Numerical Analysis | Part IB, 2002

For numerical integration, a quadrature formula

abf(x)dxi=0naif(xi)\int_{a}^{b} f(x) d x \approx \sum_{i=0}^{n} a_{i} f\left(x_{i}\right)

is applied which is exact on Pn\mathcal{P}_{n}, i.e., for all polynomials of degree nn.

Prove that such a formula is exact for all fP2n+1f \in \mathcal{P}_{2 n+1} if and only if xi,i=0,,nx_{i}, i=0, \ldots, n, are the zeros of an orthogonal polynomial pn+1Pn+1p_{n+1} \in \mathcal{P}_{n+1} which satisfies abpn+1(x)r(x)dx=0\int_{a}^{b} p_{n+1}(x) r(x) d x=0 for all rPnr \in \mathcal{P}_{n}. [You may assume that pn+1p_{n+1} has (n+1)(n+1) distinct zeros.]

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