Injectivity Sets of the Pompeiu Transform

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

Chapter Chapter 7 in Integral Geometry and Convolution Equations, 2003, pp 320-333 from Springer

Abstract: Abstract Let ϕ be a distribution with compact support in ℝ n , n ⩾ 2. For fixed λ ∊ M(n) we define the distribution λϕ acting in ɛ(ℝ n ) by the formula $$ \left\langle {\lambda \phi ,f\left( x \right)} \right\rangle = \left\langle {\phi ,f\left( {\lambda ^{ - 1} x} \right)} \right\rangle , f \in \mathcal{E}\left( {\mathbb{R}^n } \right). $$ Let $$ \mathcal{F} = \left\{ {\phi _1 , \ldots ,\phi _m } \right\} $$ be a given collection of nonzero distributions of ɛ′(ℝ n ). For an open subset $$ \mathcal{U} $$ of ℝ n such that each of sets 7.1 $$ \mathfrak{X}_j = \left\{ {\lambda \in M\left( n \right):supp \lambda \phi _j \subset \mathcal{U}} \right\}, j = 1, \ldots ,m $$ is non-empty the Pompeiu transform $$ \mathcal{P}_\mathcal{F} $$ maps $$ \mathcal{E}\left( \mathcal{U} \right) $$ into $$ \mathcal{E}\left( {\mathfrak{X}_1 } \right) \times \cdots \times \mathcal{E}\left( {\mathfrak{X}_m } \right) $$ in accordance with the formula $$ \mathcal{P}_\mathcal{F} f = \left( {f_1 , \ldots ,f_m } \right), f \in \mathcal{E}\left( \mathcal{U} \right), $$ where $$ f_j \left( \lambda \right) = \left\langle {\lambda \phi _j ,f} \right\rangle ,\lambda \in \mathfrak{X}_j ,j = 1, \ldots ,m $$

Keywords: Fourier Series; Open Subset; Unit Sphere; Open Ball; Radial Function (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-94-010-0023-9_25

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