Computation of the Fourier transform for a continuous integrable function via NFFT
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.~Gr\"{o}chenig, and A.~Klotz \cite{EhGrKl24}, we derive explicit $\ell_\infty$ uniform 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. Moreover, we prove a matching lower bound which shows that the derived rate in the truncation parameter is sharp. 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.