Past exam of the options I course of the University of Oxford / 2012 / P / P1

A polynomial $p_n\in {\Pi }_n$ is a best polynomial least-squares approximation when it minimizes the weighted error norm over all polynomials of degree at most $n$:Equivalently it minimizes ${\int }_0^1{x}^{-1/2}(f(x)-q(x){)}^{2}dx$. The weight is integrable, so the expression is finite for continuous functions; it defines a genuine norm because a continuous function with zero weighted squared integral must be identically zero.

Take any $q\in {\Pi }_n$. Write $f-q=(f-p_n)+(p_n-q)$. The second term belongs to ${\Pi }_n$, so the assumed orthogonality gives $⟨f-p_n,p_n-q⟩=0$. Expanding the squared norm yields the Pythagorean theorem in this inner product:Taking square roots provesEquality can hold only when $\Vert p_n-q\Vert =0$, hence $p_n=q$, so the best polynomial is unique. Completeness of the ambient space is unnecessary for this orthogonality argument.

Write $p_1(x)=a+bx$. It suffices to impose orthogonality of the error to the basis $1,x$ of ${\Pi }_1$. With the printed weight $x^{-1/2}$ and $f=x^{3/2}$, the normal equations areHere, for example, $⟨f,1⟩={\int }_0^{1}xdx=1/2$ and $⟨f,x⟩={\int }_0^1{x}^{2}dx=1/3$. Solving gives $a=-1/16$, $b=15/16$. ThereforeThe two normal equations make the error orthogonal to every linear polynomial, and part (b) then proves optimality, rather than merely furnishing a candidate fit.