Let $P_n(t) = p_0 + p_1 t + \cdots + p_n t^n$ be a polynomial (with real coefficients) of degree $n$ in the variable $t$. I am interested in the quantity $$\Phi_n = \min_{\sum_{i=1}^n p_i^2 = 1} \int_0^1 P_n^2(t).$$ In particular: how fast does $\Phi_n$ approach zero as $n \rightarrow \infty$? Can anything be said about which polynomials achieve the minimum, or which achieve something close to the minimal $\Phi_n$?

Note that this can be reformulated as a question about the smallest eigenvalue of a certain matrix.

Hilbert matrix$H_{jk}=1/(j+k+1)$. Googling this will produce a lot of information, including asymptotic formulae. $\endgroup$ – Christian Remling Aug 8 '15 at 18:11