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Transactions of the American Mathematical Society

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Nonlinear $ n$-term approximation of harmonic functions from shifts of the Newtonian kernel


Authors: Kamen G. Ivanov and Pencho Petrushev
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
Published electronically: February 11, 2020
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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.


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Additional Information

Kamen G. Ivanov
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
Email: kamen@math.bas.bg

Pencho Petrushev
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina
Email: pencho@math.sc.edu

DOI: https://doi.org/10.1090/tran/8071
Keywords: Nonlinear approximation, harmonic functions, Newtonian kernel, Hardy spaces, Besov spaces, frame decomposition
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.
Article copyright: © Copyright 2020 American Mathematical Society