I wrote down some proofs long time ago and I stated the following.
The (shifted) Legendre polynomial $p_i(x)$ with degree $i$ satisfies
$\max\{|p_i(x)|:0\leq x\leq 1\} = \sqrt{2i+1}$
$\max\{|p_i'(x)|:0\leq x\leq 1\} = i(i+1)\sqrt{2i+1}$
$\max\{|p_i''(x)|:0\leq x\leq 1\} = (1/2)(i-1)i(i+1)(i+2)\sqrt{2i+1}$
However, I cannot find the reference now. And I can only find
$|p_i(x)| \leq \sqrt{2i+1}$ for $x \in [0,1]$
$\sup_{x\in[0,1]} |p_i'(x)| \leq c \cdot i^{5/2}$ for some constant $ c > 0$
$\sup_{x\in[0,1]} |p_i''(x)| \leq c \cdot i^{9/2}$ for some constant $ c > 0$
May I know my equalities are correct? I would appreciate if someone could provide some references. Thanks a lot.
