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?