Injectivity Sets for Spherical Radon Transform

V. V. Volchkov
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V. V. Volchkov: Donetsk National University, Department of Mathematics

Chapter Chapter 1 in Integral Geometry and Convolution Equations, 2003, pp 339-358 from Springer

Abstract: Abstract Throughout in this chapter we assume that n ⩾ 2. Let $$ \mathcal{U} $$ be a domain in ℝ n and let $$ f \in L_{loc} \left( \mathcal{U} \right) $$ . For any $$ x \in \mathcal{U} $$ and almost all $$ r \in \left( {0,dist\left( {x,\partial \mathcal{U}} \right)} \right) $$ the spherical Radon transform of f is defined by 1.1 $$ \mathcal{R}f\left( {x,r} \right) = \frac{1} {{\omega _{n - 1} }}\int\limits_{\mathbb{S}^{n - 1} } {f\left( {x + r\eta } \right)d\omega \left( \eta \right)} . $$ (The reader should be warned that there does not seem to be a standard terminology in this area. Some authors use spherical Radon transform to refer to the transform $$\widehat f(\omega ,t)$$ defined below in Section 1.2, which we have called the spherical Radon transform on spheres).

Keywords: Harmonic Function; Algebraic Variety; Radial Function; Real Analytic Function; Nonzero Function (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-94-010-0023-9_27

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