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Mathematically, suppose I have a function $f(t)=\sum_k c_k e^{-i \omega_kt}$, where $\omega_k$ may not fall in $[0,2\pi]$. With an analytical Fourier transform, I can get a sum of delta functions centered at those frequencies. Now, I only have $N$ evenly-spaced points of $f(t_j)$ in the time domain $[0,T]$, albeit $N$ and $T$ can be freely chosen. How do I use a numerical Fourier transform to get the frequencies $\omega_k$ (e.g. for $\omega_1=1,\omega_2=10,\omega_3=-30$), not confined in $[0,2\pi]$ and with no prior knowledge of the frequency upper/lower limit?

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    $\begingroup$ You can't. Without a-priori knowledge of the upper limit, the information needed for the result you desire may have been destroyed by sampling below the Nyquist rate. $\endgroup$ – hotpaw2 Nov 12 '15 at 19:10
  • $\begingroup$ Thanks. But what if I'm interested in, say only frequency in (-wlim,+wlim)? $\endgroup$ – egwene sedai Nov 12 '15 at 19:12
  • $\begingroup$ That won't work either. For example, non-zero bandlimited functions $f_1(t)$ periodic by $T$ and $f_2(t)$ periodic by $T+\frac{T}{N-1}$ can have identical samples but they will be composed of two different finite sets of complex exponentials. $\endgroup$ – Olli Niemitalo Nov 12 '15 at 19:30

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