There's function:
$$f(x)=\mathrm{sgn}(\cos x).$$
What is it's Fourier series?
I have never worked with this class of functions actually, so I do not know where I should start and how.
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Hint: this is a piecewise constant function. – Wojowu Jun 10 '17 at 13:46
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It is a periodic function stuck into another function. Maybe it is periodic? If so, what is the period? – mathreadler Jun 10 '17 at 14:01
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@mathreadler, the period is not given, thats why I am totally confused, otherwise I would start finding $a_0$ $a_n$ and $b_n$ – M.Mass Jun 10 '17 at 14:09
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Ok but what properties do cos(x) and sgn(x) have. Maybe you can use those properties to derive periods. – mathreadler Jun 10 '17 at 14:26
1 Answers
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Hint.
You have $\mathrm{sgn}(\cos(x))=1$ if $x\in ]-\pi/2,\pi/2[$ (mod $2\pi$) and $\mathrm{sgn}(\cos(x))=1$ if $x\in ]\pi/2,3\pi/2[$ (mod $2\pi$).
The function
$$x\mapsto \mathrm{sgn}(\cos(x))=1$$
looks like this:
You can just apply your formulas for Fourier series, it will work fine. Separate the integrals according to the values $1$ or $-1$ of your function, and the computation will work fine.
E. Joseph
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