Comparing different norms of a polynomial

Question

For $m\in \mathbb{N}$ and $a=(a_0,a_1,\ldots,a_{m}) \in \mathbb{R}^{m+1}$, consider the polynomial $P_{a}$ defined by $$ P_{a} (x):= a_0 + a_1 x^2 + \ldots + a_{m}x^{2m}\text{, for $x \in \mathbb{R}$.} $$ Then there is $C_m >0$ such that for every $a \in \mathbb{R}^{m+1}$, we have $$ \int_{[0,1]} P_{a}(x)^2 dx \geq C_m \|a\|^2, $$ where $\|\cdot\|$ is the standard Euclidean norm. The existence of $C_m$ is guaranteed by the following compactness argument. Define the (continuous) function $\phi : \mathbb{R}^{m+1} \rightarrow \mathbb{R}$ by $$ \phi(a) := \int_{[0,1]} P_{a}(x)^2 dx. $$ Then for $r \in \mathbb{R}$, $\phi(ra)= r^2 \phi(a)$ and hence $\phi(a)= \|a\|^2 \phi(a/\|a\|)$. So $C_m$ can be taken to be $C_m = \min_{v \in \mathbb{S}^m} \phi(v)$.

My questions are the following:

1. Can we get a lower bound for $C_m$ (in terms of $m$)?

2. Do we know which value of $a \in \mathbb{S}^{m}$ minimizes the function $\phi$?

My guess is that the minimizer $a_{\text{min}}$ must have coordinates whose signs alternate so that it minimizes the value of $a_0 + a_1 x^2 + \ldots + a_{m}x^{2m}$, but $a_j$'s should not have the same absolute value. Rather, they should increase in some way.

Thanks!

Answer 1

(Note: there is no need to consider only polynomials of even degree $n=2m$, since the same works for any degree $n\in\mathbb N$).

The map $Q_{n+1}:\mathbb R^{n+1}\ni a\mapsto \int_0^1 \big(\sum_{k=0}^na_kx^k\big)^2dx= \sum_{h,k}\frac1{k+h+1}a_ka_h=(Ha\cdot a)$ is a positive definite quadratic form on $\mathbb R^{n+1}$, corresponding to the Hilbert matrix $H$ of order $n+1$. So you want the minimum eigenvalue $C=\min_{a\neq 0}\frac {(Ha\cdot a)}{(a\cdot a)}$ of the Hilbert matrix $H$ of order $n+1$. These are computed and tabulated (I haven't a suitable link). Note that the maximum eigenvalue gives the best bound for the opposite problem, $\max_{a\ne0} \frac{(Ha\cdot a)}{(a\cdot a)}$.