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Functional analysis and partial differential equations in spectral Barron spaces

Mourad Choulli, Shuai Lu, Hiroshi Takase

TL;DR

This work provides a rigorous functional-analytic framework for spectral Barron spaces $B^s$, defined by $\langle \xi\rangle^s \hat f \in L^1$, to bridge approximation theory and PDE analysis. It develops duality, interpolation, and embeddings (Sobolev and Hölder) for $B^s$, proves that the Laplacian is sectorial on $B^0$, and analyzes Schrödinger-type equations and boundary-value problems in both $\mathbb{R}^n$ and bounded domains, including anisotropic and nonlocal variants. Key contributions include the identification $(B^s)' \cong \tilde B^{-s}$, compact embeddings, isomorphisms via $(1-\Delta)^s$, and a spectral analysis framework for Schrödinger operators with small and nonlocal potentials, as well as a Radon-transform representation linking $B^s$ to certain function spaces. These results furnish a solid analytic basis for using spectral Barron spaces in PDE settings and reinforce their potential to support neural-network-based approximation within a principled mathematical core.

Abstract

Spectral Barron spaces, constituting a specialized class of function spaces that serve as an interdisciplinary bridge between mathematical analysis, partial differential equations (PDEs), and machine learning, are distinguished by the decay profiles of their Fourier transform. In this work, we shift from conventional numerical approximation frameworks to explore advanced functional analysis and PDE theoretic perspectives within these spaces. Specifically, we present a rigorous characterization of the dual space structure of spectral Barron spaces, alongside continuous embedding in Hölder spaces established through real interpolation theory. Furthermore, we investigate applications to boundary value problems governed by the Schrödinger equation, including spectral analysis of associated linear operators. These contributions elucidate the analytical foundations of spectral Barron spaces while underscoring their potential to unify approximation theory, functional analysis, and machine learning.

Functional analysis and partial differential equations in spectral Barron spaces

TL;DR

This work provides a rigorous functional-analytic framework for spectral Barron spaces , defined by , to bridge approximation theory and PDE analysis. It develops duality, interpolation, and embeddings (Sobolev and Hölder) for , proves that the Laplacian is sectorial on , and analyzes Schrödinger-type equations and boundary-value problems in both and bounded domains, including anisotropic and nonlocal variants. Key contributions include the identification , compact embeddings, isomorphisms via , and a spectral analysis framework for Schrödinger operators with small and nonlocal potentials, as well as a Radon-transform representation linking to certain function spaces. These results furnish a solid analytic basis for using spectral Barron spaces in PDE settings and reinforce their potential to support neural-network-based approximation within a principled mathematical core.

Abstract

Spectral Barron spaces, constituting a specialized class of function spaces that serve as an interdisciplinary bridge between mathematical analysis, partial differential equations (PDEs), and machine learning, are distinguished by the decay profiles of their Fourier transform. In this work, we shift from conventional numerical approximation frameworks to explore advanced functional analysis and PDE theoretic perspectives within these spaces. Specifically, we present a rigorous characterization of the dual space structure of spectral Barron spaces, alongside continuous embedding in Hölder spaces established through real interpolation theory. Furthermore, we investigate applications to boundary value problems governed by the Schrödinger equation, including spectral analysis of associated linear operators. These contributions elucidate the analytical foundations of spectral Barron spaces while underscoring their potential to unify approximation theory, functional analysis, and machine learning.

Paper Structure

This paper contains 24 sections, 36 theorems, 279 equations.

Table of Contents

  1. Introduction
  2. Spectral Barron spaces and their properties
  3. Spectral Barron spaces $B^0$
  4. Spectral Barron spaces $B^s$ with $s\geq 0$ and the dual spaces
  5. Embedding of spectral Barron spaces
  6. Spectral Barron spaces and interpolation
  7. Embedding by Sobolev spaces
  8. Embedding into Hölder spaces
  9. Variants of spectral Barron spaces
  10. Spectral Barron spaces in $L^p$.
  11. Spectral Barron Spaces associated with temperate weights.
  12. The Radon transform on $B^s$
  13. Schrödinger equations in spectral Barron spaces on $\mathbb{R}^n$
  14. The Laplacian operator in $B^0$
  15. Schrödinger equations
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $h\in L^1$. Then $\mathscr{F}^{-1}h\in C_b^0$, and where $(\varphi_j)$ is an arbitrary sequence in the Schwartz space $\mathscr{S}$ that converges to $h$ in the $L^1$ norm.

Theorems & Definitions (70)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.1
  • ...and 60 more