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Nonlinear approximation of harmonic functions from shifts of the Newtonian kernel in BMO

Kamen G. Ivanov, Pencho Petrushev

TL;DR

This work develops sharp nonlinear $n$-term approximation rates for harmonic functions on the unit ball from finite sums of shifts of the Newtonian kernel in harmonic BMO. By constructing highly localized frames on the sphere from Newtonian-kernel shifts and establishing dual frame decompositions for Besov and Triebel--Lizorkin spaces, the authors reduce harmonic BMO approximation to sequence-space nonlinear approximation. They identify BMO with the separable Triebel--Lizorkin space $\overset{\circ}{\mathcal{F}_\infty^{02}}$ and prove Jackson-type estimates $E_n(f)_{\operatorname{BMO}} \le c n^{-s/(d-1)} \|f\|_{\mathcal{B}^{s\tau}_{\tau}}$, with $1/\tau = s/(d-1)$; this also yields analogous results for the harmonic Besov scales and $VMO$. A Bernstein-type conjecture is proposed, and the approach enables a unified treatment via frame-based reductions to sequence spaces, highlighting the role of localization and multiscale sphere decompositions. The results advance a precise characterization of nonlinear approximation rates for harmonic functions in BMO/VMO through Newtonian-kernel shifts and Besov-type smoothness on the sphere.

Abstract

We study nonlinear n-term approximation of harmonic functions on the unit ball in $R^d$ from linear combinations of shifts of the Newtonian kernel (fundamental solution of the Laplace equation) in BMO. A sharp Jackson estimate is established that naturally involves certain Besov spaces. The method for obtaining this result is based on the construction of highly localized frames for Besov spaces and VMO on the sphere whose elements are linear combinations of a fixed number of shifts of the Newtonian kernel.

Nonlinear approximation of harmonic functions from shifts of the Newtonian kernel in BMO

TL;DR

This work develops sharp nonlinear -term approximation rates for harmonic functions on the unit ball from finite sums of shifts of the Newtonian kernel in harmonic BMO. By constructing highly localized frames on the sphere from Newtonian-kernel shifts and establishing dual frame decompositions for Besov and Triebel--Lizorkin spaces, the authors reduce harmonic BMO approximation to sequence-space nonlinear approximation. They identify BMO with the separable Triebel--Lizorkin space and prove Jackson-type estimates , with ; this also yields analogous results for the harmonic Besov scales and . A Bernstein-type conjecture is proposed, and the approach enables a unified treatment via frame-based reductions to sequence spaces, highlighting the role of localization and multiscale sphere decompositions. The results advance a precise characterization of nonlinear approximation rates for harmonic functions in BMO/VMO through Newtonian-kernel shifts and Besov-type smoothness on the sphere.

Abstract

We study nonlinear n-term approximation of harmonic functions on the unit ball in from linear combinations of shifts of the Newtonian kernel (fundamental solution of the Laplace equation) in BMO. A sharp Jackson estimate is established that naturally involves certain Besov spaces. The method for obtaining this result is based on the construction of highly localized frames for Besov spaces and VMO on the sphere whose elements are linear combinations of a fixed number of shifts of the Newtonian kernel.

Paper Structure

This paper contains 19 sections, 35 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Background and technical ground work
  3. Basic notation and simple inequalities
  4. Spherical harmonics
  5. Maximal $\delta$-nets, cubature formulas on the sphere and nested sets
  6. Spaces of functions and distributions on the sphere
  7. Besov and Triebel-Lizorkin spaces on ${{\mathbb S}^{d-1}}$
  8. Needlet frame decomposition of spaces of distributions on ${{\mathbb S}^{d-1}}$
  9. Frames in terms of shifts of the Newtonian kernel on ${{\mathbb S}^{d-1}}$
  10. The Triebel-Lizorkin spaces $\mathcal{F}^{sq}_\infty$ on ${{\mathbb S}^{d-1}}$
  11. Definition and basic properties of the spaces $\mathcal{F}^{sq}_\infty$ on ${{\mathbb S}^{d-1}}$
  12. Almost diagonal operators
  13. Frames
  14. Identification of $\operatorname{BMO}({{\mathbb S}^{d-1}})$ with $\mathcal{F}_\infty^{02}({{\mathbb S}^{d-1}})$
  15. Harmonic BMO space on the ball
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $\nu\ge 0$ and $M\in{\mathbb N}$. Assume $\lambda\in C^\infty[0, \infty)$, $\|\lambda^{(m)}\|_\infty \le \kappa$ for $0\le m\le M$ and for some $b>1$ either $\operatorname{supp} \lambda \subset [(2b)^{-1}, 2b]$ or $\operatorname{supp} \lambda \subset [0, b]$ and $\lambda(t)=1$ for $t\in [0, 1]$. and hence Furthermore, for $x, y, z\in {{\mathbb S}^{d-1}}$

Theorems & Definitions (70)

  • Theorem 2.1
  • Theorem 2.2
  • proof : Sketch of proof.
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • Definition 3.1
  • ...and 60 more