Need some good problems on Weierstrass Approximation Theorem

Question

I know the Weierstrass Approximation Theorem, and I know its proof. I however till now have not really found any good application of the theorem except in one problem where it is given that if $f$ is continuous on $[0,1]$ and $\int_0^1x^nf(x)dx=0$ for every $n$ then $f\equiv0$.

However, I am quite certain that it is an important theorem and I would like some good problems/exercises based on it. I would love some suggestions or problems you came across in some books/pdf which you liked.

Answer 1

Here are a few interesting questions:

An application of Weierstrass theorem?

Monotonic version of Weierstrass approximation theorem

approximation of a continuous function by polynomials over a strictly continuous monotone function

Application of Weierstrass theorem

Answer 2

1) You can use the Stone-Weierstrass theorem to prove the Peter-Weyl theorem.

2) You can use SW to prove the Brouwer fixed point theorem by approximating a continuous function on a disk by a polynomial. (You can find this in "Topology from a Differentiable Viewpoint").

Answer 3

At an elementary level, a typical application is the proof of the existence of an antiderivative of a function $f$ continuous on a compact interval $[a,b]\,$.

Weierstrass theorem assures the existence of a sequence of polynomials uniformly convergent to $f$ on $[a,b]\,$.

Now consider the corresponding sequence of antiderivatives having the same value in $a\,$.

So you have a sequence of polynomials convergent in $a$ and such that the corresponding sequence of derivatives converges uniformly to $f$. Then you can apply a well known theorem that assures the convergence of the sequence to a function $g$ such that $g'=f\,$.

Since there exist proofs of the Weierstrass theorem that don't use the integral (one of these is Lebesgue's debut in 1898), the above approach can well be considered alternative to Cauchy's one.