Let $\mathbb P_N$ the space polynomials of degree at most $N$ on $X=[-1,1]$. What is
$$\sup_{f\in L^\infty(X)\setminus \mathbb P_N}\frac{\|f-P_2[f]\|_{\infty}}{d_\infty(f,\mathbb P_N)},$$
where $d_\infty(f,\mathbb P_N)$ deontes the infinity-norm distance between the function and the vector space, while $P_2[f]$ denotes the projection on $\mathbb P_N$ w.r.t. the two-norm (i.e. a truncated generalized Fourier Serie). In particular, I am interested in the order in $N$: is it $O(\log(N)),O(\sqrt N)$ or something else?
Motivation and thoughts
The motivation for this question stays in the fact that finding an $L^2$ projector is much simpler: it is just a truncated Fourier serie w.r.t. the Legendre polynomials, which are a basis of $L^2$. Therefore, I was interested in understanding how much we lose with this approximation.
For now, I have found an answer for the sace where $X$ is the torus and $\mathbb P_N$ are replaced with trigonometric polynomials. Using consideration based on the norm of the Dirichlet Kernel, one finds out that the order is $\log(N)$. Unfortunately, this procedure does not generalize to our case.