Fourier interpolation is useful for many purposes. It is primarily useful for interpolating periodic functions and/or evaluating functions within a prescribed interval, such as the one for which an empirical function is given. It also exhibits at least C3 continuity, in that it provides a form from which at least the first two derivatives of the underlying function can be evaluated.
In the Fourier transform, we showed how the discrete Fourier transform is used to represent an empirical function in terms of a finite sum of sines and cosines. We thus extend that setting and use the Fourier coefficients of the transform to produce intermediate values (or interpolation points) within the original series, by letting the independent variable of the inverse Fourier transform take the value at which we want to approximate the underlying function within the original interval.
To estimate derivates of the underlying function, we simply find the derivatives of the inverse Fourier transform at any value of the independent variable within the original interval.
Notes:
| ▪ | Other methods for interpolating with the Fourier transform are possible, such as zero padding the Fourier coefficients to a length, that when inverted, returns the number of interpolating points that are desired. |
| ▪ | Extrapolation via the Fourier transform is not recommended for non-periodic functions, because the inverse Fourier transform will return periodic values when asked to compute outside the original interval. |
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