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Here is the question:

Suppose $P_0, P_1, P_2, \dots$ are polynomials orthonormal with respect to the inner product $$(f,g)=\int_a^b f(x)g(x)W(x)dx,$$ where $W(x) > 0$ is a weight function and $P_n$ is of degree $n$. Is it true that $P_n$ has $n$ distinct roots in $(a,b)$?

Clearly $P_0$ has no roots and since $(P_0,P_1)=0$, I know $P_1$ must cross the $x$-axis at least once otherwise the integral would not equal $0$, so $P_1$ has one root in $(a,b)$. However, I'm not sure how to prove this for an arbitrary $n$-value (if it is true). I would appreciate any advice.

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    The fact that $(P_0, P_k) = 0$ when $k\neq 0$ shows that all $P_k$ have a root in $(a,b)$, not just for $k = 1$.2011-10-04
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    That's true but how can I show that Pn has n distinct roots in (a,b)?2011-10-04

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