Systems of Convolution Equations

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

Chapter Chapter 4 in Integral Geometry and Convolution Equations, 2003, pp 201-211 from Springer

Abstract: Abstract Let $$ \mathcal{F} = \left\{ {\phi _1 , \ldots ,\phi _m } \right\} $$ = {ϕ 1, ..., ϕ m } be a given family of nonzero distributions from ɛ′rad(ℝ n ). IfU,U j are nonempty open subset in ℝ n such that $$ \mathcal{U}_j $$ — suppϕ j ⊂ $$ \mathcal{U} $$ and f ∈ $$ \mathcal{D}'\left( \mathcal{U} \right) $$ then convolution f*ϕ j is well defined as the distribution from $$ \mathcal{D}'\left( {\mathcal{U}_j } \right) $$ for all j = 1, ..., m. Denote $$ \mathcal{D}'_\mathcal{F} \left( \mathcal{U} \right) = \bigcap\limits_{j = 1}^m {\mathcal{D}'_{\phi j} \left( \mathcal{U} \right)} $$ . Also let $$ C_\mathcal{F}^k \left( \mathcal{U} \right) = \left( {\mathcal{D}'_\mathcal{F} \cap C^k } \right)\left( \mathcal{U} \right) $$ for k ∈ ℤ+ or k = ∞, $$ QA_\mathcal{F} \left( \mathcal{U} \right) = \left( {\mathcal{D}'_\mathcal{F} \cap QA} \right)\left( \mathcal{U} \right) $$

Keywords: Induction Hypothesis; Spherical Symmetry; Radial Function; Analogous Argument; Arbitrary Sequence (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-94-010-0023-9_17

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