An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions. (19th March 2018)

• Record Type: Journal Article
• Authors: Goncharov, F O; Novikov, R G
• Is Part Of: Inverse problems. Volume 34:Number 5(2018:May)
• Publication Date: 2018-03-19
• Subjects: weighted Radon transforms -- injectivity -- non-injectivity; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aab24d
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We consider the weighted Radon transforms R W along hyperplanes in R d, d ⩾ 3, with strictly positive weights W = W ( x, θ ), x ∈ R d, θ ∈ S d − 1 . We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions. In addition, the related weight W is infinitely smooth almost everywhere and is bounded. Our construction is based on the famous example of non-uniqueness of Boman (1993 J. d'Anal. Math .61 395–401) for the weighted Radon transforms in R 2 and on a recent result of Goncharov and Novikov (2016 Eurasian J. Math. Comput. Appl .4 23–32).