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Suppose a periodic function, $f(t)$ with period $2\pi$ has a Fourier series of

$\sum_{k=-\infty}^{\infty} c_ke^{ikt}$

Now suppose we time shift the function to obtain

$g(t) = f(t-t_0)$

My question is: what effect does this have on the Fourier series? Logically, and thinking about the graph, the series should just shift too, i.e. the Fourier series for $g(t)$ should be given by

$\sum_{k=-\infty}^{\infty} c_ke^{ik(t-t_0)}$

My problem with this chain of thought is that this makes the coefficients decrease much faster, implying a greater rate of convergence. Rearranging the above we obtain:

$\sum_{k=-\infty}^{\infty} c_ke^{-ikt_0}e^{ikt} = \sum_{k=-\infty}^{\infty} d_ke^{ikt}$ for some new set of coefficients $d_k\in\mathbb{C}$.

This makes no sense, since it implies if we just keep shifting the series by $2\pi$, we'd get the same graph, but it would converge much faster. So therefore finding the new Fourier series is more involved that just applying the same time transformation?

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It's an exponential with an imaginary exponent. Therefore, it makes a phase change of the coefficients. It doesn't make them decrease faster. –  Raskolnikov Oct 16 '12 at 8:04
    
@Raskolnikov is right. $|e^{-ikt}|$ does NOT decay as $t$ increases. –  Tunococ Oct 16 '12 at 8:10
    
Thanks! Is the rest of my reasoning correct? i.e. the series just gets shifted in time in the way I stated? –  Froskoy1 Oct 16 '12 at 8:25
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