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!



Comparing different norms of a polynomial
(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](https://en.wikipedia.org/wiki/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, $\max\_{a\ne0} \frac{(Ha\cdot a)}{(a\cdot a)}$.


