Wikipedia:Reference desk/Archives/Mathematics/2011 October 22

= October 22 =

Interesting inner product question
Consider the space of continuous functions with domain $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$ with the inner product $$ \langle f, g \rangle$$ = $$\int_\frac{-\pi}{2}^\frac{\pi}{2}f(x)g(x)dx$$. Find the function in $$span\{1,\cos x, \sin x\}$$ which is nearest to the function $$\begin{cases} 0, & \mbox{if } x < 0 \\ x, & \mbox{if } x \ge 0 \end{cases} \in C[-\frac{\pi}{2}, \frac{\pi}{2}] $$, where distance is understood to be with respect to the given inner product.

Widener (talk) 11:13, 22 October 2011 (UTC)
 * The nearest to f will be the orthogonal projection onto this subspace. So find an orthonormal basis for the subspace (hint: Gram-Schmidt). The coefficients of the orthogonal projection of f in the orthonormal basis are the inner products of f with the (new) basis vectors.  Sławomir Biały  (talk) 11:46, 22 October 2011 (UTC)