"Quantitative estimates: How well does the discrete Fourier transform approximate the Fourier transform on the real line"Martin EhlerThe discrete Fourier transform (DFT), computed via the fast Fourier transform (FFT), is a cornerstone of computational science, widely used to approximate Fourier transforms in applications ranging from signal processing to numerical PDEs. But how accurately does it approximate the true Fourier transform on the real line? We derive rigorous error estimates in terms of the function's decay and smoothness. The analysis provides an asymptotically optimal recipe of how to balance the number of samples, the sampling interval, and the grid size for a given accuracy. |
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