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

Chapter Chapter 3 in Integral Geometry and Convolution Equations, 2003, pp 12-15 from Springer

Abstract: Abstract Let $$ \mathcal{U} $$ be a non-empty open subset of ℝ n , and let ϕ be a linear form on $$ \mathcal{D}\left( \mathcal{U} \right) $$ . We denote by the value of ϕ on element f ∈ $$ \mathcal{D}\left( \mathcal{U} \right) $$ . A linear form ϕ on $$ \mathcal{D}\left( \mathcal{U} \right) $$ ) is called a distribution on $$ \mathcal{U} $$ , if for each compact set K ⊂ $$ \mathcal{U} $$ there exist a constants c > 0, m ∈ ℤ+ such that 3.1 $$ \left| {\left\langle {\phi ,F} \right\rangle } \right| \leqslant c\sum\limits_{\left| \alpha \right| \leqslant m} {\mathop {\sup }\limits_{x \in K} \left| {\left( {\partial ^\alpha f} \right)\left( x \right)} \right|} for all f \in \mathcal{D}\left( K \right). $$ This means that if the sequence f j ∈ $$ \mathcal{D}\left( \mathcal{U} \right) $$ , j = 1, 2,..., converges in $$ \mathcal{D}\left( \mathcal{U} \right) $$ to the function f then → as j → + ∞.

Keywords: Tensor Product; Linear Form; Compact Support; Domain Versus; Dirac Measure (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-010-0023-9_3

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DOI: 10.1007/978-94-010-0023-9_3

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