Uniform approximation by polynomials

The famous Weierstrass approximation theorem states that any  continuous real-valued function on a compact interval can be uniformly approximated by a polynomial with an error as small as one likes. This theorem is the first significant result in approximation theory of one real variable and plays a key role in the development of general approximation theory. There are several different ways to prove this important theorem, but in this blog, we present a proof which makes use of convolution. In particular, the convolution with an approximation to the identity is  the important case on which this theorem is based.

What is approximation theory? In this most general form, we say that approximation theory is dedicated to the description of elements in a topological space X, which can be approximated by elements in a subset of X. That is to say, given an elements x\in X, we want to approximate x by an element a of some subset A of X. The elements of A are nice'' ortractable” and we want to make x and a as closer as possible. In other words, approximation theory allows to characterize the closure of A in X.

A special case of approximation theory is the approximation of continuous function, which has significant importance in real analysis and partial differential equations. Continuous functions can be very badly behaved, for instance, they can be nowhere differentiable. One the other hand, functions such as polynomials are always very well behaved, in particular being always differentiable. Fortunately, while most continuous functions are not as well behaved as polynomials, they can always be uniformly approximated by polynomials; this important result is known as the Weierstrass approximation theorem, and is the subject of this paper. We wish to prove the following:

Theorem. (Weierstrass approximation theorem) If [a,b] is an interval, f:[a,b]\to\mathbf{R} is a continuous function, and \varepsilon>0, then there exists a polynomial P on  [a,b] such that \|P-f\|_{\infty}\leq\varepsilon (i.e., |P(x)-f(x)|\leq\varepsilon for all x\in[a,b]).

Another way of stating the theorem is as follows. Let C([a,b]\to\mathbf{R}) be the space of continuous functions from [a,b] to \mathbf{R} with uniform metric d_\infty and P([a,b]\to\mathbf{R}) be the space of all polynomials on [a,b]; this is a subspace of C([a,b]\to\mathbf{R}), since polynomials are continuous. The Weierstrass approximation theorem asserts that every continuous function is in the closure of P([a,b]\to\mathbf{R}):
\displaystyle \overline{P([a,b]\to\mathbf{R})}=C([a,b]\to\mathbf{R}).
In particular, every continuous function on [a,b] is uniform limit of polynomials. Another way of saying this is that the space of polynomials is dense in the space of continuous functions with uniform topology.

The proof is fairly easy to remember in outline, once on has mastery over its ingredients.

  •  We can composite a linear term and assume that a=0, b=1. Similarly, we can subtract a liner term and assume that f(a)=f(b)=0.
  • If one takes convolution of a continuous function on [0,1] with an approximation to the identity, then the result is a uniformly good approximation.
  •  If the approximation to the identity is a polynomial restricted to the domain [-1,1], then the result convolution is a polynomial.
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