On the range of the $ X $-ray transform of symmetric tensors compactly supported in the plane

Kamran Sadiq; Alexandru Tamasan

doi:10.3934/ipi.2022070

Article Contents

2023, Volume 17, Issue 3: 660-685. Doi: 10.3934/ipi.2022070

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On the range of the $ X $-ray transform of symmetric tensors compactly supported in the plane

Kamran Sadiq^1,2, , and
Alexandru Tamasan^3,

1.
Johann Radon Institute of Computational and Applied Mathematics (RICAM), Altenbergerstrasse 69, 4040 Linz, Austria
2.
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
3.
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

^*Corresponding author: Kamran Sadiq
^*Corresponding author: Kamran Sadiq

Received: April 2022

Revised: October 2022

Early access: December 19, 2022

Published online: December 19, 2022

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Abstract

We find the necessary and sufficient conditions for the Fourier coefficients of a function $ g $ on the torus to be in the range of the $ X $-ray transform of a symmetric tensor of compact support in the plane.

Keywords:

$ X $-ray transform,
Radon transform,
fan-beam coordinates,
Gelfand-Graev-Helgason-Ludwig moment conditions,
$ A $-analytic maps,
Hilbert transform.

Mathematics Subject Classification: Primary: 44A12, 35J56; Secondary: 45E05.

Citation:

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Figure 1. Fan-beam coordinates: $ e^{i \beta}\in \varGamma $, $ e^{i\theta}\in {{\mathbb S}^ 1} $, and $ \boldsymbol \theta = (\cos \theta, \sin \theta) $

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Figure 2. An even order $ m $-tensor field $ {\mathbf f} $ is determined by the odd negative angular modes on or above the diagonal $ k = -n $ (green region), and the odd negative angular modes (marked red) on the $ \frac{m}{2} $ red lines $ n+2k = -(m+1) $ for $ k\geq0 $. All the odd non-positive angular modes on and below the line $ n+2k = -(m+1) $, and left of the line $ n = -(m+1) $ vanish

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Figure 3. An odd order $ m $-tensor field $ {\mathbf f} $ is determined by the even negative angular modes on or above the diagonal $ k = -n $ (green region), and the even negative angular modes (marked red) on the $ \frac{m+1}{2} $ red lines $ n+2k = -(m+1) $ for $ k\geq0 $. All the even non-positive angular modes on and below the line $ n+2k = -(m+3) $, and left of the line $ n = -(m+3) $ vanish

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