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One of the most important ways in which a metric is used is in approximation. Given a function *f*, finding a sequence which converges to *f* in the metric *d*_{∞} is called **uniform approximation**. The most important result in this area is due to the German mathematician Karl Weierstrass (1815 to 1897).

*Any continuous function on a bounded interval can be uniformly approximated by polynomial functions.***Proof**

- There are several different ways of proving this important theorem. Here is a method which introduces concepts which are important in other areas of mathematics too.
**Definition**If

*f*and*g*are*suitable*functions on**R**, then the**convolution***f***g*is the function defined by*f***g*(*x*) =*f*(*x*-*t*)*g*(*t*)*dt*(*suitable*means ensuring that the integral exists).This concept of convolution is important in the theory of Laplace transforms among other places.

It turns out that the set

*X*of suitable functions is than a ring under the operations**+**and*****in much the same way that*X*forms a ring under the usual addition and multiplication.The Laplace transform is then a

*ring homomorphism*from (*X***+ ***) to (*X***+ .**)

That is: (*f*+*g*) = (*f*) + (*g*) and (*f***g*) = (*f*).(*g*).Thinking about convolution as an algebraic operation, we may ask:

*Is there an identity element for this operation*?

The answer is:*Almost*!The English mathematician Paul Dirac (1902 to 1984) one of the most important founders of Quantum Mechanics, invented the

*δ*-function.

This is a "function" with the properties:

*δ*(*x*) = 0 if*x*≠ 0 and*δ*(*x*)*dx*= 1.You should think of it as "The density function of a unit mass or charge at the origin".

Of course it is not really a function since we would have to have

*δ*(0) = ∞ in a rather special way, but it turns out that provided one only uses it in integrals everything is OK.For example,

*f***δ*(*x*) =*f*(*x*-*t*)*δ*(*t*)*dt*=*f*(*x*) since*δ*(*t*) = 0 except at*t*= 0.What we will now do is find a sequence of functions (

*K*_{n}) which approximate the*δ*-function. The sequence (*K*_{n}**f*) will then approximate*δ***f*=*f*.

**Definition**The

*n*-th**Landau kernel function***K*_{n}=*c*_{n}(1 -*x*^{2})^{n}for*x*∈ [-1, 1] and 0 otherwise, where*c*_{n}is chosen so that*K*= 1._{n}

Here are graphs of some of these functions:Note that these are the graphs of the density functions of unit masses concentrated on smaller and smaller areas.

**Lemma**

*If f is a continuous function on the interval*[-1, 1]*then K**_{n}*f is a polynomial.***Proof**

*K*_{n}**f*(*x*) =*K*_{n}(*x*-*t*)*f*(*t*)*dt*and*K*_{n}is a polynomial and so*K*_{n}(*x*-*t*) can be expanded as*g*_{0}(*t*) +*g*_{1}(*t*) = ... +*g*_{2n}(*t*)*x*^{2n}and so the integral is a polynomial in*x*.**Lemma**

*The sequence*(*K**_{n}*f*)→*f in d*._{∞}**Proof**

We need to show that*K*_{n}has "most of its area" concentrated near*x*= 0.

First we estimate how big*c*_{n}is:

(1 -*t*)^{2})^{n}*dt*= 2(1 -*t*)^{n}(1 +*t*)^{n}*dt*≥ 2 (1 -*t*)^{n}*dt*= 2/(*n*+1).

Since*K*_{n}= 1 we must have*c*_{n}≤ (*n*+1)/2.

[In fact,*c*_{n}grows like a multiple of √*n*. For large*n*,*c*_{n}is approximately 0.566√*n*.]

Look at the area under*K*_{n}which is*not*near 0.

*K*_{n}(*t*)*dt*=*c*_{n}(1 -*t*^{2})^{n}*dt*≤ (*n*+1)/2 (1 -*δ*^{2})^{n}*dt*

since*K*_{n}is decreasing on [*δ*, 1] and this is (*n*+ 1)/2 (1-*δ*^{2})^{n}(1-*δ*).

If*r*= 1 -*δ*^{2}then (*n*+ 1)*r*^{n}--> 0 as*n*→ ∞.

We are told that*f*is continuous and by a theorem we will prove in the next section we may assume that*f*is bounded by*M*(say).

If*x*∈ [0, 1] then given*ε*> 0 we can find*δ*> 0 such that if |*t*| <*δ*then |*f*(*x*-*t*) -*f*(*x*)| <*ε*.

So now look at the convolution :*K*_{n}**f*.

|*f*(*x*) -*K*_{n}**f*(*x*)| = |*f*(*x*) -*f*(*x*-*t*)|*K*_{n}(*t*)*dt*= + + .

Now on [-1, -*δ*] and on [*δ*, 1] we have*K*_{n}(*t*) is small if we choose*δ*small. In fact, we can choose*δ*so that*K*_{n}(*t*) <*ε*/*M*here and then the first and third integrals are <*ε*.

For the middle integral, |*f*(*x*) -*f*(*x*-*t*)|*K*_{n}(*t*)*dt*≤*ε**K*_{n}(*t*)*dt*<*ε*since*K*_{n}(*t*)*dt*= 1.

Thus |*f*(*x*) -*K*_{n}**f*(*x*)| is small when*n*is large and we have our convergence. This completes the proof of the Weierstrass approximation theorem.

**Remarks**- If you take a function like
*x*|*x*| with a graph:

which is continuous but which is not differentiable at*x*= 0, then we can approximate it uniformly by polynomial functions which are (of course) all differentiable.

Thus the uniform limit of differentiable functions is not necessarily differentiable.

- The Weierstrass theorem is about functions which are continuous on a closed bounded inteval like [
*a*,*b*].

Although one can define uniform limits of functions on (say) the whole real line**R**, there are continuous functions which "grow too fast" (like the exponential function*e*^{x}) or grow too slowly (like sin(*x*)) to be approximated well on the whole line. Of course such functions can all be approximated well on a "finite bit" of**R**.

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- If you take a function like