标题： 泛函分析与谱巴伦空间中的偏微分方程 显示英文标题

标题： Functional analysis and partial differential equations in spectral Barron spaces

摘要： 谱巴伦空间，构成一类特殊的函数空间，作为数学分析、偏微分方程（PDEs）和机器学习之间的跨学科桥梁，其特点是傅里叶变换的衰减特性。 在本工作中，我们从传统的数值逼近框架转向探索这些空间中的高级泛函分析和PDE理论视角。 具体而言，我们提出了谱巴伦空间对偶空间结构的严格表征，并通过实插值理论建立了在霍尔德空间中的连续嵌入。 此外，我们研究了由薛定谔方程控制的边界值问题的应用，包括相关线性算子的谱分析。 这些贡献阐明了谱巴伦空间的分析基础，同时强调了它们在统一逼近理论、泛函分析和机器学习方面的潜力。 显示英文摘要

摘要： 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 H\"older spaces established through real interpolation theory. Furthermore, we investigate applications to boundary value problems governed by the Schr\"odinger 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.