0

Let $f:[a,b] \to \mathbb{R} \in \mathbb{C}^{k}$. Show that there exists a sequence of polynomials $\{p_n\}_{n \in \mathbb{N}}$ such that $p_n^{(j)}\to f^{(j)},\forall j =0,1,\dots,k$ uniformly.

My effort:

By Weierstrass approximation Theorem, there exists a sequence of polinomials $(p_n)_{n \in \mathbb{N}}$ s.t. $p_n \to f$ uniformly. I guess this is a sequence we need, but I don't know how to prove that $p_n' \to f'$ uniformly. Do you have any suggestion?

emacs drives me nuts
  • 10,390
  • 2
  • 12
  • 31
Joãonani
  • 1,684

1 Answers1

1

I would try to approximate $ f^{(k)} $ first instead of $ f $ itself, so we get a sequence $ p^k_n \to f^{(k)} $ uniformly. Then, we can integrate each $ p^k_n $ to obtain a sequence of polynomials $ \tilde{p}^{(k-1)}_n $, to which we can add constants $ c_n $ to obtain $ \tilde{p}^{(k-1)}_n (a) + c_n = p^{(k-1)}_n (a) \to f^{(k-1)}(a) $, which, together with the unfirom convergence $ p^k_n \to f^{(k)} $, gives $ p^{(k-1)}_n \to f^{(k-1)} $ uniformly.

Then iterate.

Tim
  • 11