Abstract:We investigate the approximation of continuous Fourier transforms by
trigonometric sampling polynomials and their efficient evaluation by the
nonequispaced fast Fourier transform (NFFT). While the NFFT is traditionally
used for the evaluation of trigonometric polynomials, we show that it can also
serve as an effective computational tool for the approximation of Fourier
transform values.
Building on ideas of M.~Ehler, K.~Gröchenig, and A.~Klotz \cite{EhGrKl24}, we derive explicit
$\ell_\infty$ error bounds between the Fourier transform and suitable sampling
polynomials. The resulting estimates quantify the influence of the sampling
width and truncation parameter and provide rigorous accuracy guarantees on
entire frequency intervals. In contrast to previous analyses focusing mainly on
discrete or $L_2$-type errors, our results yield uniform approximation bounds
that are directly relevant for practical computations.
The derived theory leads to a simple algorithmic framework: first approximate
the Fourier transform by a trigonometric sampling polynomial and then evaluate this
polynomial efficiently by the NFFT. Numerical experiments confirm the
theoretical convergence rates and demonstrate that accurate approximations of
continuous Fourier transforms can be obtained with moderate computational
effort.
From: Daniel Potts [view email]
[v1]
Mon, 22 Jun 2026 07:52:23 UTC (20 KB)
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