Super-Resolution Reconstruction from Truncated Fourier Transform

We present recent theoretical and numerical results on recovering a compactly supported function v on \(\mathbb {R}^d\) , \(d\geq 1\) , from its Fourier transform \(\mathcal {F}v\) 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 \(\pi /r\) , 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).