Radon Transform Over Hyperplanes

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

Chapter Chapter 8 in Integral Geometry and Convolution Equations, 2003, pp 49-54 from Springer

Abstract: Abstract Let n ⩾ 2. Parametrize the hyperplanes in ℝ n by the unit normal vector and the distance to the origin: ξ W,d = {x ∈ ℝ n : (ω, x) = d}, where d ∈ ℝ and $$ \omega \in \mathbb{S}^{n - 1} $$ . Assume that f ∈ L(ℝ n ). Then the Radon transform R f can be regarded as a function on $$ \mathbb{S}^{n - 1} \times \mathbb{R} $$ defined by the equality 8.1 $$ Rf\left( {\omega ,d} \right) = \int\limits_{\xi w,d} {f\left( x \right)dm_{n - 1} \left( x \right)} , $$ where dm n−1 is the (n− 1)-dimensional volume. By the Fubini theorem we see that the transform R is well defined for all $$ \omega \in \mathbb{S}^{n - 1} $$ and almost all d ∈ ℝ.

Keywords: Inductive Hypothesis; Entire Function; Unit Normal Vector; Radial Function; Standard Approximation (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-94-010-0023-9_8

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