Weierstrass’s theorem — Leaving no ‘Stone’ unturned

Abstract

In this article, we discuss the basic theme of approximating functions by polynomial functions. Although it is exemplified by the classical theorem of Weierstrass, the theme goes much further. Even on the face of it, the advantage of polynomial approximations can be seen from the fact that unlike general continuous functions, it is possible to numerically feed polynomial interpolations of such functions into a computer and the justification that we will be as accurate as we want is provided by the theorems we discuss. In reality, this theme goes deep into subjects like Fourier series and has applications like separability of the space of continuous functions. Marshall Stone’s generalisation to compact Hausdorff spaces is natural and important in mathematics. Applications of the Weierstrass approximation theorem abound in mathematics — to Gaussian quadrature for instance.