Spectral Barron Spaces: Theory & Applications

• Spectral Barron Spaces are Banach spaces defined by weighted Fourier decay and integrability, providing a rigorous framework that bridges classical harmonic analysis and modern neural network approximation.
• They enable dimension-free approximation rates for both shallow and deep neural networks, outperforming classical Sobolev and Besov bounds in high-dimensional settings.
• Their applications span operator learning, PDE solution theory, and quantum many-body analysis, cementing their role in advancing both theoretical analysis and practical machine learning.

Spectral Barron spaces are Banach spaces of functions (or operator-valued objects) defined by the decay and integrability of their Fourier transform with polynomial or, more generally, weighted moments. The defining spectral norm directly characterizes the approximation complexity of functions by two-layer neural networks, yielding dimension-free theoretical rates. Spectral Barron spaces serve as a bridge between harmonic analysis, approximation theory, and PDE regularity, and have been developed and formalized in connection with deep learning, operator learning, and quantum many-body analysis.

1. Foundational Definition and Norm Structure

Let $n \in \mathbb{N}$ and $s \geq 0$. For $u : \mathbb{R}^n \to \mathbb{C}$, the (scalar) spectral Barron norm is

$$\|u\|_{\mathcal{B}^s(\mathbb{R}^n)} = \int_{\mathbb{R}^n} (1 + \|\xi\|^2)^{s/2} |\widehat{u}(\xi)|\, d\xi,$$

where $\widehat{u}$ is the Fourier transform of $u$ (as a tempered distribution). The spectral Barron space $\mathcal{B}^s(\mathbb{R}^n)$ consists of all $u$ with finite norm.

Key properties:

• $\mathcal{B}^s(\mathbb{R}^n)$ is a Banach space; $$\mathcal{B}^{s+\delta} \hookrightarrow \mathcal{B}^s$$ for $\delta > 0$.
• $\mathcal{B}^s$ contains the Schwartz class densely; for integer $m \leq s$, $\mathcal{B}^s \hookrightarrow C_0^m$ (Yserentant, 25 Feb 2025).
• The choice of weight, e.g., $(1 + |\xi|^2)^{s/2}$, is compatible with classical Sobolev/Bessel potentials, leading to embeddings into $W^{k,p}$ for suitable $s$ (Abdeljawad et al., 2023).
• For domains $\Omega \subset \mathbb{R}^d$, the seminorm is defined via extension:

$$\|f\|_{\mathcal{B}^s(\Omega)} := \inf_{f_e|_\Omega = f} \int_{\mathbb{R}^d} (1+|\xi|^2)^{s/2} |\widehat{f_e}(\xi)|\, d\xi$$

for measurable extensions $f_e$ (Siegel et al., 2021).

Extensions exist to vector-valued, operator-valued, and group-theoretic contexts: for compact group $G$ and Banach space $A$, spectral Barron spaces characterize summability of matrix-valued Fourier coefficients with respect to a group-dependent weight (Mensah et al., 13 Dec 2025, Mensah, 18 Sep 2025).

2. Core Analytic and Functional Properties

Spectral Barron spaces possess interpolation and embedding structures compatible with harmonic analysis:

• Real interpolation: For $0 \leq r < s < t$ and $\theta = (s - r)/(t - r)$,

$$\|f\|_{\mathcal{B}^s} \leq \|f\|_{\mathcal{B}^r}^{\alpha} \|f\|_{\mathcal{B}^t}^{1-\alpha}$$

(log-convexity), supporting real interpolation spaces (Choulli et al., 9 Jul 2025, Lu et al., 6 Feb 2025).

• Continuous embeddings:
  • $\mathcal{B}^s \hookrightarrow L^{\infty}$, $\mathcal{B}^s \hookrightarrow C^k$ for integer $k \leq s$ (Chen et al., 2022).
  • $$\mathcal{B}^{s_2} \hookrightarrow \mathcal{B}^{s_1}$$ for $s_2 \ge s_1$.
  • For group-theoretic generalizations, $$\mathfrak{B}_\gamma^s(G;A) \hookrightarrow H_\gamma^s(G;A)$$, where $H_\gamma^s$ is a weighted Sobolev space in Fourier coefficients (Mensah et al., 13 Dec 2025).
• Banach algebra properties: If $f, g \in \mathcal{B}^s$, then $fg \in \mathcal{B}^s$ (Feng et al., 24 Mar 2025, Mensah, 18 Sep 2025).
• Differentiation: For $f \in \mathcal{B}^{s+1}$, $\partial_k f \in \mathcal{B}^s$ (Feng et al., 24 Mar 2025).
• Duality: The dual space $\tilde{B}^{-s}$ consists of tempered distributions whose inverse Fourier transforms are essentially bounded against the reciprocal spectral weight (Choulli et al., 9 Jul 2025).

Embeddings with Besov and Sobolev Spaces

For $\mathcal{B}^s(\mathbb{R}^d)$, there are sharp dimension-independent embeddings (Liao et al., 2023): $$B^{s+\frac{d}{2}}_{2,1}(\mathbb{R}^d) \hookrightarrow \mathcal{B}^s(\mathbb{R}^d) \hookrightarrow B^s_{\infty,1}(\mathbb{R}^d)$$ Neither endpoint can be improved.

3. Neural Network Approximation and Dimension-Free Rates

Spectral Barron spaces precisely identify the function classes for which shallow and deep neural networks achieve dimension-free approximation rates:

• Shallow networks: For $f \in \mathcal{B}^s(\Omega)$, a two-layer neural network with $N$ units yields

$$\|f - f_N\|_{L^2(\Omega)} \leq C \|f\|_{\mathcal{B}^s} N^{-1/2}$$

for $s \geq 1$ (Liao et al., 9 Jul 2025, Chen et al., 2022, Feng et al., 24 Mar 2025).

• Deep networks: For networks with $L$ hidden layers and width $N$ per layer,

$\text{Error} = O(N^{-sL})$

for $0 < sL \leq 1/2$, dimension-independent and sharp (lower bounds match up to a logarithmic factor) (Liao et al., 9 Jul 2025, Liao et al., 2023).

• Uniform rates: Sup-norm errors differ only by a factor of $\sqrt{d L \ln N}$, i.e., no exponential dimension dependence (Liao et al., 9 Jul 2025).
• Approximation of PDE solutions: If a PDE solution $u$ lies in $\mathcal{B}^s$, e.g., after solving a Schrödinger equation or elliptic HJB, the Barron norm directly controls the neural approximation rate (Chen et al., 2022, Yserentant, 25 Feb 2025, Feng et al., 24 Mar 2025, Ming et al., 25 Aug 2025).

These rates surpass classical Sobolev and Besov approximation bounds, bypassing the curse of dimensionality ($O(N^{-s/d})$), provided the target function's Fourier transform is sufficiently concentrated (Liao et al., 2023).

4. Spectral Barron Spaces in Partial Differential Equations

Spectral Barron spaces are increasingly central to PDE regularity theory due to their compatibility with Fourier multipliers and their Banach-space structure:

• Regularity of Schrödinger eigenfunctions: For Coulomb or inverse-power potentials, electronic wave functions $\psi$ solve $H\psi = \lambda\psi$ and belong to $\mathcal{B}^s(\mathbb{R}^{3N})$ for $s < 1$, with optimality demonstrated by the hydrogen ground state (Yserentant, 25 Feb 2025, Ming et al., 25 Aug 2025).
• Static Schrödinger and elliptic PDEs: If $f, V$ possess Barron regularity, solutions to $(-\Delta + V)u = f$ lie in $\mathcal{B}^{s+2}$, gaining two orders of Barron regularity (Chen et al., 2022, Ming et al., 25 Aug 2025).
• Elliptic operators and boundary problems: Barron spaces provide a setting in which Fredholm-type arguments and fixed-point theorems establish well-posedness and regularity for elliptic equations, Dirichlet problems, and semi-linear PDEs directly in the Barron framework (Choulli et al., 9 Jul 2025).
• Solution theory for HJB equations: For HJB with spectral Barron coefficients, the solution sequence converges locally uniformly to a classical solution in Barron space, yielding neural network approximations at Monte-Carlo rate (Feng et al., 24 Mar 2025).

In each case, the core is the compatibility between Fourier-weighted function spaces and spectral multipliers, ensuring Banach-space regularity and compactness properties necessary for functional-analytic PDE arguments.

5. Generalizations and Group-Theoretic Variants

Spectral Barron spaces have been generalized to

• Quantum harmonic analysis: Via operator-valued quantum Fourier transforms, spectral Barron spaces are defined for trace-class operators on Hilbert space $H$ under a group representation, weighted by a frequency parameter (Mensah, 18 Sep 2025).
• Vector-valued functions and compact groups: For compact $G$, functions $f: G \to V$ possess Barron regularity if their weighted Fourier coefficients in $\widehat{G}$ are $\ell^1$-summable (Mensah et al., 13 Dec 2025).
• Anisotropic and weighted variants: For analysis of space-time PDEs, spectral Barron spaces are extended to anisotropic weighted Fourier-Lebesgue spaces with error measured in Bochner-Sobolev norms. This enables approximation in mixed space-time regimes (Abdeljawad et al., 2023).

These extensions preserve the completeness, interpolation, and embedding properties of classical spectral Barron spaces, with continuous embeddings into corresponding Sobolev or $L^\infty$ spaces.

6. Connections, Embedding Inequalities, and Related Frameworks

Spectral Barron spaces are tightly related to other functional-analytic and machine learning concepts:

• Barron spaces and parameter-based Barron norms: $\mathcal{B}_s$ (parameter-norm Barron space) and $\mathcal{F}_s$ (spectral Barron space) satisfy, for compact $\Omega$,

$$\delta \|f\|_{\mathcal{F}_{s-\delta}(\Omega)} \lesssim_s \|f\|_{\mathcal{B}_s(\Omega)} \lesssim_s \|f\|_{\mathcal{F}_{s+1}(\Omega)}$$

with dimension-free constants; exponent loss is proven optimal (Wu, 2023).

• Convex hull and variation-space representations: Spectral Barron spaces admit a convex hull/atomic representation via the dictionary of decaying Fourier modes, equivalent to a weighted $L^1$ integral norm (Siegel et al., 2021).
• Embedding and universal approximation: The spectral Barron norm precisely characterizes the class of functions for which neural networks (shallow or deep) provide universal approximation without exponential dimension dependence (Lu et al., 6 Feb 2025).

For exponential weights, Barron classes encompass the Paley-Wiener space and Gelfand-Shilov spaces, providing natural symbol classes for operator learning and approximation in Fréchet topologies (Abdeljawad et al., 27 Dec 2024).

7. Applications and Open Problems

• Operator learning and regularization: Spectral Barron spaces enable dimension-free rates for neural approximation of operator-valued mappings (symbols, solution operators) in multiple topologies, e.g., for Tikhonov regularization and inverse problems (Lu et al., 6 Feb 2025, Abdeljawad et al., 27 Dec 2024).
• Quantum chemistry and many-body physics: The regularity of electronic eigenfunctions in spectral Barron spaces reveals that high-dimensional quantum problems admit tractable neural representation, suggesting mitigation of the curse of dimensionality (Yserentant, 25 Feb 2025, Ming et al., 25 Aug 2025).
• PDE solution theory and neural operator design: The Barron setting frames solution regularity directly in terms of approximation complexity, guiding the construction of neural architectures for PDE resolution (Feng et al., 24 Mar 2025, Chen et al., 2022, Choulli et al., 9 Jul 2025).

Unresolved questions include differentiability at singularities in Barron spaces, extension to excited states or long-range potentials, further improvement via deeper networks, and the systematic development of operator-valued Barron spaces for quantum systems (Yserentant, 25 Feb 2025, Mensah, 18 Sep 2025, Mensah et al., 13 Dec 2025).

---

In summary, spectral Barron spaces provide an interdisciplinary, function-analytic framework central to the analysis of neural approximation, high-dimensional PDEs, operator learning, and quantum many-body theory, characterized by weighted Fourier integrability, Banach space structure, and explicit approximation-theoretic rates with no curse of dimensionality.