Teoria de l’aproximació en una variable

Abstract

[en] The approximation theory is concerned with how functions can best be approximated with simpler functions, in this work we will focus in approximating functions by polynomials on one variable. We will start with the Stone-Weierstrass theorem, one of the firsts in the field which states that all continuous functions in a closed interval can be approximated uniformly by polynomials. We will continue with the Müntz-Szász theorem, in which there are added restrictions in the polynomials, specifically on the degrees of the monomials. Once we have seen this theorem, we will see differents extensions and variations of the same, beginning by the so called full Müntz-Sász theorem which considers all possible behaviours of the monomials degrees and the corresponding version for the Lebesgue space of functions. Finally, we will see a theorem that can be considered equivalent to the Weierstrass theorem but in the complex domain, the Runge’s theorem.