1. Nov 8, 2012

### Pengwuino

1. The problem statement, all variables and given/known data

Why must the $\mu_i$ be the roots of $P_N(\mu_i) = 0$ to satisfy N even-moment conditions? Consider $2\pi \int d\mu \mu^{N+n}$ and write $\mu^{N+n}$ as $P_N(\mu)q_N(\mu) + R_k(\mu)$ which involves $P_N$ and the polynomials $q_N$ (quotient) and $R_k$ (remainder). If n<N, these are both of degree less than N.

2. Relevant equations

The $P_N(\mu)$ are the Legendre polynomials of order N.

3. The attempt at a solution

So my understanding is that the quadrature means that you can write some polynomial as $\int_{-1}^1 H(\mu) = \sum_{i=0}^{N} w_i P_i(\mu_i)$. I really have no idea how to go about this. Is my understanding of quadrature even correct in the first place?