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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.

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    $\begingroup$ I suggest that you fix your notation, since it is very confusing to have $p_n$ be a coefficient of the polynomial $p_n(t)$. Probably best to define $\|P(t)\|_2$ to be the $L^2$-norm of the vector formed from the coefficients of $P$, then write $\Phi_n$ as $\min_{\deg P=n,\,\|P\|=1}\int_0^1 P(t)^2\,dt$. $\endgroup$
    – Joe Silverman
    Commented Aug 8, 2015 at 18:10
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    $\begingroup$ As you already pointed out yourself, you are equivalently asking about the smallest eigenvalue of the Hilbert matrix $H_{jk}=1/(j+k+1)$. Googling this will produce a lot of information, including asymptotic formulae. $\endgroup$
    – Christian Remling
    Commented Aug 8, 2015 at 18:11
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    $\begingroup$ More specifically, the link below seems to give that $\Phi_n\sim (1+\sqrt{2})^{-4n}\sqrt{n}$: link.springer.com/article/10.1023%2FA%3A1004180718725#page-1 $\endgroup$
    – Christian Remling
    Commented Aug 8, 2015 at 18:13
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    $\begingroup$ @ChristianRemling: maybe you should elevate your comments into an answer so that this question attains "answered" status? $\endgroup$
    – Suvrit
    Commented Aug 8, 2015 at 22:11
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    $\begingroup$ Googling around, it seems that another reference for the decay of $\Phi_n$ is jstor.org/stable/2035162 ( published in the 60s) $\endgroup$
    – Bertram Q.
    Commented Aug 8, 2015 at 22:42

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This summarizes various comments from above. Write $p=(p_0,p_1,\ldots, p_n)$. Since $$ \Phi_n = \min_{\|p\|_2 =1} \sum_{j,k=0}^n \frac{p_jp_k}{j+k+1} , $$ we can also interpret $\Phi_n$ as the smallest eigenvalue of the (positive definite) Hilbert matrix $H_{jk}=1/(j+k+1)$, $j,k=0,1, \ldots , n$.

We find the following asymptotic formula in the literature: $$ \Phi_n = K\sqrt{n}\left( 1+\sqrt{2}\right)^{-4n}(1+o(1)), \quad\quad K=\frac{8\cdot 2^{3/4}\pi^{3/2}}{(1+\sqrt{2})^4} $$ See here and here.

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