We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The subspace $G = \mathbb{R}_{2m-1}^0[X]$ (polynomials of degree at most $2m-1$ vanishing at $\pm 1$). Let $(P_1, \ldots, P_{2m-2})$ be any basis of $G$. We consider the two square matrices $A = [a_{i,j}]_{1 \leq i,j \leq 2m-2}$ and $B = [b_{i,j}]_{1 \leq i,j \leq 2m-2}$ defined by $$a_{i,j} = (P_i \mid P_j), \quad b_{i,j} = (P_i' \mid P_j')$$ Determine the ratio $$\frac{\operatorname{det}(A)}{\operatorname{det}(B)}$$ as a function of $m$.
We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product
$$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$
The subspace $G = \mathbb{R}_{2m-1}^0[X]$ (polynomials of degree at most $2m-1$ vanishing at $\pm 1$).
Let $(P_1, \ldots, P_{2m-2})$ be any basis of $G$. We consider the two square matrices $A = [a_{i,j}]_{1 \leq i,j \leq 2m-2}$ and $B = [b_{i,j}]_{1 \leq i,j \leq 2m-2}$ defined by
$$a_{i,j} = (P_i \mid P_j), \quad b_{i,j} = (P_i' \mid P_j')$$
Determine the ratio
$$\frac{\operatorname{det}(A)}{\operatorname{det}(B)}$$
as a function of $m$.