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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 ${\begin{cases}0,&{\mbox{if }}x<0\\x,&{\mbox{if }}x\geq 0\end{cases}}\in C[-{\frac {\pi }{2}},{\frac {\pi }{2}}]$ , where distance is understood to be with respect to the given inner product.

Widener (talk) 11:13, 22 October 2011 (UTC)[reply]

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)[reply]