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    $\begingroup$ Well, the key-word here is Fourier series. The Fourier integral is an integral, and that's required when your function is defined everywhere and isn't periodic. But the analog of this for periodic functions is the Fourier series, which is an infinite sum over the allowed $k$'s. I'm not sure exactly what your question really is, without some examples, but perhaps looking up information about Fourier series can help you here. $\endgroup$
    – march
    Commented Jun 5 at 15:11
  • $\begingroup$ The Fourier transform integral will give you a series of Dirac delta functions, so the integral does converge but only in the sense of a distribution. In other words you need to consider the machinery of test fuctions to evaluate it rigorously. $\endgroup$
    – mike stone
    Commented Jun 5 at 16:21