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Estimates on the stability constant for the truncated Fourier transform

Mirza Karamehmedović, Martin Sæbye Carøe, Faouzi Triki

TL;DR

This work analyzes the inverse problem of recovering a compactly supported function on $[-1,1]$ from its truncated Fourier transform $F_B f$, establishing explicit Lipschitz stability estimates with stability constants that depend on the truncation parameter $B$ and the signal frequency $\omega(f)$. The authors leverage harmonic measures and the Two Constants Theorem, together with a sharpened Gagliardo-Nirenberg inequality with explicit constants, to prove that stable recovery is possible when $B$ is large enough to cover the signal’s frequency, while the stability constant blows up exponentially as $B\to0$. They also provide an explicit bound on the blow-up rate and validate the theory via numerical experiments using FRFT-based reconstructions of Laplacian eigenfunctions, observing a critical bandwidth $B_0\approx \omega(f_k)=\frac{\pi}{2}k$ beyond which reconstructions are stable. The results quantify the trade-off between measurement bandwidth and noise, offering practical guidance for designing truncation regimes in applications such as imaging and inverse scattering.

Abstract

In this paper we are interested in the inverse problem of recovering a compact supported function from its truncated Fourier transform. We derive new Lipschitz stability estimates for the inversion in terms of the truncation parameter. The obtained results show that the Lipschitz constant is of order one when the truncation parameter is larger than the spatial frequency of the function, and it grows exponentially when the truncation parameter tends to zero. Finally, we present some numerical examples of reconstruction of a compactly supported function from its noisy truncated Fourier transform. The numerical illustrations validate our theoretical results.

Estimates on the stability constant for the truncated Fourier transform

TL;DR

This work analyzes the inverse problem of recovering a compactly supported function on from its truncated Fourier transform , establishing explicit Lipschitz stability estimates with stability constants that depend on the truncation parameter and the signal frequency . The authors leverage harmonic measures and the Two Constants Theorem, together with a sharpened Gagliardo-Nirenberg inequality with explicit constants, to prove that stable recovery is possible when is large enough to cover the signal’s frequency, while the stability constant blows up exponentially as . They also provide an explicit bound on the blow-up rate and validate the theory via numerical experiments using FRFT-based reconstructions of Laplacian eigenfunctions, observing a critical bandwidth beyond which reconstructions are stable. The results quantify the trade-off between measurement bandwidth and noise, offering practical guidance for designing truncation regimes in applications such as imaging and inverse scattering.

Abstract

In this paper we are interested in the inverse problem of recovering a compact supported function from its truncated Fourier transform. We derive new Lipschitz stability estimates for the inversion in terms of the truncation parameter. The obtained results show that the Lipschitz constant is of order one when the truncation parameter is larger than the spatial frequency of the function, and it grows exponentially when the truncation parameter tends to zero. Finally, we present some numerical examples of reconstruction of a compactly supported function from its noisy truncated Fourier transform. The numerical illustrations validate our theoretical results.
Paper Structure (7 sections, 7 theorems, 66 equations, 3 figures)

This paper contains 7 sections, 7 theorems, 66 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Harmonic measures
  3. The Gagliargo-Nirenberg inequality
  4. Main result
  5. Numerical example
  6. Numerical implementation
  7. Error analysis for the numerical example

Key Result

Theorem 1

Suppose $f\in H_0^1(-1,1)$. Then In particular, if $B\geq \sqrt{\gamma}\omega(f)$ for $\gamma>1$, then with $C_\gamma = \left(1-\frac{1}{\gamma}\right)^{-1/2}$.

Figures (3)

  • Figure 1: Reconstruction error using measurements taking on a grid $\Xi_{B,h}$. For each value of the noise level $\delta$ and for each value of $B$, the reconstruction error is computed from 1000 simulations. The plots shows the average of these simulations.
  • Figure 2: Critical bandwidth, $B_0$ versus the index $k$ for the eigenfunctions $f_k$. The plot shows different cutoff errors $e_{\rm cut}$. With $e_{\rm cut } = 0.2$ and $\delta =0.05$, the critical bandwidth is $B_0 = \infty$.
  • Figure 3: Reconstruction error using noisy measurements (see \ref{['generate measurement']}) of $\widehat{f}_{15}$ taking on a grid $\Xi_{B,h}$. The black dottet lines show the sum of the error in the noise free case (blue line) and $\delta \sqrt{B}$.

Theorems & Definitions (14)

  • Theorem 1: Large Truncation
  • proof
  • Theorem 2: Solution to the Dirichlet problem, ransford Corollary 4.2.6
  • Definition 3
  • Theorem 4: Two constants theorem
  • proof
  • Theorem 5
  • proof
  • Lemma 6: Special case of the Gagliardo-Nirenberg inequality, nirenberg
  • proof
  • ...and 4 more