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Journal of Mathematical Sciences

, Volume 134, Issue 4, pp 2239–2257 | Cite as

Approximation by M. Riesz Kernels in L p for p<1

  • A. B. Aleksandrov
Article
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Abstract

Let α > 0. We consider the linear span \(\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right)\) of scalar Riesz's kernels \(\left\{ {\tfrac{1}{{\left| {x - a} \right|^\alpha }}} \right\}_{a \in \mathbb{R}^n }\) and the linear span \(\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right)\) of vector Riesz's kernels \(\left\{ {\tfrac{1}{{\left| {x - a} \right|^{\alpha + 1} }}\left( {x - a} \right)} \right\}_{a \in \mathbb{R}^n }\). We study the following problems. (1) When is the intersection \(\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n } \right)\) dense in Lp(ℝn)? (2) When is the intersection \(\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n ,\mathbb{R}^n } \right)\) dense in Lp(ℝn, ℝn)? Bibliography: 15 titles.

Keywords

Linear Span Riesz Kernel 
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, Inc. 2006

Authors and Affiliations

  • A. B. Aleksandrov
    • 1
  1. 1.St.Petersburg DepartmentSteklov Mathematical InstituteSt.PetersburgRussia

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