Made available in DSpace on 2015-03-26T14:00:06Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1695298 bytes, checksum: eb0eea0668d21ba1373ad1a2f002c072 (MD5) Previous issue date: 2013-04-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / There are many challenges that teaching and in particular the teaching of mathematics requires us today. Our work was done in view of present teachers and high school students an alter- native to the teaching of these subjects. Initially developed some considerations on teaching trigonometry, the intended objectives, target audience definition, methodological considerations and possible consequences of the same. Then the paper presents a theoretical synthesis of the subjects studied, relating algebraic situations with their graphic representations. For the application of the subject in the classroom has produced a guide to assist the student and the teacher in this task was eleborado the Teacher's Guide. This material seeks to offer learners conditions progressiver construct their learning and teachers the opportunity to address the issue differently and add to the high school study of Trigonometric Polynomials, with emphasis on building your graphics programs using dynamic geometry and note that changes in various parameters provide the respective graphs. It is worth mentioning that the proposed teaching presented is suggested as an intermediate situation between the Teaching of Trigonometry and Complex Numbers made in our schools and the most advanced processes trigonometric interpolation and Fourier series, respectively. / São muitos os desafios que o ensino e, em especial, o ensino de matemática impõe nos dia de hoje. Nosso trabalho foi elaborado na perspectiva de apresentar a professores e alunos do Ensino Médio uma alternativa para o ensino desses temas. Inicialmente foram elaborados algumas considerações sobre o ensino de trigonometria, os objetivos pretendidos, definição do público alvo, considerações metodológicas e possíveis desdobramentos do mesmo. Em seguida o trabalho apresenta uma síntese teórica dos assuntos estudados, relacionando situações algébricas com suas representações gráficas. Para a aplicação do tema em sala de aula foi elaborado um Guia do Aluno e para auxiliar o professor nesta tarefa foi eleborado o Guia do Professor. Este material procura oferecer ao educando condições de construir progressivamente a sua aprendizagem e aos professores a oportunidade de abordar o tema de forma diferenciada e de agregar ao Ensino Médio o estudo dos Polinômios Trigonométricos, com ênfase na construção de seus gráficos usando programas de geometria dinâmica e a observação que as mudanças nos diversos parâmetros proporcionam nos respectivos gráficos. Vale ressaltar que a proposta de ensino apresentada é sugerida como uma situação intermediária entre o Ensino de Trigonometria e Números Complexos realizados em nossas escolas e, respectivamente, os processos mais avançados de interpolação trigonométrica e as séries de Fourier, respectivamente.

Orientador: Eliana Xavier Linhares de Andrade / Banca: José Roberto Nogueira / Banca: Heloísa Helena Marino Silva / Resumo: O principal objetivo deste trabalho é realizar um estudo sobre polinômios algébricos e trigonométricos que possuem somente zeros reais. O Teorema de Hermite nos dá condições necessárias e su cientes para que isto aconteça. São discutidas questões relacionadas à localização dos zeros, onde a Regra de Sinais de Descartes teve grande importância. Além disso, alguns teoremas clássicos sobre zeros de polinômios algébricos e trigonométricos são apresentados. / Abstract: The main purpose of this work is to study algebraic and trigonometric poly- nomials that have only real zeros. The Hermite Theorem gives necessary and su cient conditions for this to be true. Questions concerning the locations of the zeros are discussed, where the Descarte's Rule of Signs is of great impor- tance. Furthermore, some classical theorems concerning zeros of algebraic and trigonometric polynomials are presented. / Mestre

Изучаются коэффициенты тригонометрического полинома при односторонних ограничениях на отрезках [0, π] и [0, 2π]. Для нечетных тригонометрических полиномов степени не выше двух получены точные верхние и нижние границы первого коэффициента при линейном одностороннем ограничении. Для полиномов более высоких степеней определено поведение максимальных и минимальных возможных значений при одностороннем линейном ограничении. Проведено исследование поведения максимальных и минимальных значений первых коэффициентов синус-полиномов при двусторонних линейных ограничениях. / We study coefficients of a trigonometric polynomial under a one-sided constraint on the intervals [0, π] и [0, 2π]. We obtain sharp upper and lower estimates of the first coefficient of an odd trigonometric polynomial of degree at most two under a linear constraint. For polynomials of higher degree, we find the behavior of maximal and minimal coefficients of a sine-polynomial under two-sided linear constraints.

We study the best constant C(n) in the Bernstein inequality between the L0-norm of the first derivative of a trigonometric polynomial and the L2-norm of the polynomial itself on the set of trigonometric polynomials of a given degree n ≥1 with real coefficients. We prove that on the subset of polynomials from Tn such that all zeros of the derivative of a polynomial are real, the Bernstein inequality holds with the constant n/√2. In the general case, we obtain the close two-sided estimates: n/√2≤C(n)≤n. / Изучается наилучшая константа C(n) в неравенстве Бернштейна между L0-нормой первой производной тригонометрического полинома и L2-нормой самого полинома на множестве Tn тригонометрических полиномов заданного порядка n ≥1 с вещественными коэффициентами. Показано, что на подмножестве полиномов из Tn, все нули производной которых вещественные, неравенство Бернштейна имеет место с константой n/√2. В общем случае для константы C(n) получены близкие двусторонние оценки n/√2≤C(n)≤n.

We study a Szego type inequality between the uniform norm of a fractional derivative of a conjugate trigonometric polynomial and the uniform norm of the polynomial itself. We prove that a set of extremal polynomials in the Szego inequality for the zero-order derivative on the set of trigonometric polynomials, in addition to odd polynomials found earlier by L.V.Taikov, contains even polynomials. We also describe the whole class of extremal polynomials / Изучается неравенство типа Сеге между равномерной нормой производной дробного порядка сопряженного тригонометрического полинома и равномерной нормой самого полинома. Доказано, что в неравенстве Сеге для производной нулевого порядка на множестве тригонометрических полиномов имеются как нечетные, найденные ранее Л.В.Тайковым, так и четные экстремальные полиномы. Также полностью описан класс экстремальных полиномов для данного случая.

Рассматривается задача о полиномах с фиксированными коэффициентами при старших гармониках, наименее уклоняющихся от нуля. В явном виде выписаны все тригонометрические полиномы с фиксированными коэффициентами при трех старших гармониках, наименее уклоняющиеся от нуля в интегральной норме, а также алгебраические многочлены с тремя фиксированными старшими коэффициентами, наименее уклоняющиеся от нуля в интегральной норме с весом Чебышева. / We consider the problem on polynomials with fixed higher coefficients that deviate least from zero. We find an explicit form for all trigonometric polynomials with fixed coefficients at three highest harmonics that deviate least from zero in the integral norm as well as algebraic polynomials with three fixed leading coefficients that deviate least from zero in the integral norm with the Chebyshev weight.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

Made available in DSpace on 2014-06-11T19:27:08Z (GMT). No. of bitstreams: 0 Previous issue date: 2003-02-24Bitstream added on 2014-06-13T20:08:13Z : No. of bitstreams: 1 botta_va_me_sjrp.pdf: 571155 bytes, checksum: 6e200c838e03e019c93da99a37b1515f (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / O principal objetivo deste trabalho é realizar um estudo sobre polinômios algébricos e trigonométricos que possuem somente zeros reais. O Teorema de Hermite nos dá condições necessárias e su cientes para que isto aconteça. São discutidas questões relacionadas à localização dos zeros, onde a Regra de Sinais de Descartes teve grande importância. Além disso, alguns teoremas clássicos sobre zeros de polinômios algébricos e trigonométricos são apresentados. / The main purpose of this work is to study algebraic and trigonometric poly- nomials that have only real zeros. The Hermite Theorem gives necessary and su cient conditions for this to be true. Questions concerning the locations of the zeros are discussed, where the Descarte's Rule of Signs is of great impor- tance. Furthermore, some classical theorems concerning zeros of algebraic and trigonometric polynomials are presented.

We study a Bernstein inequality for a fractional derivative of order α ≥ 0 of a trigonometric polynomial in the space L0. In the case of zero order derivative, we obtain two-sided estimates for a sharp constant in this inequality, which show its behavior with respect to n. For positive and sufficiently small α, we obtain an upper estimate for a constant in the Bernstein inequality in L0. In the second part of the dissertation, we obtain estimates for norms in the space L0 of operators that set several higher or lower coefficients of a trigonometric polynomial to be zero. / Изучается неравенство Бернштейна для дробной производной порядка α ≥ 0 тригонометрических полиномов в пространстве L0. В случае производной нулевого порядка получены двусторонние оценки точной константы в этом неравенстве, дающие порядок ее поведения по n. Для положительных, достаточно малых значений α получена оценка сверху константы в неравенстве Бернштейна в L0. Во второй части работы получены оценки норм в пространстве L0 операторов, которые зануляют несколько старших или младших коэффициентов тригонометрического полинома.