Injectivity sets of the Pompeiu transform

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 \( f_j \left( \lambda \right) = \left\langle {\lambda \phi _j ,f} \right\rangle ,\lambda \in \mathfrak{X}_j ,j = 1, \ldots ,m \)