Quantitative Estimates: How Well Does the Discrete Fourier Transform Approximate the Fourier Transform on \(\mathbb{R}?\)
Abstract
Abstract. In order to compute the Fourier transform of a function [Formula: see text] on the real line numerically, one samples [Formula: see text] on a grid and then takes the discrete Fourier transform. We derive exact error estimates for this procedure in terms of the decay and smoothness of [Formula: see text]. The analysis provides an asymptotically optimal recipe for how to relate the number of samples, the sampling interval, and the grid size.
Fields of Study & Subjects
Mathematical Analysis and Transform Methods
Mathematical Approximation and Integration
Mathematical functions and polynomials
Publication details
- Journal / source
- SIAM Review
- Publisher
- Society for Industrial & Applied Mathematics (SIAM)
- Volume
- 68
- Issue
- 1
- Pages
- 93–123
- Publication date
- 9 February 2026
- Language
- EN