Let $V(x_1, \ldots, x_n)$ be the classical Vandermonde matrix $$ V(x_1, \ldots, x_n) = \begin{bmatrix} 1 & x_1 & \ldots & x_1^{n-1} \\ \vdots & & & \vdots \\ 1 & x_n & \ldots & x_n^{n-1} \end{bmatrix} $$ For a given vector $z \in \mathbb{R}^{n}$, what is known about $\|u\|_2$, where $V u = z$? Ideally, I am interested in computing a lower bound on $\|u\|_2$ in the case where $x_j = e^{i\theta_j}$ for distinct angles $\theta_j$, $$ z = \begin{bmatrix} e^{-i \theta_1} \\ \vdots \\ e^{-i \theta_n} \end{bmatrix}. $$ It may turn out that my special case is trivial with some nice trig function tricks, but I cannot seem to get a handle on it. My conjecture is that, in my special case, $\|u\|_2 \geq 1$.
Does anybody have any ideas on how to analyze the coefficients of $u$, or point me to some references that discuss such techniques? I might be unreasonably daunted by what appears to be very messy business.
I think the general approach is not as hard as I first expected, though in general probably very messy. If we know that $Vu = z$, then without loss of generality we can rewrite the linear system so that $z$ is independent of $\theta_j$. Then $$\|u\|_2^2 = \langle V^{-1} z, V^{-1} z \rangle = \langle (VV^{\ast})^{-1} z, z \rangle $$ Then we can explicitly compute $VV^{\ast}$, and then the $L^2$-norm can be extremized by calculating the gradient of $(VV^{\ast})^{-1}$.