Painleve equations and all their associated equations produced using their Hamiltonian structures. We then use these special solutions to highlight the connection between the Painleve equations and the coefficients of some three-term recurrence relations for some specific orthogonal polynomials. The key idea of this newly developed method is the recognition of certain orthogonal polynomial moments as a particular special function. This means we can compare the matrix of moments with the Wronskian solutions, which the Painleve equations are famous for. Once this connection is found we can simply read o the all important recurrence coefficients in a closed form. In certain cases, we can even improve upon this as some of the weights allow a simplification of the recurrence coefficients to polynomials and with it, the new sequences orthogonal polynomials are simplified too.
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Munemasa, Akihiro. "Nonsymmetric P- and Q-polynomial association schemes and associated orthogonal polynomials /." The Ohio State University, 1989. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487670346874604.
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Wang, Xiangsheng. "Uniform asymptotics of the Meixner polynomials and some q-orthogonal polynomials /." access full-text access abstract and table of contents, 2009. http://libweb.cityu.edu.hk/cgi-bin/ezdb/thesis.pl?phd-ma-b30082560f.pdf.
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Thesis (Ph.D.)--City University of Hong Kong, 2009."Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves [115]-118)
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Stefánsson, Úlfar F. "Asymptotic properties of Müntz orthogonal polynomials." Diss., Georgia Institute of Technology, 2010. http://hdl.handle.net/1853/34759.
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Müntz polynomials arise from consideration of Müntz's Theorem, which is a beautiful generalization of Weierstrass's Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials on the interval of orthogonality, and in particular obtain new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics and endpoint limit asymptotics on the interval. The zero spacing behavior follows, as well as estimates for the smallest and largest zeros. This is the first time that such asymptotics have been obtained for general Müntz exponents. We also look at the asymptotic behavior outside the interval and the asymptotic properties of the associated Christoffel functions.
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Woods, Mischa Prebin. "Orthogonal polynomials and open quantum systems." Thesis, Imperial College London, 2013. http://hdl.handle.net/10044/1/17851.
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This thesis is concerned with the study of quantum systems coupled linearly to a continuous bath of oscillators examples of which are the spin-boson model and the Nelson model. The main theme throughout is to develop a better understanding of the properties of the bath of oscillators such that more efficient representations of it can be made to facilitate the understanding of the system dynamics. The main difficulty in simulating the system dynamics is that the bath of oscillators composes an infinite number of degrees of freedom. In this thesis, we investigate the mathematical properties of an approach in which the bath modes are written as a semi-infinite chain of nearest neighbour interacting harmonic oscillators such that the efficient time dependent density-matrix renormalisation group (t-DMRG) methods can be applied for simulation. In the first section, we show how there are many different ways to represent the bath as semiinfinite chains and prove that seemingly unrelated methods can all be achieved using the same mathematical formalism. We show that in an iterative process the bath can be transformed into a chain of oscillators with nearest neighbour interactions. This is achieved using the formalism of orthogonal polynomials. This allows one to define a sequence of residual spectral densities at each site along the chain. We show that this sequence of residual spectral densities is provided by the so-called ”sequence of secondary measures”. We derive a systematic procedure to obtain the spectral density of the residual bath in each step. We find that these residual spectral densities are related to an old abstract problem in mathematics known as the ”secondary measures”. We solve this problem from the field of orthogonal polynomials to give an explicit expression for the residual spectral densities and go on to prove that these functions converge under very general conditions. That is, the asymptotic part of the chain is universal, translation invariant with universal spectral density. These results suggest efficient methods for handling the numerical treatment of the residual bath. In the second section, we take a different approach. Rather than studying the properties of residual baths, we look at how system observables are affected by truncating the semi-infinite chain of harmonic oscillators to a finite length chain. By developing locality bounds for the dynamics, we derive an upper bound to the error introduced by such a truncation and show that for all finite times it can be made arbitrarily small by including a sufficient number of harmonic oscillators 3 before truncating. Furthermore, it is shown that the speed at which the system communicates with different harmonic oscillators in the chain is proportional to the maximum frequency of the environment but that this speed also depends on the particular version of the chain. These bounds are given for when the dynamics are calculated in the Interaction picture and in the Schrodinger picture.
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Bauldry, William Charles. "Orthogonal polynomials associated with exponential weights /." The Ohio State University, 1985. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487259125218568.
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Zhang, Jianxiang. "Orthogonal polynomials : Selected topics and applications /." The Ohio State University, 1995. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487862972135111.
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Musonda, John. "Orthogonal Polynomials, Operators and Commutation Relations." Licentiate thesis, Mälardalens högskola, Utbildningsvetenskap och Matematik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-35204.
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Orthogonal polynomials, operators and commutation relations appear in many areas of mathematics, physics and engineering where they play a vital role. For instance, orthogonal functions in general are central to the development of Fourier series and wavelets which are essential to signal processing. In particular, as demonstrated in this thesis, orthogonal polynomials can be used to establish the L2-boundedness of singular integral operators which is a fundamental problem in harmonic analysis and a subject of extensive investigations. The Lp-convergence of Fourier series is closely related to the Lp-boundedness of singular integral operators. Many important relations in physical sciences are represented by operators satisfying various commutation relations. Such commutation relations play key roles in such areas as quantum mechanics, wavelet analysis, representation theory, spectral theory, and many others. This thesis consists of three main parts. The first part presents a new system of orthogonal polynomials, and establishes its relation to the previously studied systems in the class of Meixner–Pollaczek polynomials. Boundedness properties of two singular integral operators of convolution type are investigated in the Hilbert spaces related to the relevant orthogonal polynomials. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that the two operators are bounded on L2-spaces, and estimates of the norms are obtained. The second part extends the investigation of the boundedness properties of the two singular integral operators to Lp-spaces on the real line, both in the weighted and unweighted spaces. It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by first proving boundedness for L2 and weak boundedness for L1, and then using interpolation to obtain boundedness for the intermediate spaces. To obtain boundedness for the remaining spaces, duality is used in the translation invariant case, while the weighted case is partly based on the methods developed by M. Riesz in his paper of 1928 for the conjugate function operator. The third and final part derives simple and explicit formulas for reordering elements in an algebra with three generators and Lie type relations. Centralizers and centers are computed as an example of an application of the formulas.Ortogonala polynom, operatorer och kommutationsrelationer förekommer i många områden av matematik, fysik och teknik där de spelar en viktig roll. Till exempel ortogonala funktioner i allmänhet är centrala för utvecklingen av Fourierserier och wavelets som är väsentliga för signalbehandling. I synnerhet, såsom visats i denna avhandling, kan ortogonala polynom användas för att fastställa L2-begränsning av singulära integraloperatorer vilket är ett fundamentalt problem i harmonisk analys och föremål för omfattande forskning. Lp-konvergensen av Fourierserien är nära relaterad till Lp-begränsning av singulära integraloperatorer. Många viktiga relationer i fysik representeras av operatorer som uppfyller olika kommutationsrelationer. Sådana kommutationsrelationer spelar nyckelroller i områden som kvantmekanik, waveletanalys, representationsteori, spektralteori och många andra. Denna avhandling består av tre huvuddelar. Den första delen presenterar ett nytt system av ortogonala polynom, och etablerar dess förhållande till de tidigare studerade systemen i klassen Meixner–Pollaczek-polynom. Begränsningsegenskaper hos två singulära integraloperatorer av faltningstyp utreds i Hilbertrum relaterade till de relevanta ortogonala polynomen. Ortogonala polynom används för att bevisa begränsning i viktade rum och Fourieranalys används för att bevisa begränsning i det translationsinvarianta fallet. Det bevisas i båda fallen att de två operatorerna är begränsade på L2-rummen, och uppskattningar av normerna tas fram. Den andra delen utvidgar till Lp-rum på reella tallinjen undersökningen av begränsningsegenskaperna hos de två singulära integraloperatorerna, både på viktade och oviktade rum. Det bevisas att de båda operatorerna är begränsade på dessa rum och uppskattningar av normerna erhålls. Detta uppnås genom att först bevisa begränsning för L2 och svag begränsning för L1, och sedan använda interpolation att erhålla begränsning för de mellanliggande rummen. För att erhålla begränsning för övriga Lp-rum används dualitet i det translationsinvarianta fallet, medan detta i det viktade fallet delvis bygger på en metod av M. Riesz i hans artikel från 1928 om konjugatfunktionsoperatorn. Den tredje och sista delen härleder enkla och explicita formler för omkastning av element i en algebra med tre generatorer och relationer av Lie-typ. Som ett exempel på en tillämpning av formlerna beräknas centralisatorer och centra.
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Wallis, David. "Modelling impact crater morphology with orthogonal polynomials." Thesis, University of Kent, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.342267.
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Webb, Marcus David. "Isospectral algorithms, Toeplitz matrices and orthogonal polynomials." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/264149.
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An isospectral algorithm is one which manipulates a matrix without changing its spectrum. In this thesis we study three interrelated examples of isospectral algorithms, all pertaining to Toeplitz matrices in some fashion, and one directly involving orthogonal polynomials. The first set of algorithms we study come from discretising a continuous isospectral flow designed to converge to a symmetric Toeplitz matrix with prescribed eigenvalues. We analyse constrained, isospectral gradient flow approaches and an isospectral flow studied by Chu in 1993. The second set of algorithms compute the spectral measure of a Jacobi operator, which is the weight function for the associated orthogonal polynomials and can include a singular part. The connection coefficients matrix, which converts between different bases of orthogonal polynomials, is shown to be a useful new tool in the spectral theory of Jacobi operators. When the Jacobi operator is a finite rank perturbation of Toeplitz, here called pert-Toeplitz, the connection coefficients matrix produces an explicit, computable formula for the spectral measure. Generalisation to trace class perturbations is also considered. The third algorithm is the infinite dimensional QL algorithm. In contrast to the finite dimensional case in which the QL and QR algorithms are equivalent, we find that the QL factorisations do not always exist, but that it is possible, at least in the case of pert-Toeplitz Jacobi operators, to implement shifts to generate rapid convergence of the top left entry to an eigenvalue. A fascinating novelty here is that the infinite dimensional matrices are computed in their entirety and stored in tailor made data structures. Lastly, the connection coefficients matrix and the orthogonal transformations computed in the QL iterations can be combined to transform these pert-Toeplitz Jacobi operators isospectrally to a canonical form. This allows us to implement a functional calculus for pert-Toeplitz Jacobi operators.
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Gishe, Jemal Emina. "A finite family of q-orthogonal polynomials and resultants of Chebyshev polynomials." [Tampa, Fla] : University of South Florida, 2006. http://purl.fcla.edu/usf/dc/et/SFE0001620.
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Musonda, John. "Three Systems of Orthogonal Polynomials and Associated Operators." Thesis, Uppsala universitet, Analys och tillämpad matematik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-175465.
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Miki, Hiroshi. "Studies on Generalized Orthogonal Polynomials and Their Applications." 京都大学 (Kyoto University), 2012. http://hdl.handle.net/2433/160977.
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DeFazio, Mark Vincent. "On the zeros of some quasidefinite orthogonal polynomials." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/NQ66344.pdf.
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Njionou, Sadjang Patrick [Verfasser]. "Moments of classical orthogonal polynomials / Patrick Njionou Sadjang." Kassel : Universitätsbibliothek Kassel, 2013. http://d-nb.info/1045763829/34.
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Vaktnäs, Marcus. "Multiple Orthogonal Polynomials & Modifications of Spectral Measures." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-453139.
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Melin, Valdemar. "Quantum Mechanical Propagators Related to Classical Orthogonal Polynomials." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-297558.
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A few quantum systems on the line with weighted classical orthogonal polynomials as eigenstates are studied. Explicit expressions of the propagators,i.e. the integral kernels of the time evolution operators, are derived. In the case of Hermite polynomials, the system is the harmonic oscillator, while forgeneralized Laguerre and Gegenbauer polynomials, the corresponding quanutum system are equivalent to two-particle Calogero-Sutherland systems.
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Spicer, Paul Edward. "On orthogonal polynomials and related discrete integrable systems." Thesis, University of Leeds, 2006. http://etheses.whiterose.ac.uk/101/.
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Orthogonal polynomials arise in many areas of mathematics and have been the subject of interest by many mathematicians. In recent years this interest has often arisen from outside the orthogonal polynomial community after their connection with integrable systems was found. This thesis is concerned with the different ways these connections can occur. We approach the problem from both perspectives, by looking for integrable structures in orthogonal polynomials and by using an integrable structure to relate different classes of orthogonal polynomials. In Chapter 2, we focus on certain classes of semi-classical orthogonal polynomials. For the classical orthogonal polynomials, the recurrence relations and differential equations are well known and easy to calculate explicitly using an orthogonality relation or generating function. However with semi-classical orthogonal polynomials, the recurrence coefficients can no longer be expressed in an explicit form, but instead obeys systems of non-linear difference equations. These systems are derived by deriving compatibility relations between the recurrence relation and the differential equation. The compatibility problem can be approached in two ways; the first is the direct approach using the orthogonality relation, while the second introduces the Laguerre method, which derives a differential system for semi-classical orthogonal polynomials. We consider some semiclassical Hermite and Laguerre weights using the Laguerre method, before applying both methods to a semi-classical Jacobi weight. While some of the systems derived will have been seen before, most of them (at least not to our knowledge) have not been acquired from this approach. Chapter 3 considers a singular integral transform that is related to the Gel’fand-Levitan equation, which provides the inverse part of the inverse scattering method (a solution method of integrable systems). These singular integral transforms constitute a dressing method between elementary (bare) solutions of an integrable system to more complicated solutions of the same system. In the context of this thesis we are interested in adapting this method to the case of polynomial solutions and study dressing transforms between different families of polynomials, in particular between certain classical orthogonal polynomials and their semi-classical deformations. In chapter 4, a new class of orthogonal polynomials are considered from a formal approach: a family of two-variable orthogonal polynomials related through an elliptic curve. The formal approach means we are interested in those relations that can be derived, without specifying a weight function. Thus, we are mainly concerned with recursive structures, particularly on their explicit derivation so that a series of elliptic polynomials can be constructed. Using generalized Sylvester identities, recurrence relations are derived and we consider the consistency of their coefficients and the compatibility between the two relations. Although the chapter focuses on the structure of the recurrence relations, some applications are also presented.
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Oakley, Steven James. "Orthogonal polynomials in the approximation of probability distributions." Diss., The University of Arizona, 1990. http://hdl.handle.net/10150/185117.
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An approach to the problem of approximating a continuous probability distribution with a series in orthogonal polynomials is presented. The approach is first motivated with a discussion of theoretical distributions which are inherently difficult to evaluate. Additionally, a practical application which involves such a distribution is developed. The three families of orthogonal polynomials that pertain to the methodology--the Hermite, Laguerre, and Jacobi--are then introduced. Important properties and characterizations of these polynomials are given to lay the mathematical framework for the orthogonal polynomial series representation of the probability density function of a continuous random variable. This representation leads to a similar series for the cumulative distribution function, which is of more practical use for computing probabilities associated with the random variable. It is demonstrated that the representations require only the moments and the domain of the random variable to be known. Relationships of the Hermite, Laguerre, and Jacobi series approximations to the normal, gamma, and beta probability distributions, respectively, are also formally established. Examples and applications of the series are given with appropriate analyses to validate the accuracy of the approximation.
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Campetti, Marcos Henrique [UNESP]. "Polinômios ortogonais e L-ortogonais associados a medidas relacionadas." Universidade Estadual Paulista (UNESP), 2011. http://hdl.handle.net/11449/94202.
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O objetivo deste trabalho é fazer um estudo das propriedades de duas sequências de polinômios, {Pϕ0 n }∞ n=0 e {Pϕ1 n }∞ n=0, ortogonais com relação, respectivamente, às medidas dϕ0 e dϕ1, relacionadas entre si, e das propriedades de duas sequências de polinômios L-ortogonais, {Bψ0 n }∞ n=0 e {Bψ1 n }∞ n=0, quando as medidas associadas, dψ0 e dψ1, est˜ao tamb´em relacionadas. Para os polinômios ortogonais, foram considerados dois casos: polinômios ortogonais associados a medidas simétricas relacionadas por dϕ1(x) = c 1 + qx2 dϕ0(x) e polinˆomios ortogonais associados a medidas relacionadas por (x − q) dϕ1(x) = c dϕ0(x). Como exemplo, os resultados foram aplicados no estudo de polinˆomios ortogonais de Sobolev associados a medidas simétricas como os de Gegenbauer e Hermite, e medidas não simétricas como as de Jacobi e Laguerre. Para os polinômios L-ortogonais, considerou-se o estudo de duas sequências de polinômios associados a medidas positivas fortes dψ0 e dψ1 relacionadas por (z − κ) dψ1(z) = c dψ0(z). Como consequência dessas propriedades, algoritmos para gerar qualquer um dos pares de coeficientes das relações de recorrência, {αψ0 n , βψ0 n } ou {αψ1 n , βψ1 n }, dado o outro, foram dados.
The main purpose of this work is to study some properties of two sequences of polynomials, {Pϕ0 n }∞ n=0 and {Pϕ1 n }∞ n=0, orthogonal, respectively, with respect to the related measures dϕ0 and dϕ1, and properties of two sequences of L-orthogonal polynomials, {Bψ0 n }∞ n=0 and {Bψ1 n }∞ n=0, when the associated measures, dψ0 and dψ1, are also related. For the orthogonal polynomials, we considered two cases: orthogonal polynomials associated with symmetric measures related to each other by dϕ1(x) = c 1 + qx2 dϕ0(x) and orthogonal polynomials associated with measures related by (x − q) dϕ1(x) = c dϕ0(x). As examples, the results are applied to obtain informations regarding Sobolev orthogonal polynomials associated with symmetric measures as Gegenbauer and Hermite measures, and non-symmetrical measures such as Jacobi and Laguerre measures. For the L-orthogonal polynomials, we considered the study of two sequences of polynomials associated with strong positive measures dψ0 and dψ1 and related to each other by (z −κ) dψ1(z) = c dψ0(z). As a consequence of these properties, algorithms to generate any pair of coefficients of the recurrence relations, {αψ0 n , βψ0 n } or {αψ1 n , βψ1 n }, given the other, were given.
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Martins, Fabiano Alan. "Polinômios para-ortogonais e análise de freqüência /." São José do Rio Preto : [s.n.], 2005. http://hdl.handle.net/11449/94292.
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Orientador: Cleonice Fátima BraccialiBanca: Walter dos Santos Motta Junior
Banca: Eliana Xavier Linhares de Andrade
Resumo: O objetivo deste trabalho é estudar uma aplicação de polinômios conhecidos, como polinômios para-ortogonais, na solução do problema de análise de freqüência. Para isto, estudamos os polinômios de Szegö que são ortogonais no cýrculo unitário e que dão origem aos polinômios para-ortogonais. Estudamos casos especiais de polinômios para-ortogonais que, através de uma transformação do cýrculo unitário no intervalo [-1, 1], estão associados a certos polinômios ortogonais. Apresentamos também uma abordagem do problema de análise de freqüência utilizando esses polinômios ortogonais em [-1, 1].
Abstract: The purpose of this work is to study an application of some polynomials, known as para-orthogonal polynomials, in the solution of the frequency analysis problem. We study the Szeguo polynomials that are orthogonal polynomials on the unit circle and give origin to the para-orthogonal polynomials. We investigate some special cases of para-orthogonal polynomials that are associate with certain orthogonal polynomials on [-1, 1] through a transformation from the unit circle to the real interval [-1, 1]. We also present an approach of the frequency analysis problem using these orthogonal polynomials on [-1, 1].
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Kamioka, Shuhei. "Combinatorial Aspects of Orthogonal Polynomials and Discrete Integrable Systems." 京都大学 (Kyoto University), 2008. http://hdl.handle.net/2433/123829.
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Haq, Nazmus Saqeeb. "Orthogonal polynomials, perturbed Hankel determinants and random matrix models." Thesis, Imperial College London, 2013. http://hdl.handle.net/10044/1/21759.
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In this thesis, for a given weight function w(x), supported on [A,B]\subseteq\mathbb{R}, we consider the sequence of monic polynomials orthogonal with respect to w(x), and the Hankel determinant D_n=\det(\mu_{j+k})_{j,k=0}^{n-1}, generated from the moments \mu_j of w(x). A motivating factor for studying such objects is that by observing the Andreief-Heine identity, these determinants represent the partition function of a Hermitian random matrix ensemble. It is well known that the Hankel determinant can be computed via the product of L^2 norms over [A,B]\subseteq\mathbb{R} of the orthogonal polynomials associated with w(x). Since such polynomials satisfy a three-term recurrence relation, we also study the behaviour of the recurrence coefficients, denoted by \alpha_n and \beta_n, as these are intimately related to the behaviour of D_n. We consider orthogonal polynomials and Hankel determinants associated with the following two weight functions: First, we consider a deformation of the Jacobi weight, given by w(x)=(1-x^2)^{\apha} (1-k^2x^2)^{\beta}, x\in[-1,1], \alpha>-1, \beta\in\mathbb{R}, k^2\in(0,1). This is a generalization of a system of orthogonal polynomials studied by C. J. Rees in 1945. Such orthogonal polynomials are of great interest because the corresponding Hankel determinant is related to the \tau-function of a Painlevé VI differential equation, the special cases of which are related to enumerative problems arising from String theory. For finite n---we employ the orthogonal polynomial ladder operators (formulae that raise or lower the index of the polynomial) to find equations for auxiliary quantities defined by the corresponding orthogonal polynomials, from which we derive differential identities satisfied by the Hankel determinant, and differential-difference identities for the recurrence coefficients \alpha_n and \beta_n. Making use of the ladder operators, we find that the recurrence coefficient \beta_n(k^2), n=1,2,...; and P_1(n,k^2), the coefficient of x^{n-2} of the corresponding monic orthogonal polynomials, satisfy second order non-linear difference equations. The large n expansion based on the difference equations when combined Toda-type differential relations satisfied by the associated Hankel determinant yields a complete asymptotic expansion of D_n. The finite n representation of D_n in terms of a particular Painlevé VI equation is also discussed as well as the generalization of the linear second order differential equation found by Rees. Second, we consider the deformed Laguerre weight: w(x)=x^{\alpha} e^{-x} (t+x)^{N_s} (T+x)^{-N_s}, x\in[0,\infty), \alpha>-1, T, t, N_s>0. This weight is of interest since it appears in the study of a multiple-antenna wireless communication scenario. The key quantity determining system performance is the statistical properties of the signal-to-noise ratio (SNR) \gamma, which recent work has characterized through its moment generating function, in terms of the Hankel determinant generated via our deformed Laguerre weight. We make use of the ladder operators to give an exact finite n characterization of the Hankel determinant in terms of a two-variable generalization of a Painlevé V differential equation, which reduces to Painlevé V under certain limits. We also employ Dyson's Coulomb fluid theory to derive an approximation for D_n---in the limit where n is large. The finite and large n characterizations are then used to compute closed-form (non-determinantal) expressions for the cumulants of the distribution of \gamma, and to compute wireless communication performance quantities of engineering interest.
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Balderrama, Cristina. "Orthogonal polynomials with hermitian matrix argument and associated semigroups." Angers, 2009. http://www.theses.fr/2009ANGE0035.
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Dans ce travail, nous construisons et étudions des familles de polynômes orthogonaux généralisés définis dans l'espace des matrices hermitiennes qui sont associées à une famille de polynômes orthogonaux sur R. Nous considérons plusieurs normalisations pour ces polynômes, et obtenons des formules classiques à partir des formules correspondantes pour des polynômes définis sur R. Nous construisons également des semi-groupes d'opérateurs associés aux polynômes orthogonaux généralisés, et donnons l'expression du générateur infinitésimal de ce semi-groupe ; nous prouvons que ce semi-groupe est markovien dans les cas classiques. En ce qui concerne les expansions d-dimensionnelles de Jacobi nous étudions les notions d'intégrale fractionnelle (potentiel de Riesz), de potentiel de Bessel et de dérivées fractionnelles. Nous donnons une nouvelle décomposition de l'espace L2 associé à la mesure de Jacobi d-dimensionnelle, et obtenons un analogue du théorème du multiplicateur de Meyer dans ce cadre. Nous étudions aussi les espaces de Jacobi-SobolevIn this work we construct and study families of generalized orthogonal polynomials with hermitian matrix argument associated to a family of orthogonal polynomials on R. Different normalizations for these polynomials are considered and we obtain some classical formulas for orthogonal polynomials from the corresponding formulas for the one–dimensional polynomials. We also construct semigroups of operators associated to the generalized orthogonal polynomials and we give an expression of the infinitesimal generator of this semigroup and, in the classical cases, we prove that this semigroup is also Markov. For d–dimensional Jacobi expansions we study the notions of fractional integral (Riesz potentials), Bessel potentials and fractional derivatives. We present a novel decomposition of the L2 space associated with the d–dimensional Jacobi measure and obtain an analogous of Meyer's multiplier theorem in this setting. Sobolev Jacobi spaces are also studied
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Mbuyi, Cimwanga Norbert. "Interlacing zeros of linear combinations of classical orthogonal polynomials." Thesis, University of Pretoria, 2009. http://hdl.handle.net/2263/25258.
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Stoiciu, Mihai Simon Barry. "Zeros of random orthogonal polynomials on the unit circle /." Diss., Pasadena, Calif. : California Institute of Technology, 2005. http://resolver.caltech.edu/CaltechETD:etd-05272005-110242.
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Zhang, Lun. "Global asymptotics of orthogonal polynomials via Riemann-Hilbert approach /." access full-text access abstract and table of contents, 2009. http://libweb.cityu.edu.hk/cgi-bin/ezdb/thesis.pl?phd-ma-b23749453f.pdf.
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Thesis (Ph.D.)--City University of Hong Kong, 2009."Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves [95]-100)
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Webb, Grayson. "Biorthogonal Polynomials." Thesis, Linköpings universitet, Matematik och tillämpad matematik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-140733.
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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.
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Yahaya, Daud. "Polynomial interpolation on a triangular region." Thesis, University of St Andrews, 1994. http://hdl.handle.net/10023/13887.
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It is well known that given f there is a unique polynomial of degree at most n which interpolates f on the standard triangle with uniform nodes (i, j), i, j ≥ 0, i + j ≤n. This leads us to the study of polynomial interpolation on a "triangular" domain with the nodes, S = {([i], [j]): i, j ≥ 0, i + j ≤n}, [k] = [k][sub]q = (1-qk)/(1-q), q > 0, which includes the standard triangle as a special case. In Chapter 2 of this thesis we derive a forward difference formula (of degree at most n) in the x and y directions for the interpolating polynomial P[sub]n on S. We also construct a Lagrange form of an interpolating polynomial which uses hyperbolas (although its coefficients are of degree up to 2n) and discuss a Neville-Aitken algorithm. In Chapter 3 we derive the Newton formula for the interpolating polynomial P[sub]n on the set of distinct points {(xi, y[sub]j): i, j ≥ 0, i + j ≤n}. In particular if xi = [i][sub]p and y[sub]j = [j]q, we show that Newton's form of P[sub]n reduces to a forward difference formula. We show further that we can express the interpolating polynomial on S itself in a Lagrange form and although its coefficients Ln/ij are not as simple as those of the first Lagrange form, they all have degree n. Moreover, Ln/ij can all be expressed in terms of Lm/0,0, 0 ≤ m ≤ n. In Chapter 4 we show that P[sub]n has a limit when both p, q → 0. We then verify that the interpolation properties of the limit form depend on the appropriate partial derivatives of f(x, y). In Chapter 5 we study integration rules I[sub]n of interpolatory type on the triangle S[sub] = {(x, y): 0 ≤ x ≤y ≤ [n]). For 1 ≤ n ≤5, we calculate the weights wn/ij for I[sub]n in terms of the parameter q and study certain general properties which govern wn/ij on S[sub]n. Finally, Chapter 6 deals with the behaviour of the Lebesgue functions λ[sub]n(x, y; q) and the corresponding Lebesgue constant. We prove a property concerning where λ[sub]n takes the value 1 at points other than the interpolation nodes. We also analyse the discontinuity of the directional derivative of λ[sub]n on S[sub]n.
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Bracciali, Cleonice Fátima. "Some consequences of symmetry in strong Stieltjes distributions." Thesis, University of St Andrews, 1998. http://hdl.handle.net/10023/13881.
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Ali, A. Hamid A. Hussain. "Some aspects of the Jacobian conjecture : the geometry of automorphisms of C2." Thesis, University of St Andrews, 1987. http://hdl.handle.net/10023/13878.
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We consider the affine varieties which arise by considering invertible polynomial maps from C2 to itself of less than or equal to a given-degree. These varieties arise naturally in the investigation of the long-standing Jacobian Conjecture. We start with some calculations in the lower degree cases. These calculations provide a proof of the Jacobian conjecture in these cases and suggest how the investigation in the higher degree cases should proceed. We then show how invertible polynomial maps can be decomposed as products of what we call triangular maps and we are able to prove a uniqueness result which gives a stronger version of Jung's theorem [j] which is one of the most important results in this area. Our proof also gives a new derivation of Jung's theorem from Segre's lemma. We give a different decomposition of an invertible polynomial map as a composition of "irreducible maps" and we are able to write down standard forms for these irreducibles. We use these standard forms to give a description of the structure of the varieties of invertible maps. We consider some interesting group actions on our varieties and show how these fit in with the structure we describe. Finally, we look at the problem of identifying polynomial maps of finite order. Our description of the structure of the above varieties allows us to solve this problem completely and we are able to show that the only elements of finite order are those which arise from conjugating linear elements of finite order.
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Kim, Yong Y. "Flexural-Torsional Coupled Vibration of Rotating Beams Using Orthogonal Polynomials." Thesis, Virginia Tech, 2000. http://hdl.handle.net/10919/32616.
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Dynamic behavior of flexural-torsional coupled vibration of rotating beams using the Rayleigh-Ritz method with orthogonal polynomials as basis functions is studied. The present work starts from a review of the development and analysis of four basic types of beam theories: the Euler-Bernoulli, Rayleigh, Shear and Timoshenko and goes over to a study of flexural-torsional coupled vibration analysis using basic beam theories. In obtaining natural frequencies, orthogonal polynomials used in the Rayleigh-Ritz method are studied as an efficient way of getting results. The study is also performed for both non-rotating and rotating beams. Orthogonal polynomials and functions studied in the present work are : Legendre, Chebyshev, integrated Legendre, modified Duncan polynomials, the eigenfunctions of a pinned-free uniform beam, and the special trigonometric functions used in conjunction with Hermite cubics. Studied cases are non-rotating and rotating Timoshenko beams, bending-torsion coupled beam with free-free boundary conditions, a cantilever beam, and a rotating cantilever beam. The obtained natural frequencies and mode shapes are compared to those available in various references and results for coupled flexural-torsional vibrations are compared to both previously available references and with those obtained using NASTRAN finite element package.Master of Science
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Wade, Jeremy. "Summability of Fourier orthogonal expansions and a discretized Fourier orthogonal expansion involving Radon projections for functions on the cylinder /." Connect to title online (Scholars' Bank) Connect to title online (ProQuest), 2009. http://hdl.handle.net/1794/10245.
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Tcheutia, Daniel Duviol [Verfasser]. "Algorithmic Methods for Mixed Recurrence Equations, Zeros of Classical Orthogonal Polynomials and Classical Orthogonal Polynomial Solutions of Three-Term Recurrence Equations / Daniel Duviol Tcheutia." Kassel : Universitätsbibliothek Kassel, 2019. http://d-nb.info/1195722036/34.
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Campetti, Marcos Henrique. "Polinômios ortogonais e L-ortogonais associados a medidas relacionadas /." São José do Rio Preto : [s.n.], 2011. http://hdl.handle.net/11449/94202.
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Orientador: Eliana Xavier Linhares de AndradeBanca: Fernando Akira Kurokawa
Banca: Cleonice Fátima Bracciali
Resumo: O objetivo deste trabalho é fazer um estudo das propriedades de duas sequências de polinômios, {Pϕ0 n }∞ n=0 e {Pϕ1 n }∞ n=0, ortogonais com relação, respectivamente, às medidas dϕ0 e dϕ1, relacionadas entre si, e das propriedades de duas sequências de polinômios L-ortogonais, {Bψ0 n }∞ n=0 e {Bψ1 n }∞ n=0, quando as medidas associadas, dψ0 e dψ1, est˜ao tamb'em relacionadas. Para os polinômios ortogonais, foram considerados dois casos: polinômios ortogonais associados a medidas simétricas relacionadas por dϕ1(x) = c 1 + qx2 dϕ0(x) e polinˆomios ortogonais associados a medidas relacionadas por (x − q) dϕ1(x) = c dϕ0(x). Como exemplo, os resultados foram aplicados no estudo de polinˆomios ortogonais de Sobolev associados a medidas simétricas como os de Gegenbauer e Hermite, e medidas não simétricas como as de Jacobi e Laguerre. Para os polinômios L-ortogonais, considerou-se o estudo de duas sequências de polinômios associados a medidas positivas fortes dψ0 e dψ1 relacionadas por (z − κ) dψ1(z) = c dψ0(z). Como consequência dessas propriedades, algoritmos para gerar qualquer um dos pares de coeficientes das relações de recorrência, {αψ0 n , βψ0 n } ou {αψ1 n , βψ1 n }, dado o outro, foram dados.
Abstract: The main purpose of this work is to study some properties of two sequences of polynomials, {Pϕ0 n }∞ n=0 and {Pϕ1 n }∞ n=0, orthogonal, respectively, with respect to the related measures dϕ0 and dϕ1, and properties of two sequences of L-orthogonal polynomials, {Bψ0 n }∞ n=0 and {Bψ1 n }∞ n=0, when the associated measures, dψ0 and dψ1, are also related. For the orthogonal polynomials, we considered two cases: orthogonal polynomials associated with symmetric measures related to each other by dϕ1(x) = c 1 + qx2 dϕ0(x) and orthogonal polynomials associated with measures related by (x − q) dϕ1(x) = c dϕ0(x). As examples, the results are applied to obtain informations regarding Sobolev orthogonal polynomials associated with symmetric measures as Gegenbauer and Hermite measures, and non-symmetrical measures such as Jacobi and Laguerre measures. For the L-orthogonal polynomials, we considered the study of two sequences of polynomials associated with strong positive measures dψ0 and dψ1 and related to each other by (z −κ) dψ1(z) = c dψ0(z). As a consequence of these properties, algorithms to generate any pair of coefficients of the recurrence relations, {αψ0 n , βψ0 n } or {αψ1 n , βψ1 n }, given the other, were given.
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Barros, Michele Carvalho de. "Comportamento assintótico dos polinômios ortogonais de Sobolev-Jacobi e Sobolev-Laguerre /." São José do Rio Preto : [s.n.], 2008. http://hdl.handle.net/11449/94284.
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Orientador: Eliana Xavier Linhares de AndradeBanca: Ana Paula Peron
Banca: Alagacone Sri Ranga
Resumo: Sejam Sn(x); n ¸ 0; os polinômios de Sobolev, ortogonais com relação ao produto interno hf; giS = ZR f(x)g(x)dÃ0(x) + ¸ ZR f0(x)g0(x)dÃ1(x); ¸ > 0; onde fdÃ0; dÃ1g forma um par coerente de medidas relacionadas às medidas de Jacobi ou de Laguerre. Denotemos por PÃ0 n (x) e PÃ1 n (x); n ¸ 0; os polinômios ortogonais com respeito a dÃ0 e dÃ1; respectivamente. Neste trabalho, estudamos o comportamento assintótico, quando n ! 1; das razões entre os polinômios de Sobolev, Sn(x); e os polinômios ortogonais PÃ0 n (x) e PÃ1 n (x); além do comportamento limite da razão entre esses dois últimos polinômios. Propriedades assintóticas para os coeficientes da relação de recorrência satisfeita pelos polinômios de Sobolev também foram estudadas.
Abstract: Let Sn(x); n ¸ 0; be the Sobolev polynomials, orthogonal with respect to the inner product hf; giS = ZR f(x)g(x)dÃ0(x) + ¸ ZR f0(x)g0(x)dÃ1(x); ¸ > 0; where fdÃ0; dÃ1g forms a coherent pair of measures related to the Jacobi measure or Laguerre measure. Let PÃ0 n (x) and PÃ1 n (x); n ¸ 0; denote the orthogonal polynomials with respect to dÃ0 and dÃ1; respectively. In this work we study the asymptotic behaviour, as n ! 1; of the ratio between the Sobolev polynomials, Sn(x); and the ortogonal polynomials PÃ0 n (x) and PÃ1 n (x); as well as the limit behaviour of the ratio between the last two polynomials. Furthermore, we also give asymptotic results for the coefficients of the recurrence relation satisfied by the Sobolev polynomials.
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Rafaeli, Fernando Rodrigo. "Zeros de polinomios ortogonais na reta real." [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306958.
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Orientadores: Dimitar Kolev Dimitrov, Roberto AndreaniTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
Made available in DSpace on 2018-08-15T04:39:55Z (GMT). No. of bitstreams: 1 Rafaeli_FernandoRodrigo_D.pdf: 1231425 bytes, checksum: 33a23775a69f9b2b36c516f7cfcb0d0f (MD5) Previous issue date: 2010
Resumo: Neste trabalho são obtidos resultados sobre o comportamento de zeros de polinômios ortogonais. Sabe-se que todos eles são reais e distintos e fazem papel importante de nós das mais utilizadas fórmulas de integração numérica, que são as fórmulas de quadratura de Gauss. São obtidos resultados sobre a localização e a monotonicidade dos zeros, considerados como funções dos correspondentes parâmetros, dos polinômios ortogonais clássicos. Apresentaremos também vários resultados que tratam da localização, monotonicidade e da assintótica de zeros de certas classes de polinômios ortogonais relacionados com as medidas clássicas
Abstract: Results concerning the behaviour of zeros of orthogonal polynomials are obtained. It is known that they are real and distinct and play as important role as node of the most frequently used rules for numerical integration, the Gaussian quadrature formulae. Result about the location and monotonicity of the zeros, considered as functions of parameters involved in the measure, are provided. We present various results that treat questions about location, monotonicity and asymptotics of zeros of certain classes of orthogonal polynomials with respect to measure that are closely related to the classical ones
Doutorado
Analise Aplicada
Doutor em Matemática Aplicada
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Luo, Yu. "Studies on generalizations of the classical orthogonal polynomials where gaps are allowed in their degree sequences." Kyoto University, 2020. http://hdl.handle.net/2433/253419.
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Jacq, Thomas Soler. "Asymptotic spectral analysis of growing graphs and orthogonal matrix-valued polynomials." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2016. http://hdl.handle.net/10183/143939.
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Neste trabalho abordaremos a an alise espectral de grafos por dois estudos: técnicas de probabilidade quântica e por polinômios ortogonais com valores em matrizes. No Capítulo 1, consideraremos a matriz de adjacência do grafo tal como um operador linear e sua decomposição quântica permitir a uma an alise espectral que produzir a um teorema do limite central para tal grafo. No Capítulo 2, consideraremos uma medida com valores em matrizes induzida por polinômios ortogonais com valores em matrizes. Sob certas condições, e possível exibir explicitamente uma expressão de tal medida. Algumas aplicações em teoria dos grafos são dadas quando nos restringimos as matrizes estoc asticas e com valores em 0-1. Do nosso conhecimento, os cálculos e exemplos obtidos nas seçõoes 0.3.2, 0.3.3, 2.4 e 2.5 são novos.In this work we focus on the spectral analysis of graphs via two studies: quantum probabilistic techniques and by orthogonal matrix-valued polynomials. In Chapter 1 we consider the adjacency matrix of a graph as a linear operator, and its quantum decomposition will allow a spectral analysis that will produce a central limit theorem for such graph. In Chapter 2, we consider a matrix-valued measure induced by orthogonal matrix-valued polynomials. Under certain conditions, it is possible to display an explicit expression for such measure. Some applications to combinatorics and graph theory are given when we restrict to the stochastic and 0-1 matrices. Up to our knowledge, the calculations and examples obtained in sections 0.3.2, 0.3.3, 2.4 and 2.5 are new.
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Pinter, Ferenc J. "Perturbation of orthogonal polynomials on an arc of the unit circle /." The Ohio State University, 1995. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487862399450495.
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Sousa, Vítor Luís Pereira Morais de. "The Riemann-Hilbert method applied to the theory of orthogonal polynomials." Doctoral thesis, Universidade de Aveiro, 2011. http://hdl.handle.net/10773/3871.
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Takata, Tomohiro. "Certain multiple orthogonal polynomials and a discretization of the Bessel equation." 京都大学 (Kyoto University), 2006. http://hdl.handle.net/2433/144354.
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Kyoto University (京都大学)0048
新制・課程博士
博士(理学)
甲第11975号
理博第2955号
新制||理||1442(附属図書館)
23788
UT51-2006-C655
京都大学大学院理学研究科数学・数理解析専攻
(主査)教授 上野 健爾, 教授 井川 満, 教授 河野 明
学位規則第4条第1項該当
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Parra, Ferrada Ivan [Verfasser]. "Planar orthogonal polynomials and two dimensional Coulomb gases / Ivan Parra Ferrada." Bielefeld : Universitätsbibliothek Bielefeld, 2020. http://d-nb.info/1224313127/34.
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Nunes, Josiani Batista. "Limitantes para os zeros de polinômios gerados por uma relação de recorrência de três termos /." São José do Rio Preto : [s.n.], 2009. http://hdl.handle.net/11449/94251.
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Orientador: Eliana Xavier Linhares de AndradeBanca: Alagacone Sri Ranga
Banca: Andre Piranhe da Silva
Resumo: Este trabalho trata do estudo da localização dos zeros dos polinômios gerados por uma determinada relação de recorrência de três termos. O objetivo principal é estudar limitantes, em termos dos coeficientes da relação de recorrência, para as regiões onde os zeros estão localizados. Os zeros são explorados atravé do problema de autovalor associado a uma matriz de Hessenberg. As aplicações são consideradas para polinômios de Szeg"o fSng, alguns polinômios para- ortogonais ½Sn(z) + S¤n (z) 1 + Sn(0) ¾ e ½Sn(z) ¡ S¤n (z) 1 ¡ Sn+1(0) ¾, especialmente quando os coeficientes de reflexão são reais. Um outro caso especial considerado são os zeros do polinômio Pn(z) = n Xm=0 bmzm, onde os coeficientes bm; para m = 0; 1; : : : ; n, são complexos e diferentes de zeros.
Abstract: In this work we studied the localization the zeros of polynomials generated by a certain three term recurrence relation. The main objective is to study bounds, in terms of the coe±cients of the recurrence relation, for the regions where the zeros are located. The zeros are explored through an eigenvalue representation associated with a Hessenberg matrix. Applications are considered to Szeg}o polynomials fSng, some para-orthogonal polyno- mials ½Sn(z) + S¤n (z) 1 + Sn(0) ¾and ½Sn(z) ¡ S¤n (z) 1 ¡ Sn+1(0) ¾, especially when the re°ection coe±cients are real. As another special case, the zeros of the polynomial Pn(z) = n Xm=0 bmzm, where the non-zero complex coe±cients bm for m = 0; 1; : : : ; n, were considered.
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