Let $f:\mathbb{R}\to\mathbb{R}$ be continuously differentiable,a sequence of polynomials $p_n$ s.t. $p_n\to f$ and $p_n'\to f'$, uniformly on $[a,b]$

Question

Let $f:\mathbb{R}\to\mathbb{R}$ be continuously differentiable. Show that for every bounded interval $[a,b]$,there exists a sequence of polynomials $p_n$ such that $p_n\to f$ and $p_n'\to f'$, uniformly on $[a,b]$.

My solution:

I know that by using Stone-Weierstrass Theorem, there exists a sequence of polynomials $p_n$ such that $p_n\to f$ uniformly on $[a,b]$, and the promble maybe how to prove $p_n$ defined above also reaches $p_n'\to f'$.

By define a function series $(q_n)=(p_n-f)$,we have $q_n\to 0,~$then we have $q_n'=p_n'-f'\to 0$, so $p_n'\to f$, uniformly on $[a,b]$.

I guess this is obvious wrong, but I can't figure out why it is wrong and I'm stuck in how to solve this question correctly.

Answer 1

For a solution, see the following:

Choose by Weierstrass a sequence $(q_{n})$ of polynomials such that $q_{n}\rightarrow f'$ uniformly.

Let $p_{n}(x)=\int_{a}^{x}q_{n}(t)dt+f(a)$. Note that $p_{n}$ is still a polynomial with $p_{n}'=q_{n}$. Now we have \begin{align*} |p_{n}(x)-f(x)|&=\left|\int_{a}^{x}q_{n}(t)dt-\int_{a}^{x}f'(t)dt\right|\\ &\leq\int_{a}^{x}|q_{n}(t)-f'(t)|dt\\ &\leq(b-a)\sup_{t\in[a,b]}|q_{n}(t)-f'(t)|\\ &\rightarrow 0 \end{align*} uniformly in $x\in[a,b]$.