3. Electronic Theses and Dissertations (ETDs) - All submissions
Permanent URI for this communityhttps://wiredspace.wits.ac.za/handle/10539/45
Browse
3 results
Search Results
Item Approximation theory for exponential weights.(1998) Kubayi, David Giyani.Much of weighted polynomial approximation originated with the famous Bernstein qualitative approximation problem of 1910/11. The classical Bernstein approximation problem seeks conditions on the weight functions \V such that the set of functions {W(x)Xn};;"=l is fundamental in the class of suitably weighted continuous functions on R, vanishing at infinity. Many people worked on the problem for at least 40 years. Here we present a short survey of techniques and methods used to prove Markov and Bernstein inequalities as they underlie much of weighted polynomial approximation. Thereafter, we survey classical techniques used to prove Jackson theorems in the unweighted setting. But first we start, by reviewing some elementary facts about orthogonal polynomials and the corresponding weight function on the real line. Finally we look at one of the processes (If approximation, the Lagrange interpolation and present the most recent results concerning mean convergence of Lagrange interpolation for Freud and Erdos weights.Item Darboux-crum transformations of orthogonal polynomials and associated boundary conditions(2013-07-30) Rademeyer, Maryke CarleenLinear 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.Item Orthogonal polynomials and the moment problem(2012-10-01) Steere, Henry RolandThe classical moment problem concerns distribution functions on the real line. The central feature is the connection between distribution functions and the moment sequences which they generate via a Stieltjes integral. The solution of the classical moment problem leads to the well known theorem of Favard which connects orthogonal polynomial sequences with distribution functions on the real line. Orthogonal polynomials in their turn arise in the computation of measures via continued fractions and the Nevanlinna parametrisation. In this dissertation classical orthogonal polynomials are investigated rst and their connection with hypergeometric series is exhibited. Results from the moment problem allow the study of a more general class of orthogonal polynomials. q-Hypergeometric series are presented in analogy with the ordinary hypergeometric series and some results on q-Laguerre polynomials are given. Finally recent research will be discussed.