Estimates on the stability constant for the truncated Fourier transform

TL;DR

This paper derives new stability estimates for reconstructing a compactly supported function from its truncated Fourier transform, showing how the stability constant varies with the truncation parameter, and validates these findings with numerical examples.

Contribution

The paper provides novel Lipschitz stability estimates for the inverse truncated Fourier transform problem, highlighting the dependence on the truncation parameter.

Findings

Lipschitz constant is of order one when truncation exceeds the spatial frequency.

Lipschitz constant grows exponentially as truncation parameter approaches zero.

Numerical examples confirm the theoretical stability estimates.

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.