Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 6B / a / Solution

The nonzero orthogonal polynomials $Q_0,\dots ,Q_n$ have distinct degrees and form an orthogonal basis of ${\mathcal{P}}_n$. DefineFor each $j\le n$,Thus the residual $f-p_n^{\ast }$ is orthogonal to all of ${\mathcal{P}}_n$.

For any $p\in {\mathcal{P}}_n$, writeThe two terms are orthogonal, so the Pythagorean theorem in an inner-product space givesEquality holds only for $p=p_n^{\ast }$. This proves the formula and uniqueness of the least-squares polynomial in an orthogonal-polynomial basis.