Inverse scattering – Fourier analysis – super-resolution

The paper “Stability estimates for reconstruction from the Fourier transform on the ball” was published in “Journal of Inverse and Ill-Posed Problems” (Q1 WoS) journal. Doctor of physical and mathematical sciences Novikov R.G. is one of the authors of this article.

The work develops methods for recovering a function f with compact support (or very fast decay at infinity) from its Fourier transform restricted to a ball of fixed radius r. Using the Chebyshev polynomial theory, the authors give reconstruction formulas and Hölder-logarithmic stability estimates for the problem under consideration.

However, the issues of the numerical implementation of this type of reconstruction led the authors to further results, theoretical part of which is published in the preprint M. Isaev, R.G. Novikov, Reconstruction from the Fourier transform on the ball via prolate spheroidal wave functions // more detailed // and the numerical part, obtained jointly with G. Sabinin, is in preparation for publication. In particular, the results obtained give super-resolution reconstruction, that is, they allow recovering details beyond the diffraction limit, that is, details of size less than π/r, where r is the radius of the ball mentioned above.

Source: M. Isaev, R.G. Novikov, Stability estimates for reconstruction from the Fourier transform on the ball // Journal of Inverse and Ill-Posed Problems. 2021. V. 29. No. 3. P. 421–433. DOI: 10.1515/jiip-2020-0106