Mathematical reminiscences
Jan Boman
Inverse Problems & Imaging 2010, 4(4): 571-577 doi: 10.3934/ipi.2010.4.571
Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform
Jan Boman
Inverse Problems & Imaging 2010, 4(4): 619-630 doi: 10.3934/ipi.2010.4.619
Using a vanishing theorem for microlocally real analytic distributions and a theorem on flatness of a distribution vanishing on infinitely many hyperplanes we give a new proof of an injectivity theorem of Bélisle, Massé, and Ransford for the ray transform on $\R^n$. By means of an example we show that this result is sharp. An extension is given where real analyticity is replaced by quasianalyticity.
keywords: analytic wave front set. ray transform
A local uniqueness theorem for weighted Radon transforms
Jan Boman
Inverse Problems & Imaging 2010, 4(4): 631-637 doi: 10.3934/ipi.2010.4.631
We consider a weighted Radon transform in the plane, $R_m(\xi, \eta) = \int_{\R} f(x, \xi x + \eta) m(x,\xi,\eta) dx$, where $m(x,\xi,\eta)$ is a smooth, positive function. Using an extension of an argument of Strichartz we prove a local injectivity theorem for $R_m$ for essentially the same class of $m(x,\xi,\eta)$ that was considered by Gindikin in his article in this issue.
keywords: weighted Radon transform. Radon transform
Stability estimates in tensor tomography
Jan Boman Vladimir Sharafutdinov
Inverse Problems & Imaging 2018, 12(5): 1245-1262 doi: 10.3934/ipi.2018052

We study the X-ray transform $I$ of symmetric tensor fields on a smooth convex bounded domain $Ω\subset{\mathbb R}^n$. The main result is the stability estimate $\|^{s}f\|_{L^2}≤ C\|If\|_{H^{1/2}}$, where $^{s}f$ is the solenoidal part of the tensor field $f$. The proof is based on a comparison of the Dirichlet integrals for the exterior and interior Dirichlet problems and on a generalization of the Korn inequality to symmetric tensor fields of arbitrary rank.

keywords: Tensor tomography X-ray transform the Dirichlet principle the Korn inequality

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