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I have an oscillatory, decaying function that can be decomposed as

$$\sum_k e^{iz_kt} $$where $z_k$ are complex. What I want is the imaginary parts of all of the $z_k$'s with some range of real parts. I could do a fourier transform and extract the widths of the lines, but I don't know how to handle the case that the lines overlap (which I expect to happen in at least a few cases).

Is there an algorithm to extract the imaginary parts of the $z_k$, like the Fouier transform would extract the real parts?

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Check out harmonic inversion. There is software for it.

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