Weighted approximation in R[superscript]n

The main body of this thesis is divided into three parts. Part 1, consisting of the first two chapters, contains all the required background material. We provide an overview of weighted potential theory and its application to the question of uniformly approximating continuous functions by functions of the form wnPn for a given weight w and a polynomial Pn of degree at most n. We also include the relevant pluripotential theory using Klimek's monograph as our primary reference. The basic results in weighted pluripotential theory were proven by Siciak, Bloom and by Bloom and Levenberg. Chapters 3 and 4 form Part 2 of the thesis. Chapter 3 consists of further results in weighted pluripotential theory. In the first section of Chapter 3 we establish some properties of the weighted pluricomplex Green function VΣ,Q. These include results concerning the behaviour of VΣ,Q under polynomial mappings, taking limits and on product sets. In the second section we prove some general results on the support Sw of the equilibrium measure. One notable result obtained is a characterization of the points in Sw. The focus of Chapter 4 is a generalized Weierstrass approximation problem. Specifically, we consider the question of uniform approximation by weighted multivariable polynomials on compact sets in [special characters omitted]. The final part of the thesis is a chapter on the approximation of continuous functions by multivariable incomplete polynomials in [special characters omitted]. Incomplete polynomials were first defined by G. G. Lorentz in the 1970s on the unit interval [0, 1] and they have been the subject of intense study in the general area of approximation theory. Recently, Kroó started to extend the theory of incomplete polynomials to [special characters omitted]. He obtained results on a restricted class of convex domains in [special characters omitted] which includes parallelograms with a vertex at the origin and sections of the unit disk with central angle less than π/2. We succeed in proving some general results in [special characters omitted] for n > 1 using weighted pluripotential techniques.

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