Weierstrass Approximation Theorem

[adpc]

Homework Statement

Show that if f is continuously differentiable on [a, b], then there is a sequence of
polynomials pn converging uniformly to f such that p'n converge uniformly to f' as
well.

Homework Equations

The Attempt at a Solution

Let pn(t) = cn t^n
Use uniform convergence and integrate from a to x to get:
$$f(x) - f(a) = \lim_{n \rightarrow \infty} \left[ \frac{c_n}{n+1} ( x^{n+1} - a^{n+1}) \right]$$

Now what is next?? How do I show that f(a) converges to the right side of the limit? so that then i can conclude that f(x) = lim cn x^(n+1) / (n+1)

 

[ystael]

You seem to be confused. The Weierstrass approximation theorem tells you that, given any continuous function $$g$$ on $$[a, b]$$, there exists some sequence $$(p_n)$$ of polynomials which converges uniformly to $$g$$ on $$[a, b]$$. But it does not tell you anything about the form of the $$p_n$$; in particular, you cannot assume that they have the form $$p_n(t) = c_n t^n$$. So this approach will not get you anywhere.

Your intuition that you should be approximating $$f'$$ and integrating is, however, correct. Think about what integration does with accumulated errors in an estimate: if $$|p_n(t) - f'(t)| < \varepsilon$$, uniformly in $$t \in [a, b]$$, can you use that to construct a polynomial that estimates $$f$$ with some error related somehow to $$\varepsilon$$?