First,we define the scalar product $ <P;Q> = \int_{I} P(t)Q(t)w(t)dt $ where $w(t)$ is a weight function.
Then, we denote $ (P_n)_{n \in \mathbb{N}} $ the orthonormal basis of $(X^n)_{n \in \mathbb{N}}$ using Gram-Schmidt Process.
I have shown that $ \forall n \in \mathbb{N}, P_n $ has $ n $ distinct roots ana beetween two roots of $P_{n+1}$ there's only one root of $P_n$. Now I want to proove that $ \forall n \in \mathbb{N}$ and $x_0 \in I $ such as $P_n(x_0) \neq 0$, the number of roots of $P_n$ greater to $x_0$ is equal to the number of sign's change in the finite sequence $(P_k(x_o))_{k\leq n}$.
Thought I've Failed In:
- Sturm Sequences : Since there's a sign counting, it was the first thing I jumped on. I've read a book which is stated the in a Sturm sequence $(P_k)_{k \leq n}$ we must have $P_n (x) \neq 0 , \forall x \in I$.
I don't have any idea to think of,Is their a theorem involing sign change that could inform me more ? Is their an alternative to Sturm Sequence ?
Thanks For Reading it.