Fourier interpolation
June 19, 2019 — June 19, 2019
a.k.a. spectral resampling/differentiation/integration.
Rick Lyons, How to Interpolate in the Time-Domain by Zero-Padding in the Frequency Domain. Also more classic Rick Lyons: FFT Interpolation Based on FFT Samples: A Detective Story With a Surprise Ending.
Steven Johnson’s Notes on FFT-based differentiation is all I need; it points out a couple of subtleties about DTFT-based differentiation of functions.
A common numerical technique is to differentiate some sampled function y(x) via fast Fourier transforms (FFTs). Equivalently, one differentiates an approximate Fourier series. Equivalently, one differentiates a trigonometric interpolation. These are also known as spectral differentiation methods. However, the implementation of such methods is prey to common confusions due to the aliasing phenomenon inherent in sampling, and the treatment of the maximum-frequency (Nyquist) component is especially tricky. In this note, we review the basic ideas of spectral differentiation (for equally spaced samples)…
Specifically, if I have some function sampled on
and I will invert it
Note that I will also be assuming, informally put, that
1 Minimum curvature interpolant
So far so normal. When we wish to interpolate and/or differentiate, things get a little less obvious; we wish to preserve minimum curvature for the interpolant
For a trivial example of the pathologies, all possible
Some calculation reveals that a minimum mean-square-derivative interpolant has a relatively simple form for even
2 Derivatives
Now, suppose at some equally-spaced points
Unless we know something special about
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