Past exam of the approximation theory and finite element methods course of the University of Oxford / 2010 / A / 2 / c / Solution

The moments are ${\int }_0^1{x}^{j+2}dx=1/(j+3)$. The Gram-Schmidt process first gives the constant ${\varphi }_0=\sqrt{3}$. Removing the constant component of $x$ gives $r_1=x-3/4$, with squared norm $1/80$, hence ${\varphi }_1=4\sqrt{5}(x-3/4)$.

For the quadratic, imposing orthogonality of $r_2=x^2+ax+b$ to one and $x$ gives $1/5+a/4+b/3=0$ and $1/6+a/5+b/4=0$. Thus $r_2=x^2-4x/3+2/5$, whose squared norm is $1/1575$. Taking positive leading coefficients, the first three orthonormal polynomials are