Approximating Lipschitz and continuous functions by polynomials; Jackson’s theorem

Abstract

The celebrated theorem of Weierstrass, dating from 1885, states that continuous functions can be uniformly approximated by polynomials on any bounded, closed interval. But just how well can we approximate by polynomials of a certain degree? Let us introduce some notation to facilitate the discussion. For an interval I (which will usually be ), denote by C (I) the space of continuous functions on I, and write for (the notation is often used). Uniform convergence of fn to f (on I) equates to the statement that . Denote by the space of polynomials of degree not more than n. This is a linear subspace of C (I) of dimension n + 1. We write