Past exam of the numerical analysis course of the University of Oxford / 2014 / 2 / b

Write $p_n(x)={\sum }_{j=0}^n{c}_j{x}^j$. The allowed orthogonal projection characterization, applied to the basis $1,x,\dots ,x^n$, gives the normal equationsExplicitly $G_{ij}=0$ if $i+j$ is odd and $G_{ij}=2/(i+j+1)$ if it is even. These equations can also be derived by differentiating $\int (u-\sum c_j{x}^j{)}^{2}dx$ with respect to each coefficient.

The Gram matrix $G$ is a symmetric positive-definite matrix. For any nonzero real coefficient vector $z$,The polynomial inside the square is nonzero, so its square is positive on some interval. Hence $G$ has zero kernel and is invertible. Thus the normal equations have exactly one solution, giving exactly one best least-squares polynomial. This proves both existence and uniqueness. For completeness, if the residual is orthogonal to ${\Pi }_n$, then for every $q\in {\Pi }_n$ the Pythagorean theorem gives $\Vert u-q{\Vert }^2=\Vert u-p_n{\Vert }^2+\Vert p_n-q{\Vert }^2$, directly verifying minimality and uniqueness.