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IPI

Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of such a field $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics in $M$. Under the assumption that the metric $g$ is real-analytic, it is shown that there exists a vector field $v$ satisfying $f=dv$ on the set of points lying on these geodesics and $v=0$ on the intersection of this set with the boundary ∂$ M$ of the manifold $M$. Using this result, a Helgason's type of a support theorem for the geodesic ray transform is proven. The approach is based on analytic microlocal techniques.

IPI

In this article, we analyze the microlocal properties of the
linearized forward scattering operator $F$ and the reconstruction
operator $F^{*}F$ appearing in bistatic synthetic aperture radar
imaging. In our model, the radar source and detector travel along a
line a fixed distance apart. We show that $F$ is a Fourier integral
operator, and we give the mapping properties of the projections from
the canonical relation of $F$, showing that the right projection is a
blow-down and the left projection is a fold. We then show that
$F^{*}F$ is a singular FIO belonging to the class $I^{3,0}$.

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