Let it be given that $A$ is a real valued $m$ by $n$ matrix with entries $p_j\left(\frac{i}{m-1}\right)_{i,j=0}^{m-1,n-1}$, where $p_j$ is a Legendre polynomial. Show that $$\frac{\|{x}\|_2}{2} \leq \sqrt{\frac{2}{m}}\|{Ax}\|_2\leq 3\frac{\|{x}\|_2}{2}$$ when $m\geq Cn^2$ for $C\in\mathbb{R}$ and for all $x\in\mathbb{C}^n$.

I am given a hint:

$$\int_{-1}^1 |p'(x)|^2dx\leq cn^4\int_{-1}^{1}|p(x)|^2dx$$ for all $p\in\mathbb{P}_n$ and $c>0$.

Would someone be able to help me with this? I am somewhat stuck and do not really know where to start.

I know that the Legendre polynomials are degined as polynomials $p_1,p_1,\dots$ satisfying $p_n\in\mathbb{P}_n$ and

$$\int_0^1p_n(x)p_m(x)dx = \delta_{n,m}$$

for $n,m=0,1,2,\dots$.


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