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Momentum ray transforms

The first author was supported by US NSF grant DMS 1616564 and a SERB Matrics Grant, MTR/2017/000837.
The second author was supported by SERB National Postdoctoral fellowship, PDF/2017/002780.
The first three authors were supported by Airbus Corporate Foundation Chair grant "Mathematics of Complex Systems" established at TIFR CAM and TIFR ICTS, Bangalore, India.
The work was started when the last author visited TIFR CAM January 2017. The author is grateful to the institute for the support and hospitality.
The last author was supported by RFBR, Grant 17-51-150001.
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  • The momentum ray transform $ I^k $ integrates a rank $ m $ symmetric tensor field $ f $ over lines in $ \mathbb{R}^n $ with the weight $ t^k $: $ (I^k\!f)(x,\xi) = \int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\, \mathrm{d} t. $ In particular, the ray transform $ I = I^0 $ was studied by several authors since it had many tomographic applications. We present an algorithm for recovering $ f $ from the data $ (I^0\!f,I^1\!f,\dots, I^m\!f) $. In the cases of $ m = 1 $ and $ m = 2 $, we derive the Reshetnyak formula that expresses $ \|f\|_{H^s_t({\mathbb R}^n)} $ through some norm of $ (I^0\!f,I^1\!f,\dots, I^m\!f) $. The $ H^{s}_{t} $-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.

    Mathematics Subject Classification: Primary: 44A12, 65R32; Secondary: 46F12.

    Citation:

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  • [1] A. Abhishek and R. K. Mishra, Support theorems and an injectivity result for integral moments of a symmetric m-tensor field, https://arXiv.org/abs/1704.02010, Journal of Fourier Analysis and Applications, 2018, 1–26. doi: 10.1007/s00041-018-09649-7.
    [2] P. Kuchment and F. Terizoglu, Inversion of weighted divergent beam and cone transforms, Inverse Problems and Imaging, 11 (2017), 1071-1090.  doi: 10.3934/ipi.2017049.
    [3] W. Lionheart and V. A. Sharafutdinov, Reconstruction algorithm for the linearized polarization tomography problem with incomplete data, in Imaging Microstructures: Mathematical and Computational Challenges, Ed. Habib Ammari and Hyeonbae Kang, Contemporary Mathematics, 494 (2009), 137–159. doi: 10.1090/conm/494/09648.
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    [6] V. A. Sharafutdinov and J.-N. Wang, Tomography of small residual stresses, Inverse Problems, 28 (2012), 065017, 17 pp. doi: 10.1088/0266-5611/28/6/065017.
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