Nonlinear $n$-term approximation of harmonic functions from shifts of the Newtonian kernel
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- by Kamen G. Ivanov and Pencho Petrushev PDF
- Trans. Amer. Math. Soc. 373 (2020), 3117-3176 Request permission
Abstract:
A basic building block in classical potential theory is the fundamental solution of the Laplace equation in $\mathbb {R}^d$ (Newtonian kernel). The main goal of this article is to study the rates of nonlinear $n$-term approximation of harmonic functions on the unit ball $B^d$ from shifts of the Newtonian kernel with poles outside $\overline {B^d}$ in the harmonic Hardy spaces. Optimal rates of approximation are obtained in terms of harmonic Besov spaces. The main vehicle in establishing these results is the construction of highly localized frames for Besov and Triebel-Lizorkin spaces on the sphere whose elements are linear combinations of a fixed number of shifts of the Newtonian kernel.References
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Additional Information
- Kamen G. Ivanov
- Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
- MR Author ID: 92095
- Email: kamen@math.bas.bg
- Pencho Petrushev
- Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina
- MR Author ID: 138805
- Email: pencho@math.sc.edu
- Received by editor(s): August 27, 2018
- Received by editor(s) in revised form: May 28, 2019
- Published electronically: February 11, 2020
- Additional Notes: The first author was supported by Grant DN 02/14 of the Fund for Scientific Research of the Bulgarian Ministry of Education and Science. The second author was supported by NSF Grant DMS-1714369.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 3117-3176
- MSC (2010): Primary 41A17, 41A25; Secondary 42C15, 42C40, 42B35, 42B30
- DOI: https://doi.org/10.1090/tran/8071
- MathSciNet review: 4082235