# Inverse scattering – Fourier analysis – super-resolution

The paper “Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions” was published in “Inverse Problems” (Q1 WoS) journal. Doctor of physical and mathematical sciences R.G. Novikov   is one of the authors of this article.

The work implements numerically formulas of the theoretical article M. Isaev, R.G. Novikov (2022 J. Math. Pures Appl. 163, 318–333) for finding a compactly supported function v on R^d from its Fourier transform  given within the ball of radius r. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions, which arise, in particular, in the singular value decomposition of the aforementioned band-limited Fourier transform for d = 1. In multidimensions, these formulas also include inversion of the Radon transform. In particular, the work gives numerical examples of super-resolution, that is, 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, G.V. Sabinin, Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions//Inverse Problems. 2022. V. 38. № 10.  Article 105002 (17pp)   DOI:10.1088/1361-6420/ac87cb