A presentation of the Weierstrass approximation theorem and the Stone-Weierstrass theorem and a comparison of these two theorems are the objects of this thesis.

The theory of Riemann surfaces is quite old, consequently it is well developed. Riemann surfaces originated in complex analysis as a means of dealing with the problem of multi-valued functions. Such multi-valued functions occur because the analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function. The theory splits in two parts; the compact and the non-compact case. The function theory developed on these cases are quite dissimilar. The main difficulty one encounters in the compact case is the scarcity of global holomorphic functions, which limits one’s study to meromorphic functions. This however is not an issue in non-compact Riemann surfaces, where one enjoys a vast variety of global holomorphic functions. While the function theory of compact Riemann surfaces is centered around the Riemann-Roch theorem, which essentially tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles, the function theory developed on non-compact Riemann surface engages tools for approximation of functions on certain subsets by holomorphic maps on larger domains. The most powerful tool in this regard is the Runge’s approximation theorem. An intriguing application of this is the Gunning-Narasimhan theorem, which says that every connected open Riemann surface has an immersion into the complex plane. The main goal of this project is to prove Runge’s approximation theorem and illustrate its effectiveness in proving the Gunning-Narasimhan theorem. Finally we look at an analogue of Gunning-Narasimhan theorem in the case of a compact Riemann surface.

This thesis takes a closer look at three fundamental Classical Theorems in Real Analysis. First, for the Bolzano Weierstrass Theorem, we will be interested in constructing a convergent subsequence from a non-convergent bounded sequence. Such a subsequence is guaranteed to exist, but it is often not obvious what it is, e.g., if an = sin n. Next, the H¨older Inequality gives an upper bound, in terms of p ∈ [1,∞], for the the integral of the product of two functions. We will find the value of p that gives the best (smallest) upper-bound, focusing on the Beta and Gamma integrals. Finally, for the Weierstrass Polynomial Approximation, we will find the degree of the approximating polynomial for a variety of functions. We choose examples in which the approximating polynomial does far worse than the Taylor polynomial, but also work with continuous non-differentiable functions for which a Taylor expansion is impossible.

Let K be a compact subset of Cn . The polynomially convex hull of K is defined as The compact set K is said to be polynomially convex if = K. A closed subset is said to be locally polynomially convex at if there exists a closed ball centred at z such that is polynomially convex. The aim of this thesis is to derive easily checkable conditions to detect polynomial convexity in certain classes of sets in This thesis begins with the basic question: Let S1 and S2 be two smooth, totally real surfaces in C2 that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is locally polynomially convex at the origin? If then it is a folk result that the answer is, “Yes.” We discuss an obstruction to the presumed proof, and use a different approach to provide a proof. When dimR it turns out that the positioning of the complexification of controls the outcome in many situations. In general, however, local polynomial convexity of also depends on the degeneracy of the contact of T0Sj with We establish a result showing this. Next, we consider a generalization of Weinstock’s theorem for more than two totally real planes in C2 . Using a characterization, recently found by Florentino, for simultaneous triangularizability over R of real matrices, we present a sufficient condition for local polynomial convexity at of union of finitely many totally real planes is C2 . The next result is motivated by an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc generated by z and h — where h is a nowhereholomorphic harmonic function on D that is continuous up to ∂D — equals . The abstract tools used by Axler and Shields make harmonicity of h an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if h is replaced by h+ R, where R is a nonharmonic perturbation whose Laplacian is “small” in a certain sense. Ideas developed for the latter result, especially the role of plurisubharmonicity, lead us to our final result: a characterization for compact patches of smooth, totallyreal graphs in to be polynomially convex.

Die vorliegende Arbeit beschäftigt sich mit drei Klassen ausgewählter nichtlinearer Probleme, die Forschungsgegenstand der angewandten Mathematik sind. Diese Probleme behandeln die Minimierung von Integralen in der Variationsrechnung (Kapitel 3), das Lösen partieller Differentialgleichungen (Kapitel 4) und das Lösen nichtlinearer Optimierungsaufgaben (Kapitel 5). Mit deren Hilfe lassen sich unterschiedlichste Phänomene der Natur- und Ingenieurwissenschaften sowie der Ökonomie mathematisch modellieren. Als konkretes Beispiel werden mathematische Modelle der Theorie elastischer Festkörper betrachtet. Das Ziel der vorliegenden Arbeit besteht darin, ein gegebenes nichtlineares Problem durch polynomiale Probleme zu approximieren. Um dieses Ziel zu erreichen, beschäftigt sich ein großer Teil der vorliegenden Arbeit mit der polynomialen Approximation von nichtlinearen Funktionen. Den Ausgangspunkt dafür bildet der Weierstraßsche Approximationssatz. Auf der Basis dieses bekannten Satzes und eigener Sätze wird als Hauptresultat der vorliegenden Arbeit gezeigt, dass im Übergang von einer gegebenen Funktion zum approximierenden Polynom wesentliche Eigenschaften der gegebenen Funktion erhalten werden können. Die wichtigsten Eigenschaften, für die dies bisher nicht bekannt war, sind: Quasikonvexität im Sinne der Variationsrechnung, Quasimonotonie im Zusammenhang mit partiellen Differentialgleichungen sowie Quasikonvexität im Sinne der nichtlinearen Optimierung (Theoreme 3.16, 4.10 und 5.5). Schließlich wird gezeigt, dass die zu den untersuchten Klassen gehörenden nichtlinearen Probleme durch polynomiale Probleme approximiert werden können (Theoreme 3.26, 4.16 und 5.8). Die dieser Approximation zugrunde liegende Konvergenz garantiert sowohl eine Approximation im Parameterraum als auch eine Approximation im Lösungsraum. Für letztere werden die Konzepte der Gamma-Konvergenz (Epi-Konvergenz) und der G-Konvergenz verwendet. / In this thesis, we are concerned with three classes of non-linear problems that appear naturally in various fields of science, engineering and economics. In order to cover many different applications, we study problems in the calculus of variation (Chapter 3), partial differential equations (Chapter 4) as well as non-linear programming problems (Chapter 5). As an example of possible applications, we consider models of non-linear elasticity theory. The aim of this thesis is to approximate a given non-linear problem by polynomial problems. In order to achieve the desired polynomial approximation of problems, a large part of this thesis is dedicated to the polynomial approximation of non-linear functions. The Weierstraß approximation theorem forms the starting point. Based on this well-known theorem, we prove theorems that eventually lead to our main result: A given non-linear function can be approximated by polynomials so that essential properties of the function are preserved. This result is new for three properties that are important in the context of the considered non-linear problems. These properties are: quasiconvexity in the sense of the calculus of variation, quasimonotonicity in the context of partial differential equations and quasiconvexity in the sense of non-linear programming (Theorems 3.16, 4.10 and 5.5). Finally, we show the following: Every non-linear problem that belongs to one of the three considered classes of problems can be approximated by polynomial problems (Theorems 3.26, 4.16 and 5.8). The underlying convergence guarantees both the approximation in the parameter space and the approximation in the solution space. In this context, we use the concepts of Gamma-convergence (epi-convergence) and of G-convergence.