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Quantitative estimates: How well does the discrete Fourier transform approximate the Fourier transform on $\mathbb{R}$

Martin Ehler, Karlheinz Gröchenig, Andreas Klotz

TL;DR

This work addresses how accurately the discrete Fourier transform (via FFT) approximates the Fourier transform on the real line when data are only finitely sampled. It develops non-asymptotic, explicit error bounds that depend on both time-domain and frequency-domain decay properties of the input function, using Wiener amalgam spaces to capture sampling behavior. The authors establish optimal parameter relationships among the sampling interval $p$, grid spacing $h$, and sample count $n$, across polynomial, sub-exponential, and mixed decay regimes, and they validate the theory with numerical experiments. The results extend rigorous error analysis beyond compactly supported or analytic functions, providing practical guidance for tuning FFT-based Fourier transforms in applications with realistic decay profiles. The work thus enhances the reliability and efficiency of numerical Fourier analysis in signal processing, PDEs, and related fields.

Abstract

In order to compute the Fourier transform of a function $f$ on the real line numerically, one samples $f$ 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 $f$. The analysis provides a new recipe of how to relate the number of samples, the sampling interval, and the grid size.

Quantitative estimates: How well does the discrete Fourier transform approximate the Fourier transform on $\mathbb{R}$

TL;DR

This work addresses how accurately the discrete Fourier transform (via FFT) approximates the Fourier transform on the real line when data are only finitely sampled. It develops non-asymptotic, explicit error bounds that depend on both time-domain and frequency-domain decay properties of the input function, using Wiener amalgam spaces to capture sampling behavior. The authors establish optimal parameter relationships among the sampling interval , grid spacing , and sample count , across polynomial, sub-exponential, and mixed decay regimes, and they validate the theory with numerical experiments. The results extend rigorous error analysis beyond compactly supported or analytic functions, providing practical guidance for tuning FFT-based Fourier transforms in applications with realistic decay profiles. The work thus enhances the reliability and efficiency of numerical Fourier analysis in signal processing, PDEs, and related fields.

Abstract

In order to compute the Fourier transform of a function on the real line numerically, one samples 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 . The analysis provides a new recipe of how to relate the number of samples, the sampling interval, and the grid size.
Paper Structure (25 sections, 18 theorems, 109 equations, 7 figures, 1 table)

This paper contains 25 sections, 18 theorems, 109 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Questions
  3. State-of-the-art
  4. Our contributions
  5. Error measures
  6. Polynomial decay and finite smoothness
  7. Sub-exponential decay
  8. Mixed conditions
  9. Interpolation
  10. Minimal requirements for convergence
  11. Other error measures and noisy samples
  12. Methods
  13. Numerical simulations
  14. Polynomial decay in time and in frequency
  15. Exponential decay in time and polynomial decay in frequency
  16. ...and 10 more sections

Key Result

Theorem 2.1

\newlabeltm:intro00 Let $a,b>1$. If $f$ and $\hat{f}$ are continuous and satisfy polynomial decay of the form $\sup_{x\in\mathbb{R}}|f(x)|(1+|x|)^{a}<\infty$ and $\sup_{\xi\in\mathbb{R}}|\hat{f}(\xi)|(1+|\xi|)^{b}<\infty$, then, for all $\alpha < a-\frac{1}{2}$ and $\beta <b-\frac{1}{2}$, there is

Figures (7)

  • Figure 1: Logarithmic plots of $E^{[n]}_{h}(f^{a,b})$ for various $a$ and $b$. Points are computed numerically for $n=2^l$, $l=10,\ldots, 20$. Reference lines represent the theoretical slopes predicted by Theorem \ref{['tm:intro0']} and listed in Table \ref{['table:unique']}.
  • Figure 1: Visualization of the construction of the Wiener amalgam norm of a continuous function $f$.
  • Figure 2: For $f^{2,4}$, we compare the error curve for the standard choice $p = \sqrt{n}$, $h = 1/\sqrt{n}$ with the error curve for the improved choice $p = n^{7/10}$, $h = n^{-3/10}$. The reference lines indicate the theoretical error decay rates: $n^{-3/4}$ for the standard choice and $n^{-21/20}$ for the optimal one.
  • Figure 3: Comparison of error curves for $f^{2,5}$.
  • Figure 4: Comparison of error curves for $f^{3,5}$.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3: sub-exponential decay
  • Theorem 2.4: mixed decay
  • Corollary 2.5
  • Proposition 2.6
  • Remark 3.1
  • Lemma 4.1
  • Proof 1
  • Lemma 4.2
  • ...and 25 more