Related papers: Momentum Ray Transforms, II: Range Characterizatio…
The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines with the weight $t^k$: $ (I^k\!f)(x,\xi)=\int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\,dt. $ In particular, the ray transform $I=I^0$…
The ray transform $I$ integrates symmetric $m$-tensor field in $\mathbb{R}^n$ over lines. This transform in Sobolev spaces was studied in our earlier work where higher order Reshetnyak formulas (isometry relations) were established. The…
The momentum ray transform $I_m^k$ integrates a rank $m$ symmetric tensor field $f$ on ${\mathbb R}^n$ over lines with the weight $t^k$, $I_m^kf(x,\xi)=\int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\,\mathrm{d}t$. Let…
The ray transform $I_m$ integrates a symmetric $m$ rank tensor field $f$ on $\mathbb{R}^n$ over lines. In the case of $n\ge3$, the range characterization of the operator $I_m$ on weighted Sobolev spaces $H^{s}_t({{\mathbb R}}^n;S^m{{\mathbb…
In this article, we study Momentum Light Ray Transform (MLRT) on symmetric tensor fields. MLRT is an integral transform in time-space domain ($(t,x)\in \mathbb{R}^{1+n}$), which integrates a scalar function or a tensor field along the light…
We establish range characterizations, or data consistency conditions, for an integral transform that maps a function to its weighted integrals over conical surfaces in $\mathbb{R}^n$. We consider two different geometries for the cone…
A central objective in inverse problems arising in integral geometry is to understand the kernel characterization, inversion formulas, stability estimates, range characterization, and unique continuation properties of integral transforms.…
In this article, we study various aspects of the mixed ray transform of $(k + \ell)$-tensor fields that are symmetric in its first $k$ and last $\ell$ indices. As a first result, we derive an inversion algorithm to recover the solenoidal…
Let $I_{m}$ denote the Euclidean ray transform acting on compactly supported symmetric $m$-tensor field distributions $f$, and $I_{m}^{*}$ be its formal $L^2$ adjoint. We study a unique continuation result for the normal operator…
In this paper we study the local magnetic ray transform of symmetric tensor fields up to rank two on a Riemannian manifold of dimension $\geq 3$ with boundary. In particular, we consider the magnetic ray transform of the combinations of…
The monograph contains a systematic treatment of a circle of problems in analysis and integral geometry related to inversion of the Radon transform on the space of real rectangular matrices. This transform assigns to a function $f$ on the…
We study a solenoidal-potential type decomposition of a symmetric $m$-tensor field in $\Rb^2$, and its implications to injectivity questions for the momentum and elastic ray transforms. For symmetric tensor fields, a general decomposition…
Entanglement between different regions in momentum space is studied for ground states of some spin-chain Hamiltonians: the XY model, the Ising model in a transverse field (ITF) and the XXZ models. In the XY and ITF cases, entanglement only…
The paper is devoted to the range description of the Radon type transform that averages a function over all spheres centered on a given sphere. Such transforms arise naturally in thermoacoustic tomography, a novel method of medical imaging.…
In this article we characterize the range of the attenuated and non-attenuated $X$-ray transform of compactly supported symmetric tensor fields in the Euclidean plane. The characterization is in terms of a Hilbert-transform associated with…
We find necessary and sufficient conditions on the Fourier coefficients of a function $g$ on the torus to be in the range of the $X$-ray transform of functions with compact support in the plane, and establish the connection between the…
We propose a thorough analysis of the tensor tomography problem on the Euclidean unit disk parameterized in fan-beam coordinates. This includes, for the inversion of the Radon transform over functions, using another range characterization…
Weighted V-line transforms map a symmetric tensor field of order $m\ge0$ to a linear combination of certain integrals of those fields along two rays emanating from the same vertex. A significant focus of current research in integral…
We investigate the Radon transform for double fibrations of the horocycle spaces for the semisimple symmetric spaces with respect to the inclusion incidence relations. We present the inversion formula, support theorem and the range theorem…
We present here the linear regime of the Einstein's field equations in the characteristic formulation. Through a simple decomposition of the metric variables in spin-weighted spherical harmonics, the field equations are expressed as a…