Fourier analysis of periodic Radon transforms

Abstract

We study reconstruction of an unknown function from its $d$-plane Radon transform on the flat $n$-torus when $1\le d\le n-1$. We prove new reconstruction formulas and stability results with respect to weighted Bessel potential norms. We solve the associated Tikhonov minimization problem on $H^s$ Sobolev spaces using the properties of the adjoint and normal operators. One of the inversion formulas implies that a compactly supported distribution on the plane with zero average is a weighted sum of its X-ray data.

23 pages; final version

References in corpus (1)

• Summability of Multi-Dimensional Trigonometric Fourier Series

Cited by in corpus (3)

• Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems
• Broken ray transform for twisted geodesics on surfaces with a reflecting obstacle
• Integral geometry and unique continuation principles