Hilbert introduced his famous matrix when he studied the following problem. How small can the integral
$$\int_{a}^b|p(x)|^2dx $$
become for a non-zero polynomial $p$ with integer coefficients? He answered this question by showing that the discriminant of the associated quadratic form on the space of polynomials of degree at most $n$ is of the from $c_n\left(\frac{b-a}4\right)^{n^2}$ for $c_n$ tending to 1. He then applied Minkowski's theorem to deduce that, if $(b-a) < 4$, then indeed we can make the $L^2$ norm on the interval $[a,b]$ as small as we like. My question is whether the converse is true. So, if $b-a > 4$ then the $L^2$ norm is bounded from below by a positive constant. My second question, is whether there are generalizations of his result to integrals over sets other than intervals.
The article I am referring to is: ``Ein Beitrag zur Theorie des Legendre'schen Polynoms." Thanks!
1 Answer
A stronger result is true: if $b-a>4$, then the infimum of $\int |p|^2$ is positive when taken over all monic polynomials $p(x)=x^n+c_{n-1}x^{n-1}+\ldots +c_0$.
This follows from facts about orthogonal polynomials and their associated Jacobi matrices. Here we are interested in the (spectral) measure $d\rho(x)=\chi_{(a,b)}(x)\, dx$. (To make sure that the standard framework from below applies, I should actually work with the normalized version $\rho/(b-a)$, but of course this does not affect the question under consideration here.)
Consider the associated orthogonal polynomials $p_n$. These satisfy a three term (Jacobi) recurrence relation $$ a_n p_{n+1}(x) + a_{n-1}p_{n-1}(x) + b_n p_n(x) = xp_n(x) . $$ If we start with the initial values $p_0=0$, $a_0p_1=1$, then these are normalized in the sense that $\int |p_n|^2\, d\rho =1$. (This is not immediately obvious but follows from the standard machinery because $u\mapsto \sum u_n p_n(x)$ is a unitary map from $\ell^2$ to $L^2(\rho)$ and $p_n$ is the image of $e_n\in\ell^2$ and $\|e_n\|=1$.)
We can now expand a general $p$ as $p=\sum_{n=1}^N c_n p_n$. By orthogonality, $\|p\|^2_{L^2(\rho)}=\sum |c_n|^2$. Moreover, the recursion shows that the leading coefficient of $p_n$ is $1/(a_1a_2\cdots a_{n-1})$. So if $p$ is monic, then $\|p\|\ge |c_N|\ge a_1\cdots a_{N-1}$.
So it now suffices to show that $\liminf_{n\to\infty} (a_1\cdots a_n)>0$. This looks plausible because at least in friendly situations, $A=\lim (a_1 \cdots a_n)^{1/n}$, if it exists, is the logarithmic capacity of the spectrum $E$ of $\rho$.
Theorem 1.4 of my paper here establishes the stronger result that $\liminf a_n\ge (b-a)/4$ in a different way. It also gives references on how to use potential theoretic tools to obtain the weaker (but more general) version.
Finally, for an intuitive explanation of why $b-a=4$ is the critical value, recall that the capacity of an interval $[a,b]$ equals one when $b-a=4$.
