$ n, d \in \mathbb R , V = \{f ∈ R[X] \mid \deg(f) ≤ n\}$ and $x_0,\ldots,x_d \in \mathbb R$ are distinct.
Prove $ \langle f,g\rangle := \sum_{i=0}^d f(x_i)g(x_i)$ positive definite is, when $d ≥ n$.
I don't see the relation between $d$ and $n$. $d$ only increases the number of additions. Is there something I'm missing? Is there something with the distinctness?
Any help would be appreciated!
The fact is that, for $k+1$ distinct points $x_0,\cdots, x_k$ and a natural number $n>k$, there always exists a (necessarily non-zero) polynomial $p$ such that $\deg p=n$ and $\forall j,\ p(x_j)=0$, namely $p(x)=x^{n-k-1}(x-x_0)\cdots(x-x_k)$.
Clearly, $\sum\limits_{j=0}^k p(x_j)q(x_j)=0$ for all polynomials $q$, so the "at least $d$ distinct points and $d\ge n$" hypothesis is necessary for positive definiteness. It turns out to be sufficient as well.