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

  • *Corresponding author: Kamran Sadiq

    *Corresponding author: Kamran Sadiq 
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  • 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.

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


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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) $

    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

    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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