| Summary: | We propose AAA rational approximation as a method for interpolating or approximating smooth functions from equispaced samples. Although
it is always better to approximate from large numbers of samples if they are
available, whether equispaced or not, this method often performs impressively
even when the sampling grid is coarse. In most cases it gives more accurate
approximations than other methods. We support this claim with a review and
discussion of nine classes of existing methods in the light of general properties of approximation theory as well as the “impossibility theorem” for equispaced approximation. We make careful use of numerical experiments, which
are summarized in a sequence of nine figures. Among our new contributions is
the observation, summarized in Figure 4.5, that methods such as polynomial
least-squares and Fourier extension may be either exponentially accurate and
exponentially unstable, or less accurate and stable, depending on implementation.
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