I'm using the Jacobsen interpolation to get a more precise frequency of my signal. To get the corresponding DFT coefficient I'm doing:
$$X_{f} = \sum_{n=0}^{N}{x_{n} e^{-2\pi ifn}}$$
Where $x_{n}$ is the samples of my signal and $f$ is the frequency found by the interpolation (that is not one of those in the DFT bins). This is a quite heavy computation for large signals.
Is there a method to interpolate the coefficient corresponding to that frequency without having to compute the sum?