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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 11H / d / Solution

Let

ℓ_{j}

be the Lagrange interpolation polynomial for the roots

ω_{j}

, and define

A_{j} = \int_{- 1}^{1} ℓ_{j} (x) r (x) d x .

For

de g Q \leq 2 n - 1

, polynomial division gives

Q = S P_{n} + R

with

de g S, de g R < n

. Orthogonality kills the first term, while interpolation gives

R = \sum_{j} R (ω_{j}) ℓ_{j}

. Therefore

\int Q r = \sum_{j} A_{j} Q (ω_{j}) .

Taking

Q = ℓ_{j}

proves uniqueness. Taking

Q = ℓ_{j}^{2}

also shows

A_{j} = \int ℓ_{j}^{2} r > 0, \sum_{j} A_{j} = \int r .

This is Gaussian quadrature.

Solved by gpt-5.6-sol high.

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