$\{1,\cos nx,\sin nx\}_{n=1}^{\infty}$ is orthogonal on $L_2(-\pi,\pi)$ ?
The following sequence is orthogonal on $L_2(-\pi,\pi)$?
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real-analysis
functional-analysis
orthogonality
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0what is definition of orthogonality? – 2017-01-08
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0@MyGlasses: The usual one in $L^2$, of course: $\langle f, g \rangle = \frac 1 {2 \pi} \int _{-\pi} ^\pi f(x) \overline {g(x)} \ \Bbb d x$ and $f \perp g \iff \langle f, g \rangle = 0$. – 2017-01-08
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0@AlexM. $f\neq g$ – 2017-01-08
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0@AlexM. It's possible that MyGlasses was trying to see if OP was willing to interact at all. – 2017-01-08
1 Answers
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Clearly, $ 1 \bot \cos(nx)$ and $1 \bot \sin(nx)$
So, we have to prove that
$$\int_{-\pi}^\pi \sin(kx) \cos(nx) dx = 0$$
It suffice to remark that $f_{k,n}(x) = \sin(kx) \cos(nx)$ is an odd function and that $(-\pi, \pi)$ is symetric. This will imply that the integral is equal to $0$.