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Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 87-88). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 87-88). Also available on the Internet.

Orientador: Dimitar Kolev Dimitrov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-16T01:35:19Z (GMT). No. of bitstreams: 1 Lucas_FabioRodrigues_D.pdf: 837192 bytes, checksum: 1cec40a06f620203e95cbca6134fd41a (MD5) Previous issue date: 2010 / Resumo: Nesta tese abordamos alguns problemas relacionados com zeros de polinômios e de funções inteiras. Estabelecemos fórmulas explícitas para os polinômios da sequência de Sturm, gerada por um polinômio e pela sua derivada. Como consequência, obtemos condições necessárias e suficientes para que um polinômio sem zeros múltiplos tenha somente zeros reais. Provamos também a veracidade de algumas condições necessárias para a hipótese de Riemann, estendendo desta forma um resultado anterior de Csordas, Norfolk e Varga que estabelecem uma conjectura de Pólya / Abstract: In this thesis we approach problems concerning zeros of polynomials and entire functions. We establish explicit formula for the polynomial in the Sturm sequence, generated by a polynomial and its derivative. As a consequence, we obtain necessary and sufficient conditions for a polynomial without multiple zeros to possess only real zeros. We prove also the truth of certain necessary conditions for the Riemann Hypothesis, thus extending a previous result of Csordas, Norfolk and Varga who established a conjecture of Pôlya / Doutorado / Analise Aplicada / Doutor em Matemática Aplicada

This dissertation investigates two problems in the function theory of one complex variable. In Chapter 1, we study the asymptotics and zero distribution of solutions of the differential equation wⁿ + A(z)w = 0, where A(z) is a transcendental entire function of very slow growth. The result parallels the classical case when A(z) is assumed to be a polynomial. An analogue concerning the case when A(z) is a transcendental entire function whose series expansion satisfies the Hadamard gap condition is given. In Chapter 2, we give upper bounds for the number of deficient functions of entire functions of completely regular growth and entire functions whose zeros have angular densities. In particular, the bound is 2λ + 1 if the entire function is of completely regular growth with order λ, 0 < λ < ∞. / Ph. D.

Ce mémoire contient quelques résultats sur l'intégration numérique. Ils sont liés à la célèbre formule de quadrature de K. F. Gauss. Une généralisation très intéressante de la formule de Gauss a été obtenue par P. Turán. Elle est contenue dans son article publié en 1948, seulement quelques années après la seconde guerre mondiale. Étant données les circonstances défavorables dans lesquelles il se trouvait à l'époque, l'auteur (Turán) a laissé beaucoup de détails à remplir par le lecteur. Par ailleurs, l'article de Turán a inspiré une multitude de recherches; sa formule a été étendue de di érentes manières et plusieurs articles ont été publiés sur ce sujet. Toutefois, il n'existe aucun livre ni article qui contiennent un compte-rendu détaillé des résultats de base, relatifs à la formule de Turán. Je voudrais donc que mon mémoire comporte su samment de détails qui puissent éclairer le lecteur tout en présentant un exposé de ce qui a été fait sur ce sujet. Voici comment nous avons organisé le contenu de ce mémoire. 1-a. La formule de Gauss originale pour les polynômes - L'énoncé ainsi qu'une preuve. 1-b. Le point de vue de Turán - Compte-rendu détaillé des résultats de son article. 2-a. Une formule pour les polynômes trigonométriques analogue à celle de Gauss. 2-b. Une formule pour les polynômes trigonométriques analogue à celle de Turán. 3-a. Deux formules pour les fonctions entières de type exponentiel, analogues à celle de Gauss pour les polynômes. 3-b. Une formule pour les fonctions entières de type exponentiel, analogue à celle de Turán. 4-a. Annexe A - Notions de base sur les polynômes de Legendre. 4-b. Annexe B - Interpolation polynomiale. 4-c. Annexe C - Notions de base sur les fonctions entières de type exponentiel. 4-d. Annexe D - L'article de P. Turán. / This mémoire contains some results about numerical integration. They are related to the famous quadrature formula of K. F. Gauss. A very interesting generalization of the formula of Gauss was obtained by P.Turán. It is contained in a paper that was published in 1948, only a few years after the second world war. Due to adverse circunstances he was in at the time, the author (Turán) left many details for the reader to fill in. Otherwise, the article of Turán inspired a multitude of research, and his formula has been extended in many ways and several papers have been written on this subject. However, there is no single book or paper where one can nd a clear and comprehensive account of the basic results pertaining to Turán's formula. Thus, I would like my Master's mémoire to contain enough details that can enlighten the reader and present an exposition of much that has been done on this subject. Here is how we have arranged the contents of the mémoire. 1-a. The original formula of Gauss for polynomials - statement along with a proof. 1-b. Turán's point of view - detailed account of the results contained in his paper. 2-a. A formula for trigonometric polynomials analogous to that of Gauss. 2-b. A formula for trigonometric polynomials analogous to that of Turán. 3-a. Two formulae for entire functions of exponential type, analogous to the one of Gauss for polynomials. 3-b. A formula for entire functions of exponential type, analogous to that of Turán. 4-a. Annexe A - Basic facts about Legendre polynomials. 4-b. Annexe B - Polynomial interpolation. 4-c. Annexe C - Basic facts about entire functions of exponential type. 4-d. Annexe D - Paper of P. Turán.

In this thesis, we examined some Dirichlet type problems of the form: \begin{eqnarray*} \triangle u & = & 0\ {\rm in\ } \mathbb{R}^n \\ u & = & f\ {\rm on\ } \psi = 0, \end{eqnarray*} and we were particularly interested in finding entire solutions when entire data was prescribed. This is an extension of the work of D. Siegel, M. Mouratidis, and M. Chamberland, who were interested in finding polynomial solutions when polynomial data was prescribed. In the cases where they found that polynomial solutions always existed for any polynomial data, we tried to show that entire solutions always existed given any entire data. For half space problems we were successful, but when we compared this to the heat equation, we found that we needed to impose restrictions on the type of data allowed. For problems where data is prescribed on a pair of intersecting lines in the plane, we found a surprising dependence between the existence of an entire solution and the number theoretic properties of the angle between the lines. We were able to show that for numbers $\alpha$ with $\omega_1$ finite according to Mahler's classification of transcendental numbers, there will always be an entire solution given entire data for the angle $2\alpha\pi$ between the lines. We were also able to construct an uncountable, dense set of angles of measure 0, much in the spirit of Liouville's number, for which there will not always be an entire solution for all entire data. Finally, we investigated a problem where data is given on the boundary of an infinite strip in the plane. We were unable to settle this problem, but we were able to reduce it to other {\it a priori} more tractable problems.

In this thesis, we examined some Dirichlet type problems of the form: \begin{eqnarray*} \triangle u & = & 0\ {\rm in\ } \mathbb{R}^n \\ u & = & f\ {\rm on\ } \psi = 0, \end{eqnarray*} and we were particularly interested in finding entire solutions when entire data was prescribed. This is an extension of the work of D. Siegel, M. Mouratidis, and M. Chamberland, who were interested in finding polynomial solutions when polynomial data was prescribed. In the cases where they found that polynomial solutions always existed for any polynomial data, we tried to show that entire solutions always existed given any entire data. For half space problems we were successful, but when we compared this to the heat equation, we found that we needed to impose restrictions on the type of data allowed. For problems where data is prescribed on a pair of intersecting lines in the plane, we found a surprising dependence between the existence of an entire solution and the number theoretic properties of the angle between the lines. We were able to show that for numbers $\alpha$ with $\omega_1$ finite according to Mahler's classification of transcendental numbers, there will always be an entire solution given entire data for the angle $2\alpha\pi$ between the lines. We were also able to construct an uncountable, dense set of angles of measure 0, much in the spirit of Liouville's number, for which there will not always be an entire solution for all entire data. Finally, we investigated a problem where data is given on the boundary of an infinite strip in the plane. We were unable to settle this problem, but we were able to reduce it to other {\it a priori} more tractable problems.

We prove several inequalities involving the Fourier transform of functions which are compactly supported. The constraint that the functions have compact support is a simplifying feature which is desirable in applications, but there is a trade-off in control of other relevant quantities-- such as the mass of the function. With applications in mind, we prove inequalities which quantify these types of trade-offs. / text

Ce mémoire contient quelques résultats sur l'intégration numérique. Ils sont liés à la célèbre formule de quadrature de K. F. Gauss. Une généralisation très intéressante de la formule de Gauss a été obtenue par P. Turán. Elle est contenue dans son article publié en 1948, seulement quelques années après la seconde guerre mondiale. Étant données les circonstances défavorables dans lesquelles il se trouvait à l'époque, l'auteur (Turán) a laissé beaucoup de détails à remplir par le lecteur. Par ailleurs, l'article de Turán a inspiré une multitude de recherches; sa formule a été étendue de di érentes manières et plusieurs articles ont été publiés sur ce sujet. Toutefois, il n'existe aucun livre ni article qui contiennent un compte-rendu détaillé des résultats de base, relatifs à la formule de Turán. Je voudrais donc que mon mémoire comporte su samment de détails qui puissent éclairer le lecteur tout en présentant un exposé de ce qui a été fait sur ce sujet. Voici comment nous avons organisé le contenu de ce mémoire. 1-a. La formule de Gauss originale pour les polynômes - L'énoncé ainsi qu'une preuve. 1-b. Le point de vue de Turán - Compte-rendu détaillé des résultats de son article. 2-a. Une formule pour les polynômes trigonométriques analogue à celle de Gauss. 2-b. Une formule pour les polynômes trigonométriques analogue à celle de Turán. 3-a. Deux formules pour les fonctions entières de type exponentiel, analogues à celle de Gauss pour les polynômes. 3-b. Une formule pour les fonctions entières de type exponentiel, analogue à celle de Turán. 4-a. Annexe A - Notions de base sur les polynômes de Legendre. 4-b. Annexe B - Interpolation polynomiale. 4-c. Annexe C - Notions de base sur les fonctions entières de type exponentiel. 4-d. Annexe D - L'article de P. Turán. / This mémoire contains some results about numerical integration. They are related to the famous quadrature formula of K. F. Gauss. A very interesting generalization of the formula of Gauss was obtained by P.Turán. It is contained in a paper that was published in 1948, only a few years after the second world war. Due to adverse circunstances he was in at the time, the author (Turán) left many details for the reader to fill in. Otherwise, the article of Turán inspired a multitude of research, and his formula has been extended in many ways and several papers have been written on this subject. However, there is no single book or paper where one can nd a clear and comprehensive account of the basic results pertaining to Turán's formula. Thus, I would like my Master's mémoire to contain enough details that can enlighten the reader and present an exposition of much that has been done on this subject. Here is how we have arranged the contents of the mémoire. 1-a. The original formula of Gauss for polynomials - statement along with a proof. 1-b. Turán's point of view - detailed account of the results contained in his paper. 2-a. A formula for trigonometric polynomials analogous to that of Gauss. 2-b. A formula for trigonometric polynomials analogous to that of Turán. 3-a. Two formulae for entire functions of exponential type, analogous to the one of Gauss for polynomials. 3-b. A formula for entire functions of exponential type, analogous to that of Turán. 4-a. Annexe A - Basic facts about Legendre polynomials. 4-b. Annexe B - Polynomial interpolation. 4-c. Annexe C - Basic facts about entire functions of exponential type. 4-d. Annexe D - Paper of P. Turán.