An example of non-uniqueness for Radon transforms with continuous positive rotation invariant weights

2017/09/28 by Fedor Goncharov, Goncharov, Fedor, Roman Novikov +1
Earth and Planetary Sciences · Mathematics · Medicine · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Medical Imaging Techniques and Applications #Numerical methods in inverse problems #Seismic Imaging and Inversion Techniques #math-ph #math.FA #math.MP

paper · pdf · doi:10.48550/arxiv.1709.09954

openalex publication_date 2017/09/28 · arxiv created 2018/06/26 · arxiv updated 2018/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider weighted Radon transforms R_W along hyperplanes in R³ with strictly positive weights W. We construct an example of such a transform with non-trivial kernel KerR_W in the space of infinitely smooth compactly supported functions and with continuous weight. Moreover, in this example the weight W is rotation invariant. In particular, by this result we continue studies of Quinto (1983), Markoe, Quinto (1985), Boman (1993) and Goncharov, Novikov (2017). We also extend our example to the case of weighted Radon transforms along two-dimensional planes in R^d , d ≥ 3.

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