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

Let $e=u-p$. If $p$ minimizes the squared L2 norm error, then for any $q\in A$ the real quadratic $\Vert e-tq{\Vert }_2^2=\Vert e{\Vert }_2^2-2t(e,q)+t^2\Vert q{\Vert }_2^2$ has its minimum at $t=0$. Its derivative there is zero, giving $(e,q)=0$.

Conversely, if $e$ is orthogonal to $A$, then for every $q\in A$, $p-q\in A$ and the Pythagorean theorem for inner product spaces givesThis proves the equivalence and identifies $p$ as an orthogonal projection. It characterizes a minimizer whenever one exists; a general nonclosed vector subspace need not admit a projection for every $u$.

The orthogonal projection is $p_n={\sum }_{r=0}^n(u,{\varphi }_r){\varphi }_r$. Subtract it from $u$ and use the orthonormal basis relations to obtainThis is the requested Bessel inequality.

For the infinite tail, use completeness of the orthonormal polynomial family in the usual weighted L2 space. The relevant density facts are that continuous functions are dense for the finite measure $w(x)dx$, and the Weierstrass approximation theorem makes polynomials uniformly dense in continuous functions on the compact interval. Since $\int w<\infty$, uniform approximation also gives weighted L2 norm approximation. Parseval's identity therefore givesHere finiteness of the weight integral is also necessary for the constant polynomial to belong to the stated Hilbert space. Without completeness of the polynomial family, an additional squared component orthogonal to every polynomial would remain; Bessel inequality alone does not justify equality.

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

For $n\ge 1$, ${\varphi }_n$ is orthogonal to every polynomial in $P_{n-1}$, so $(u,{\varphi }_n)=(u-p_{n-1}^{\ast },{\varphi }_n)$. The Cauchy-Schwarz inequality givesUsing the tail identity from part (b) and shifting the index,For the stated power law and $n>1$, the decreasing-function integral test givesConsequentlyThe assumption $a>1$ is sufficient; this summation estimate itself only needs $a>1/2$.