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Injectivity Sets of the Pompeiu Transform

  • V. V. Volchkov

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
$$ \mathfrak{X}_j = \left\{ {\lambda \in M\left( n \right):supp \lambda \phi _j \subset \mathcal{U}} \right\}, j = 1, \ldots ,m $$
(7.1)
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 \( f_j \left( \lambda \right) = \left\langle {\lambda \phi _j ,f} \right\rangle ,\lambda \in \mathfrak{X}_j ,j = 1, \ldots ,m</Para> \)

Keywords

Fourier Series Open Subset Unit Sphere Open Ball Radial Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • V. V. Volchkov
    • 1
  1. 1.Department of MathematicsDonetsk National UniversityDonetskUkraine

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