Let $f : [0,1] \to \mathbb{R}$ be a continuous function that vanishes at $x = 1.$ Show that there exists a sequence of polynomials, each vanishing at $x = 1$, which converges to $f$ uniformly on $[0,1].$
It feels like a Stone-Weierstrass question, but after looking over the Stone-Weierstrass proof several times I am not sure if it truly applies.
My second thought is Arzela-Ascoli – namely that the family of functions you consider are the collection of polynomials, call this collection $P$, that vanish at $x = 1.$ My problem then becomes that this family is not uniformly bound over the interval, and I do not know if we can assert that a sequence of these polynomials converges to a specific $f,$ I think Arzela-Ascoli only proves that a uniformly convergent sequence in $P$ exists.
Anyone have any insight? Thanks in advance.
Best Answer
Take the Stone Weierstrass polynomials $\{p_n\}$, and modify them. Let $p_n(1) = a_n$ and $q_n(x) = p_n(x) - a_n$. Since $a_n = p_n(1) \rightarrow f(1) = 0$, $\{q_n\}$ are convergent to $f$