I'm trying to get my head around the proof that an orthogonal polynomial ($P_n$ say) has at least n distinct roots. My understanding of the proof http://en.wikipedia.org/wiki/Orthogonal_polynomials#Existence_of_real_roots so far is that (by contradiction):
- Assume we have $m \le n$ roots. We'll show $m=n$
- Let $\displaystyle S(x) = \prod_{j=1}^m (x-x_j)$
- Gives us that $S(x)$ is an nth degree polynomial
- $S(x)$ changes sign at each of the $x_j$
My problem is this statement:
$S(x)P_n(x)$ is therefore strictly positive, or strictly negative, everywhere except at the $x_j$.
The $x_j$? What $x_j$? The lecture notes I have also say "except at $x_i$" so I'm pretty confused.
If someone can help me out here I'd greatly appreciate it. Thank you!