Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 62). Also available in electronic version.

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The fundamental theorem, as far as this work is concerned, is Weierstrass' theorem (1885) on the approximability of continuous functions by polynomials. Since the time of Weierstrass (1815-97) and his equally important contemporary Chebyshev (1821-94), the topic of approximation has grown enormously into a subject of considerable interest to both pure and applied mathematicians. The subject matter of this thesis, being exclusively concerned with polynomial approximations to a single-valued, function of one real variable, is on the side of 'applied' side of approximation theory. The first chapter lists the definitions and theorems required subsequently. Chapter is devoted to estimates for the maximum error in minimax polynomial approximations. Extensions of this are used to obtain crude error estimates for cubic spline approximations. The following chapter extends the minimax results to deal also with best L[sub]p polynomial approximations, which include beat least squares (L2) and best modulus of integral (L1) approximations as special cases. Chapter 4 is different in character. It is on the practical problem of approximating to convex or nearly convex data.

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.

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.

Orientadores: Dimitar Kolev Dimitrov, Roberto Andreani / Tese (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

This thesis deals mainly with geometric properties of generalized Bernstein polynomials which replace the single Bernstein polynomial by a one-parameter family of polynomials. It also provides a triangular decomposition and 1-banded factorization of the Vandermonde matrix. We first establish the generalized Bernstein polynomials for monomials, which leads to a definition of Stirling polynomials of the second kind. These are q-analogues of Stirling numbers of the second kind. Some of the properties of the Stirling numbers are generalized to their q-analogues. We show that the generalized Bernstein polynomials are monotonic in degree n when the function ƒ is convex ... Shape preserving properties of the generalized Bernstein polynomials are studied by making use of the concept of total positivity. It is proved that monotonic and convex functions produce monotonic and convex generalized Bernstein polynomials. It is also shown that the generalized Bernstein polynomials are monotonic in the parameter q for the class of convex functions. Finally, we look into the degree elevation and degree reduction processes on the generalized Bernstein polynomials.

A dissertation submitted to the Faculty of Science, School of Mathematics University of the Witwatersrand Johannesburg South Africa / Linear second order ordinary di erential boundary value problems feature prominently in many scienti c eld, such as physics and engineering. Solving these problems is often riddled with complications though a myriad of techniques have been devised to alleviate these di culties. One such method is by transforming a problem into a more readily solvable form or a problem which behaves in a manner which is well understood. The Darboux-Crum transformation is a particularly interesting transformation characterised by some surprising properties, and an increase in the number of works produced in the last few years related to this transformation has prompted this investigation. The classical orthogonal polynomials, namely those of Jacobi, Legendre, Hermite and Laguerre, have been nominated as test candidates and this work will investigate how these orthogonal families are a ected when transformed via Darboux-Crum transformations.

A thesis submitted to the Faculty of science, University of Witwatersrand, Johannesburg in fulfilment of the requirements of the degree of Doctor of Philosophy. Johannesburg 1995. / We investigate Mean Convergence of Lagrange Interpolation and Rates of Approximation for Erdo's Weights on the Real line. An Erdos Weight is of the form, W = exp[-Q], where typically Q is even, continous and is of faster than polynomial growth at infinity. Concerning Lagrange Interpolation, we first investigate the problem of formulating and proving the correct Jackson Theorems for Erdos Weights. [ Abbreviated abstract : Open document to view full version] / GR2017

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