Super-Resolution Reconstruction from Truncated Fourier Transform

We present recent theoretical and numerical results on recovering a compactly supported function v on R-d, d >= 1, from its Fourier transform Fv given within the ball B-r. We proceed from known results on the prolate spheroidal wave functions and on the Radon transform. The most interesting point of our numerical examples consists in super-resolution, that is, in recovering details beyond the diffraction limit, that is, details of size less than pidividedbyr.pi r, where r is the radius of the ball mentioned above. This short review is based on the works Isaev, Novikov (2022 J. Math. Pures Appl. 163 318-333) and Isaev, Novikov, Sabinin (2022 Inverse Problems 38 105002).