fourier functions form an orthonormal set

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Proposition

The subset of functions ${\frac{e^{i n x}}{\sqrt{2 π}}}_{n \in Z}$ is an orthonormal subset of $L^{2} ([- π, π])$

Note

if we don't like working with complex exponentials we can use

e^{i x} = \cos x + i \sin x

And work out everything we need.

Proof

Note that

\begin{aligned} ⟨ e^{i n x}, e^{i m x} ⟩ & = \int_{- π}^{π} e^{i n x} \overset{―}{e^{i m x}} d x \\ = \int_{- π}^{π} e^{i (n - m) x} d x \\ = {\begin{cases} 2 π, & m = n \\ \frac{1}{i (n - m)} e^{i (n - m) x} |_{- π}^{π} = 0 & m \neq n \end{cases} \end{aligned}

Since the exponential is periodic in $2 π$ . Normalizing by $2 π$ yields

⟨ \frac{e^{i n x}}{\sqrt{2 π}}, \frac{e^{i m x}}{\sqrt{2 π}} ⟩ = {\begin{cases} 1 & m = n \\ 0 & m \neq n \end{cases}

Giving us our orthonormal set.

References

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fourier functions form an orthonormal set

References

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