15 - Best Approximation by Orthogonal Projection

Best Approximation by Orthogonal Projection

Let

$$⟨f,g⟩={\int }_a^{b}w(x)f(x)g(x)dx\text{ and }||f|{|}_2^2=⟨f,f⟩$$

where $w(x)>0$ for all $(a,b)$

If $f\in L_w^2(a,b)$ and $p_n\in {\Pi }_n$ such that

$$⟨f-p_n,r⟩=0\forall r\in {\Pi }_n$$

Then

$$||f-p_n|{|}_2\le ||f-r|{|}_2\forall r\in {\Pi }_n$$

so $p_n$ is a best (weighted) lest-squares approximation to $f$ on $[a,b]$

Proof

$$\begin{array}{rl}||f-p_n|{|}_2^2& =⟨f-p_n,f-p_n⟩\\ & =⟨f-p_n,f-r⟩+⟨f-p_n,r-p_n⟩\forall r\in {\Pi }_n\\ & =⟨f-p_n,f-r⟩\text{ as }⟨f-p_n,r-p_n⟩=0\text{ by assumption}\\ & \le ||f-p_n|{|}_2||f-r|{|}_2\\ & \\ ⟹||f-p_n|{|}_2& \le ||f-r|{|}_2\end{array}$$