- #1

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I'm starting with a function

[tex] f(x) = \sum_{k = - \infty}^\infty g(x + 2\pi k) [/tex]

which means that f is periodic in periods of two pi and thus can be explanded as a complex fourier series

[tex] f(x) = \sum_{n = -\infty}^\infty c_n e^{inx}[/tex]

with coefficients

[tex]c_n = \frac{1}{2\pi } \int_0^{2\pi} f(x) e^{-inx} dx[/tex]

This can manipulated further into

[tex]c_n = \frac{1}{2\pi } \int_0^{2\pi} f(x) e^{-inx} dx =\frac{1}{2\pi } \sum_{k=-\infty}^{\infty} \int_0^{2\pi} g(x + 2 \pi k) e^{-inx} dx = \frac{1}{2\pi } \sum_{k=-\infty}^{\infty} \int_{2\pi k}^{2\pi(k+1)}g(x) e^{-inx} dx = \frac{1}{2\pi} \int_{-\infty} ^\infty g(x) e^{-i nx} dx [/tex]

and that is all nice. However, how does one take the step from this and to the resulting

[tex] \sum_{n = -\infty}^\infty g(n) = \sum_{k = - \infty}^\infty \int_{-\infty}^\infty g(x)e^{-2\pi i k x} dx?[/tex]