Looking at a Fourier expansion,(adsbygoogle = window.adsbygoogle || []).push({});

[tex]f(x) = a_0 + a_1 \cos{x} + b_1 \sin{x} + a_2 \cos{2 x} + b_2 \sin{2 x} + ...[/tex],

I would expect the coefficients to be the projections of [tex]f(x)[/tex] on the different Fourier functions, which should form an orthonormal basis. This doesn't seem to be the case, though. The formulae for the coefficients say

[tex]a_0 = \frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) 1 dx[/tex];

[tex]a_n = \frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) 2 \cos{n x} dx[/tex];

[tex]b_n = \frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) 2 \sin{n x} dx[/tex].

Now, assuming that the inner product is defined as

[tex]<f,g> = \frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) g(x) dx[/tex],

these coefficients are not simply the projections, since the sines and cosines get a factor two. Further more, these sines and cosines do not form an orthonormal set. So what exactly is going on? Are my assumptions about the nature of the coefficients or about the expected orthonormality of the functions wrong?

I also noticed that if you use the exponential form of the Fourier series, all the functions are perfectly orthonormal and the coefficients are indeed the projections on these functions. Perhaps something happens when changing from exponential form to sine and cosine form?

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# Orthonormality of Fourier functions

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