I'm doing exercises for analysis and got stuck, here is the whole problem:
(1)suppose $C_n=\left[\int_{-1}^1\left(1-x^2\right)^n \mathrm{~d} x\right]^{-1}$ ，show that： $C_n<\sqrt{n}$ ；
(2) suppose $f(x)$ is a continuous function on $[0,1]$ , with $f(0)=f(1)=0$. when $x \notin[0,1]$ , define $f(x)=$ 0 . take $Q_n(x)=C_n\left(1-x^2\right)^n$. show that : $$ P_n(x)=\int_{-1}^1 f(x+t) Q_n(t) \mathrm{d} t, \quad 0 \leqslant x \leqslant 1 $$ is a polynomial , and $$ \lim _{n \rightarrow \infty} P_n(x)=f(x) $$
uniformly converges on $[0,1]$ ;
(3) when $f(0)=f(1)=0$ does not occur, proof Stone-Weierstrass THM.
for (1)I've tried induction but when I make it from n to n+1,I don't know whether can I split up the integration.
and I also can't get the whole view of the proof . can anyone help? thanks a lot!
The post provides a proof of the Weierstrass theorem. Concerning $C_n$ the integral is used in proving the Wallis formula. In order to obtain the estimate we make use of the Bernoulli inequality $(1+a)^n\ge 1+na,$ $a>-1,$ and proceed as follows. $$ \int\limits_{-1}^1(1-x^2)^n\,dx =2 \int\limits_0^1(1-x^2)^n\,dx\ge 2\int\limits_{0}^{n^{-1/2}}(1-x^2)^n\,dx \\ \ge2\int\limits_{0}^{n^{-1/2}}(1-nx^2)\,dx=2[n^{-1/2}- {\textstyle{1\over 3}}n^{-1/2}]=\textstyle{4\over 3}n^{-1/2} $$ Thus $C_n\le {3\over 4}n^{1/2}.$
The formula $(2)$ holds since the mass of $Q_n$ is equal $1$ and it is concentrated near $0$ for large values of $n.$ Namely for any $0<\delta<1$ we have $$\lim_n\int\limits_{\delta\le |x|\le 1}Q_n(t)\,dt=0$$