Numerical reconstruction from the Fourier transform on the ball using prolate spheroidal wave functions. (1st October 2022)

• Record Type: Journal Article
• Authors: Isaev, Mikhail; Novikov, Roman G; Sabinin, Grigory V
• Is Part Of: Inverse problems. Volume 38:Number 10(2022)
• Publication Date: 2022-10-01
• Subjects: ill-posed inverse problems -- band-limited Fourier transform -- prolate spheroidal wave functions -- Radon transform -- super-resolution; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ac87cb
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We implement numerically formulas of Isaev and Novikov (2022 J. Math. Pures Appl. 163 318–33) for finding a compactly supported function v on R d, d ⩾ 1, from its Fourier transform F [ v ] given within the ball B 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, we give numerical examples of super-resolution, that is, recovering details beyond the diffraction limit.