The John equation for tensor tomography in three-dimensions *The second author was supported by RFBR, Grant 15-01-05929-a. (9th September 2016)

• Record Type: Journal Article
• Authors: Nadirashvili, Nikolai S; Sharafutdinov, Vladimir A; Vlăduţ, Serge G
• Is Part Of: Inverse problems. Volume 32:Number 10(2016:Oct.)
• Publication Date: 2016-09-09
• Subjects: tensor fields tomograhy -- John equation -- Radon transform; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/32/10/105013
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: John proved that a function φ on the manifold of lines in R 3 belongs to the range of the x-ray transform if and only if φ satisfies some second order differential equation and obeys some smoothness and decay conditions. We generalize the John equation to the case of the x-ray transform on arbitrary rank symmetric tensor fields: a function φ on the manifold of lines in R 3 belongs to the range of the x-ray transform on rank m symmetric tensor fields if and only if φ satisfies some differential equation of order 2 ( m + 1 ) and obeys some smoothness and decay conditions.