Fourier Analysis of Periodic Radon Transforms

Abstract

AbstractWe study reconstruction of an unknown function from its d-plane Radon transform on the flat torus $${\mathbb {T}}^n = {\mathbb {R}}^n /{\mathbb {Z}}^n$$ T n = R n / Z n when $$1 \le d \le n-1$$ 1 ≤ d ≤ 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 $$H^s$$ 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.