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

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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 $⟨f,g⟩$ = $\underset{\frac{-\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}f(x)g(x)dx$. Find the function in $span\{1,\cos x,\sin x\}$ which is nearest to the function $\{\begin{array}{cc}0,& \text{if }x<0\\ x,& \text{if }x\ge 0\end{array}\in C[-\frac{\pi }{2},\frac{\pi }{2}]$, 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)