# Solving an inequality containing Legendre polynomials

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