Density in Approximation Theory

TL;DR

This paper surveys key results and methods related to the density of function families in approximation theory, focusing on their ability to approximate target functions.

Contribution

It provides a comprehensive overview of main density theorems and techniques in approximation theory, highlighting foundational results.

Findings

Summary of main density theorems

Overview of density methods

Clarification of approximation possibilities

Abstract

Approximation theory is concerned with the ability to approximate functions by simpler and more easily calculated functions. The first question we ask in approximation theory concerns the {\it possibility of approximation}. Is the given family of functions from which we plan to approximate dense in the set of functions we wish to approximate? In this work we survey some of the main density results and density methods.