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


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.


Linear Span Riesz Kernel 
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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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