One-sided weighted approximation

This thesis is devoted to weighted approximation by polynomials on the real line and deals with what is called one-sided weighted approximation in Approximation Theory. More precisely, we investigate the existence of a sequence of polynomials which lie above (or below) a given function and approximate it arbitrarily well. We present an overview of this subject in the Lp-spaces, starting with the early works of Freud and ending by the latest results in this area. This branch of weighted approximation was developed in the last 30-40 years and now contains a big variety of techniques and approaches. We develop new methods on one-sided approximation in the Linfinity-norm and obtain several results. We also discuss a number of open questions and describe possible directions of future research in this area.