I have a problem where I need to solve this maximization problem,
$$ \max_{\phi \in \mathbb{R}^{p+1}} \int_0^1 \left(\sum_{i=0}^p x^i \phi_i \right)^2 dx $$ $$ \textsf{subject to} \quad ||\phi||_2 = 1$$ In particular, I am interested in finding a nontrivial upper bound for the maximum value as a function of $p$. Does anyone have any leads?