pith. sign in

arxiv: 2512.15992 · v2 · submitted 2025-12-17 · 🧮 math.NA · cs.IT· cs.LG· cs.NA· math.IT

Time-Frequency Analysis for Neural Networks

Ahmed Abdeljawad , Elena Cordero This is my paper

Pith reviewed 2026-05-16 21:07 UTC · model grok-4.3

classification 🧮 math.NA cs.ITcs.LGcs.NAmath.IT
keywords time-frequency analysismodulation spacesneural network approximationSobolev normsshallow networksdimension-independent ratesweighted modulation spaces
0
0 comments X

The pith

Shallow neural networks using time-frequency localized units achieve N^{-1/2} approximation rates in Sobolev norms for functions in modulation spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a quantitative approximation theory for shallow neural networks by incorporating time-frequency analysis into the construction of network units. It proves that for functions in weighted modulation spaces, networks formed by combining standard activations with localized time-frequency windows deliver dimension-independent error bounds of order N to the power of negative one half in Sobolev norms on bounded domains, with all constants made explicit. This approach also yields global approximation results on the full space and extends to several related function spaces, with numerical tests showing improved performance over standard ReLU networks.

Core claim

For any function f in the weighted modulation space M^{p,q}_m(R^d), there exist shallow neural networks f_N with N units such that the Sobolev norm error ||f - f_N||_{W^{n,r}(Ω)} is bounded by a constant multiple of N^{-1/2} times the modulation norm of f on bounded domains Ω. The units are built by pairing standard activations with localized time-frequency windows to form dictionaries that satisfy the covering and localization properties required in modulation space theory.

What carries the argument

Modulation space dictionaries formed by combining standard activations with localized time-frequency windows, which provide the covering and localization properties that deliver the N^{-1/2} rate.

If this is right

  • Global approximation theorems hold on the whole space R^d using weighted modulation dictionaries.
  • The results apply directly to Feichtinger's algebra, Fourier-Lebesgue spaces, and Barron spaces.
  • Modulation-based networks achieve substantially better Sobolev approximation than standard ReLU networks in one- and two-dimensional numerical tests.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same dictionary construction could be layered in deeper architectures to preserve the rate while increasing expressivity.
  • Explicit constants in the bounds make it feasible to compare this construction against other function-space approaches to neural approximation.
  • Higher-dimensional numerical tests would directly check whether the theoretical dimension independence appears in practice.

Load-bearing premise

The neural network units can be formed by combining standard activations with localized time-frequency windows such that the resulting dictionary satisfies the necessary covering and localization properties in the modulation space to deliver the stated rate.

What would settle it

A function in a modulation space for which the best approximation error by any network built from such time-frequency units remains larger than order N^{-1/2} in the target Sobolev norm on a bounded domain would falsify the main rate.

Figures

Figures reproduced from arXiv: 2512.15992 by Ahmed Abdeljawad, Elena Cordero.

Figure 1
Figure 1. Figure 1: Visualizing the tiling of the time-frequency plane. Left: Modulation spaces use a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visual representation of the admissible regions for the weight indices [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Training loss over epochs for the modulation and plain ReLU networks (1201 [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of plain and modulation model predictions on unseen one-dimensional [PITH_FULL_IMAGE:figures/full_fig_p035_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Training loss over epochs for the modulation and plain ReLU networks (1801 [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of plain and modulation model predictions on unseen two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of plain and modulation model predictions on unseen two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of plain and modulation model predictions on unseen one-dimensional [PITH_FULL_IMAGE:figures/full_fig_p036_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Loss (in log scale) versus epochs for the approximation of [PITH_FULL_IMAGE:figures/full_fig_p036_9.png] view at source ↗
read the original abstract

We develop a quantitative approximation theory for shallow neural networks using tools from time-frequency analysis. Working in weighted modulation spaces $M^{p,q}_m(\mathbf{R}^{d})$, we prove dimension-independent approximation rates in Sobolev norms $W^{n,r}(\Omega)$ for networks whose units combine standard activations with localized time-frequency windows. Our main result shows that for $f \in M^{p,q}_m(\mathbf{R}^{d})$ one can achieve \[ \|f - f_N\|_{W^{n,r}(\Omega)} \lesssim N^{-1/2}\,\|f\|_{M^{p,q}_m(\mathbf{R}^{d})}, \] on bounded domains, with explicit control of all constants. We further obtain global approximation theorems on $\mathbf{R}^{d}$ using weighted modulation dictionaries, and derive consequences for Feichtinger's algebra, Fourier-Lebesgue spaces, and Barron spaces. Numerical experiments in one and two dimensions confirm that modulation-based networks achieve substantially better Sobolev approximation than standard ReLU networks, consistent with the theoretical estimates.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a quantitative approximation theory for shallow neural networks using tools from time-frequency analysis. Working in weighted modulation spaces M^{p,q}_m(R^d), it proves dimension-independent approximation rates of order N^{-1/2} in Sobolev norms W^{n,r}(Ω) on bounded domains, with explicit control of all constants, realized by combining standard activations with localized time-frequency windows. Additional results cover global approximation on R^d using weighted modulation dictionaries and consequences for Feichtinger's algebra, Fourier-Lebesgue spaces, and Barron spaces. Numerical experiments in one and two dimensions show that modulation-based networks outperform standard ReLU networks in Sobolev approximation, consistent with the theory.

Significance. If the central claims hold, the work is significant for establishing explicit, dimension-independent rates that connect time-frequency analysis directly to neural network approximation, avoiding hidden dimension dependence. The provision of explicit constants, atomic decomposition or greedy selection arguments, and reproducible numerical validation in 1D/2D are particular strengths that make the result falsifiable and practically relevant for high-dimensional settings.

minor comments (3)
  1. §3 (main theorem): the statement that constants are fully explicit would be strengthened by including a brief remark on their dependence (or independence) on the weight m and the parameters p,q,r,n; this is a local clarification rather than a load-bearing gap.
  2. Numerical experiments section: the description of how standard activations are combined with localized TF windows to form the dictionary could be expanded with one concrete example (e.g., the explicit form of a single unit) to aid reproducibility.
  3. Notation: the transition from the modulation-space norm on R^d to the Sobolev norm on bounded Ω is clear in the abstract but would benefit from an explicit statement of the restriction operator in the statement of Theorem 1.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance in connecting time-frequency analysis to neural network approximation theory, and the recommendation for minor revision. No major comments were raised in the report, so we have no specific points requiring detailed rebuttal or revision at this stage. We remain available to address any minor suggestions or clarifications that may arise.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds from established atomic decompositions and covering properties of modulation spaces M^{p,q}_m to the stated Sobolev approximation rate via N-term selection of time-frequency atoms realized with standard activations. The main inequality follows directly from these independent space properties and greedy selection arguments without any step that defines the target rate in terms of itself or renames a fitted quantity as a prediction. No load-bearing self-citation chain is required for the central claim; the constants are stated to be explicit and controlled by the modulation norm alone. The construction on bounded domains and global extensions are self-contained against the cited time-frequency literature.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the established theory of weighted modulation spaces and the assumption that time-frequency localized activations can be realized in neural network units; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • standard math Standard properties of weighted modulation spaces M^{p,q}_m(R^d) and their embeddings into Sobolev spaces
    The approximation result is stated directly in these spaces, relying on their known time-frequency localization and norm equivalences.
  • domain assumption Existence and approximation properties of dictionaries formed by combining standard activations with localized time-frequency windows
    The network construction presupposes that such hybrid units generate a dictionary capable of delivering the N^{-1/2} rate with explicit constants.

pith-pipeline@v0.9.0 · 5484 in / 1459 out tokens · 38208 ms · 2026-05-16T21:07:44.087086+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages · 1 internal anchor

  1. [1]

    Uniform approximation with quadratic neural networks,

    A. Abdeljawad, “Uniform approximation with quadratic neural networks,”Neural Net- works, vol. 192, p. 107742, Dec. 2025.doi:10.1016/j.neunet.2025.107742

    work page doi:10.1016/j.neunet.2025.107742 2025

  2. [2]

    Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey-type pseudo-differential operators,

    A. Abdeljawad, S. Coriasco, and J. Toft, “Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey-type pseudo-differential operators,”Analysis and Ap- plications, vol. 18, no. 04, pp. 523–583, Jul. 2020.doi:10.1142/S0219530519500143

    work page doi:10.1142/s0219530519500143 2020

  3. [3]

    Abdeljawad and T

    A. Abdeljawad and T. Dittrich,Space-Time Approximation with Shallow Neural Net- works in Fourier Lebesgue Spaces, arXiv:2312.08461 [cs], Dec. 2023

    work page arXiv 2023

  4. [4]

    Do large language models perform latent multi-hop reasoning without exploiting shortcuts? abs/2411.16679, 2024

    A. Abdeljawad and T. Dittrich,Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains, Version Number: 1, 2024.doi:10.48550/ARXIV. 2411.04108

    work page internal anchor Pith review doi:10.48550/arxiv 2024

  5. [5]

    Approximations with deep neural networks in Sobolev time-space,

    A. Abdeljawad and P. Grohs, “Approximations with deep neural networks in Sobolev time-space,”Analysis and Applications, vol. 20, no. 03, pp. 499–541, May 2022.doi: 10.1142/S0219530522500014

    work page doi:10.1142/s0219530522500014 2022

  6. [6]

    Barron, a.e.: Universal approximation bounds for superpositions of a sigmoidal function

    A. Barron, “Universal approximation bounds for superpositions of a sigmoidal func- tion,”IEEE Transactions on Information Theory, vol. 39, no. 3, pp. 930–945, May 1993.doi:10.1109/18.256500

    work page doi:10.1109/18.256500 1993

  7. [7]

    Subexponential decay and regularity estimates for eigenfunctions of localization operators,

    F. Bastianoni and N. Teofanov, “Subexponential decay and regularity estimates for eigenfunctions of localization operators,”Journal of Pseudo-Differential Operators and Applications, vol. 12, no. 1, p. 19, Mar. 2021.doi:10.1007/s11868-021-00383-1

    work page doi:10.1007/s11868-021-00383-1 2021

  8. [8]

    Fundamentals of Enzyme Kinetics: Michaelis-Menten and Non-Michaelis- Type (Atypical) Enzyme Kinetics

    Á. Bényi and K. A. Okoudjou,Modulation Spaces: With Applications to Pseudod- ifferential Operators and Nonlinear Schrödinger Equations(Applied and Numerical Harmonic Analysis). New York, NY: Springer New York, 2020.doi:10.1007/978-1- 0716-0332-1

    work page doi:10.1007/978-1- 2020

  9. [9]

    Generalized Anti-Wick Operators with Symbols in Distributional Sobolev spaces,

    P. Boggiatto, E. Cordero, and K. Gröchenig, “Generalized Anti-Wick Operators with Symbols in Distributional Sobolev spaces,”Integral Equations and Operator Theory, vol. 48, no. 4, pp. 427–442, Apr. 2004, Publisher: Springer Science and Business Media LLC.doi:10.1007/s00020-003-1244-x

    work page doi:10.1007/s00020-003-1244-x 2004

  10. [10]

    Stochastic partial differential equations in M-type 2 Banach spaces,

    Z. Brzeźniak, “Stochastic partial differential equations in M-type 2 Banach spaces,” Potential Analysis, vol. 4, no. 1, pp. 1–45, Feb. 1995.doi:10.1007/BF01048965 40

    work page doi:10.1007/bf01048965 1995

  11. [11]

    Neural Network Approximation and Estimation of Classifiers with Classification Boundary in a Barron Class,

    A. Caragea, P. Petersen, and F. Voigtlaender, “Neural Network Approximation and Estimation of Classifiers with Classification Boundary in a Barron Class,”The Annals of Applied Probability, vol. 33, no. 4, Aug. 2023.doi:10.1214/22-AAP1884

    work page doi:10.1214/22-aap1884 2023

  12. [12]

    Deep relaxation: Partial differential equations for optimizing deep neural networks,

    P. Chaudhari, A. Oberman, S. Osher, S. Soatto, and G. Carlier, “Deep relaxation: Partial differential equations for optimizing deep neural networks,”Research in the Mathematical Sciences, vol. 5, no. 3, p. 30, Sep. 2018.doi:10.1007/s40687- 018- 0148-y

    work page doi:10.1007/s40687- 2018

  13. [13]

    Neural ordinary differential equations,

    R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud, “Neural ordinary differential equations,” inAdvances in Neural Information Processing Systems, S. Ben- gio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, Eds., vol. 31, Curran Associates, Inc., 2018

    work page 2018

  14. [14]

    A Regularity Theory for Static Schrödinger Equa- tions on{R} d in Spectral Barron Spaces,

    Z. Chen, J. Lu, Y. Lu, and S. Zhou, “A Regularity Theory for Static Schrödinger Equa- tions on{R} d in Spectral Barron Spaces,”SIAM Journal on Mathematical Analysis, vol. 55, no. 1, pp. 557–570, Feb. 2023.doi:10.1137/22M1478719

    work page doi:10.1137/22m1478719 2023

  15. [15]

    Optimal Stable Nonlinear Approximation,

    A. Cohen, R. DeVore, G. Petrova, and P. Wojtaszczyk, “Optimal Stable Nonlinear Approximation,”Foundations of Computational Mathematics, vol. 22, no. 3, pp. 607– 648, Jun. 2022.doi:10.1007/s10208-021-09494-z

    work page doi:10.1007/s10208-021-09494-z 2022

  16. [16]

    Time–Frequency analysis of localization operators,

    E. Cordero and K. Gröchenig, “Time–Frequency analysis of localization operators,” Journal of Functional Analysis, vol. 205, no. 1, pp. 107–131, Dec. 2003.doi:10.1016/ S0022-1236(03)00166-6

    work page 2003

  17. [17]

    Cordero and L

    E. Cordero and L. Rodino,Time-Frequency Analysis of Operators. De Gruyter, Sep. 2020.doi:10.1515/9783110532456

    work page doi:10.1515/9783110532456 2020

  18. [18]

    Approximation by superpositions of a sigmoidal function.Mathematics of Control, Signals and Systems, 2:303–314, 1989

    G. Cybenko, “Approximation by Superpositions of a Sigmoidal Function,”Mathe- matics of Control, Signals, and Systems, vol. 2, no. 4, pp. 303–314, Dec. 1989.doi: 10.1007/BF02551274

    work page doi:10.1007/bf02551274 1989

  19. [19]

    Weighted variation spaces and approximation by shallow ReLU networks,

    R. DeVore, R. D. Nowak, R. Parhi, and J. W. Siegel, “Weighted variation spaces and approximation by shallow ReLU networks,”Applied and Computational Harmonic Analysis, vol. 74, p. 101713, Jan. 2025.doi:10.1016/j.acha.2024.101713

    work page doi:10.1016/j.acha.2024.101713 2025

  20. [20]

    Nonlinear Approximation,

    R. A. DeVore, “Nonlinear Approximation,”Acta Numerica, vol. 7, pp. 51–150, Jan. 1998.doi:10.1017/S0962492900002816

    work page doi:10.1017/s0962492900002816 1998

  21. [21]

    The Barron Space and the Flow-Induced Function Spaces for Neural Network Models,

    W. E, C. Ma, and L. Wu, “The Barron Space and the Flow-Induced Function Spaces for Neural Network Models,”Constructive Approximation, vol. 55, no. 1, pp. 369–406, Feb. 2022.doi:10.1007/s00365-021-09549-y

    work page doi:10.1007/s00365-021-09549-y 2022

  22. [22]

    The Deep Ritz Method: A Deep Learning-Based Numerical Algo- rithm for Solving Variational Problems,

    W. E and B. Yu, “The Deep Ritz Method: A Deep Learning-Based Numerical Algo- rithm for Solving Variational Problems,”Communications in Mathematics and Statis- tics, vol. 6, no. 1, pp. 1–12, Mar. 2018.doi:10.1007/s40304-018-0127-z

    work page doi:10.1007/s40304-018-0127-z 2018

  23. [23]

    Deep Neural Network Approximation Theory,

    D. Elbrachter, D. Perekrestenko, P. Grohs, and H. Bolcskei, “Deep Neural Network Approximation Theory,”IEEE Transactions on Information Theory, vol. 67, no. 5, pp. 2581–2623, May 2021.doi:10.1109/TIT.2021.3062161

    work page doi:10.1109/tit.2021.3062161 2021

  24. [24]

    Wilson Bases and Modulation Spaces,

    H. G. Feichtinger, K. Gröchenig, and D. Walnut, “Wilson Bases and Modulation Spaces,”Mathematische Nachrichten, vol. 155, no. 1, pp. 7–17, Jan. 1992.doi:10. 1002/mana.19921550102 41

    work page 1992

  25. [25]

    Atomic characterizations of modulation spaces through Gabor- type representations,

    H. G. Feichtinger, “Atomic characterizations of modulation spaces through Gabor- type representations,”The Rocky Mountain Journal of Mathematics, vol. 19, no. 1, pp. 113–125, 1989, Publisher: Rocky Mountain Mathematics Consortium

    work page 1989

  26. [26]

    Modulation spaces on locally compact abelian groups,

    H. G. Feichtinger, “Modulation spaces on locally compact abelian groups,” inTechnical report, University of Vienna, 1983; also in Proceedings of the international conference on wavelets and applications, R. Radha, M. Krishna, and S. Thangavelu, Eds., New Delhi: Allied Publishers, 2003, pp. 1–56

    work page 1983

  27. [27]

    Time-frequency analysis on modulation spacesMp,q m , 0<p, q≤ ∞,

    Y. V. Galperin and S. Samarah, “Time-frequency analysis on modulation spacesMp,q m , 0<p, q≤ ∞,”Applied and Computational Harmonic Analysis, vol. 16, no. 1, pp. 1–18, Jan. 2004, Publisher: Elsevier BV.doi:10.1016/j.acha.2003.09.001

    work page doi:10.1016/j.acha.2003.09.001 2004

  28. [28]

    Nonlinear Approximation with Local Fourier Bases,

    K. Gröchenig and S. Samarah, “Nonlinear Approximation with Local Fourier Bases,” Constructive Approximation, vol. 16, no. 3, pp. 317–331, Jul. 2000.doi:10. 1007 / s003659910014

    work page 2000

  29. [29]

    Foundations of Time-Frequency Analysis

    K. Gröchenig,Foundations of Time-Frequency Analysis(Applied and Numerical Har- monic Analysis), J. J. Benedetto, Ed. Boston, MA: Birkhäuser Boston, 2001.doi: 10.1007/978-1-4612-0003-1

    work page doi:10.1007/978-1-4612-0003-1 2001

  30. [30]

    Proof of the Theory-to-Practice Gap in Deep Learning Via Sampling Complexity Bounds for Neural Network Approximation Spaces,

    P. Grohs and F. Voigtlaender, “Proof of the Theory-to-Practice Gap in Deep Learning Via Sampling Complexity Bounds for Neural Network Approximation Spaces,”Foun- dations of Computational Mathematics, Jul. 2023.doi:10.1007/s10208-023-09607- w

    work page doi:10.1007/s10208-023-09607- 2023

  31. [31]

    Sharp weighted convolution inequalities and some applications,

    W. Guo, D. Fan, H. Wu, and G. Zhao, “Sharp weighted convolution inequalities and some applications,”Studia Mathematica, vol. 241, no. 3, pp. 201–239, 2018.doi:10. 4064/sm8583-5-2017

    work page 2018

  32. [32]

    Inclusion relations between modulation and Triebel- Lizorkin spaces,

    W. Guo, H. Wu, and G. Zhao, “Inclusion relations between modulation and Triebel- Lizorkin spaces,”Proceedings of the American Mathematical Society, vol. 145, no. 11, pp. 4807–4820, May 2017.doi:10.1090/proc/13614

    work page doi:10.1090/proc/13614 2017

  33. [33]

    Katznelson,An Introduction to Harmonic Analysis, 3rd ed

    Y. Katznelson,An Introduction to Harmonic Analysis, 3rd ed. Cambridge University Press, Jan. 2004.doi:10.1017/CBO9781139165372

    work page doi:10.1017/cbo9781139165372 2004

  34. [34]

    Approximation by Combinations of ReLU and Squared ReLU Ridge Functions withℓ1 andℓ 0 Controls,

    J. M. Klusowski and A. R. Barron, “Approximation by Combinations of ReLU and Squared ReLU Ridge Functions withℓ1 andℓ 0 Controls,”IEEE Transactions on In- formation Theory, vol. 64, no. 12, pp. 7649–7656, Dec. 2018.doi:10.1109/TIT.2018. 2874447

    work page doi:10.1109/tit.2018 2018

  35. [35]

    The inclusion relation between Sobolev and modu- lation spaces,

    M. Kobayashi and M. Sugimoto, “The inclusion relation between Sobolev and modu- lation spaces,”Journal of Functional Analysis, vol. 260, no. 11, pp. 3189–3208, Jun. 2011.doi:10.1016/j.jfa.2011.02.015

    work page doi:10.1016/j.jfa.2011.02.015 2011

  36. [36]

    A Theoretical Analysis of Deep Neural Networks and Parametric PDEs,

    G. Kutyniok, P. Petersen, M. Raslan, and R. Schneider, “A Theoretical Analysis of Deep Neural Networks and Parametric PDEs,”Constructive Approximation, vol. 55, no. 1, pp. 73–125, Feb. 2022.doi:10.1007/s00365-021-09551-4

    work page doi:10.1007/s00365-021-09551-4 2022

  37. [37]

    Lagaris, Aristidis Likas, and Dimitrios I

    I. Lagaris, A. Likas, and D. Fotiadis, “Artificial neural networks for solving ordinary and partial differential equations,”IEEE Transactions on Neural Networks, vol. 9, no. 5, pp. 987–1000, Sep. 1998.doi:10.1109/72.712178 42

    work page doi:10.1109/72.712178 1998

  38. [38]

    Spectral Barron Space and Deep Neural Network Approxima- tion,

    Y. Liao and P. Ming, “Spectral Barron Space and Deep Neural Network Approxima- tion,” 2023, Publisher: arXiv tex.version: 1.doi:10.48550/ARXIV.2309.00788

    work page doi:10.48550/arxiv.2309.00788 2023

  39. [39]

    Neural network approximations of PDEs beyond linearity: A representational perspective,

    T. Marwah, Z. C. Lipton, J. Lu, and A. Risteski, “Neural network approximations of PDEs beyond linearity: A representational perspective,” inProceedings of the 40th international conference on machine learning, A. Krause, E. Brunskill, K. Cho, B. Engelhardt, S. Sabato, and J. Scarlett, Eds., ser. Proceedings of machine learning research, vol. 202, PMLR, J...

    work page 2023

  40. [40]

    Modulation Spaces and the Curse of Dimensionality,

    R. Parhi and M. Unser, “Modulation Spaces and the Curse of Dimensionality,” in 2023 International Conference on Sampling Theory and Applications (SampTA), New Haven, CT, USA: IEEE, Jul. 10, 2023, pp. 1–5.doi:10.1109/SampTA59647.2023. 10301395

    work page doi:10.1109/sampta59647.2023 2023

  41. [41]

    Optimal approximation of piecewise smooth func- tions using deep ReLU neural networks,

    P. Petersen and F. Voigtlaender, “Optimal approximation of piecewise smooth func- tions using deep ReLU neural networks,”Neural Networks, vol. 108, pp. 296–330, 2018. doi:10.1016/j.neunet.2018.08.019

    work page doi:10.1016/j.neunet.2018.08.019 2018

  42. [42]

    Micro-Local Analysis in Fourier Lebesgue and Modulation Spaces: Part II,

    S. Pilipović, N. Teofanov, and J. Toft, “Micro-Local Analysis in Fourier Lebesgue and Modulation Spaces: Part II,”Journal of Pseudo-Differential Operators and Applica- tions, vol. 1, no. 3, pp. 341–376, Sep. 2010.doi:10.1007/s11868-010-0013-2

    work page doi:10.1007/s11868-010-0013-2 2010

  43. [43]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,”Journal of Computational Physics, vol. 378, pp. 686– 707, Feb. 2019.doi:10.1016/j.jcp.2018.10.045

    work page doi:10.1016/j.jcp.2018.10.045 2019

  44. [44]

    M. A. Shubin,Pseudodifferential Operators and Spectral Theory. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001.doi:10.1007/978-3-642-56579-3

    work page doi:10.1007/978-3-642-56579-3 2001

  45. [45]

    Approximation Rates for Neural Networks With General Activation Functions,

    J. W. Siegel and J. Xu, “Approximation Rates for Neural Networks With General Activation Functions,”Neural Networks, vol. 128, pp. 313–321, Aug. 2020.doi:10. 1016/j.neunet.2020.05.019

    work page 2020

  46. [47]

    Characterization of the Variation Spaces Corresponding to Shallow Neural Networks,

    J. W. Siegel and J. Xu, “Characterization of the Variation Spaces Corresponding to Shallow Neural Networks,”Constructive Approximation, Feb. 2023.doi:10 . 1007 / s00365-023-09626-4

    work page 2023

  47. [48]

    Sharp Bounds on the Approximation Rates, Metric Entropy, and n-Widths of Shallow Neural Networks,

    J. W. Siegel and J. Xu, “Sharp Bounds on the Approximation Rates, Metric Entropy, and n-Widths of Shallow Neural Networks,”Foundations of Computational Mathemat- ics, vol. 24, no. 2, pp. 481–537, Apr. 2024.doi:10.1007/s10208-022-09595-3

    work page doi:10.1007/s10208-022-09595-3 2024

  48. [49]

    An algebra of pseudodifferential operators,

    J. Sjöstrand, “An algebra of pseudodifferential operators,”Mathematical Research Let- ters, vol. 1, no. 2, pp. 185–192, 1994, Publisher: International Press of Boston.doi: 10.4310/mrl.1994.v1.n2.a6

    work page doi:10.4310/mrl.1994.v1.n2.a6 1994

  49. [50]

    Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: Optimal rate and curse of dimensionality,

    T. Suzuki, “Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: Optimal rate and curse of dimensionality,” inInternational Conference on Learning Representations, 2019. 43

    work page 2019

  50. [51]

    Tartar,An Introduction to Sobolev Spaces and Interpolation Spaces(Lecture Notes of the Unione Matematica Italiana)

    L. Tartar,An Introduction to Sobolev Spaces and Interpolation Spaces(Lecture Notes of the Unione Matematica Italiana). Berlin, Heidelberg: Springer Berlin Heidelberg, 2007, vol. 3.doi:10.1007/978-3-540-71483-5

    work page doi:10.1007/978-3-540-71483-5 2007

  51. [52]

    Modulation Spaces, Gelfand-Shilov Spaces and Pseudodifferential Op- erators,

    N. Teofanov, “Modulation Spaces, Gelfand-Shilov Spaces and Pseudodifferential Op- erators,”Sampling Theory in Signal and Image Processing, vol. 5, no. 2, pp. 225–242, May 2006.doi:10.1007/BF03549452

    work page doi:10.1007/bf03549452 2006

  52. [53]

    Continuity Properties for Modulation Spaces, with Applications to Pseudo- Differential Calculus, II,

    J. Toft, “Continuity Properties for Modulation Spaces, with Applications to Pseudo- Differential Calculus, II,”Annals of Global Analysis and Geometry, vol. 26, no. 1, pp. 73–106, Aug. 2004.doi:10.1023/B:AGAG.0000023261.94488.f4

    work page doi:10.1023/b:agag.0000023261.94488.f4 2004

  53. [54]

    Lp Sampling Numbers for the Fourier-Analytic Barron Space,

    F. Voigtlaender, “Lp Sampling Numbers for the Fourier-Analytic Barron Space,”arXiv preprint arXiv:2208.07605, 2022, Publisher: arXiv tex.version: 1

    work page arXiv 2022

  54. [55]

    Optimal Rates of Approximation by Shallow ReLUk Neural Networks and Applications to Nonparametric Regression,

    Y. Yang and D.-X. Zhou, “Optimal Rates of Approximation by Shallow ReLUk Neural Networks and Applications to Nonparametric Regression,”Constructive Approxima- tion, vol. 62, no. 2, pp. 329–360, Oct. 2025.doi:10.1007/s00365-024-09679-z

    work page doi:10.1007/s00365-024-09679-z 2025

  55. [56]

    Error Bounds for Approximations with Deep ReLU Networks,

    D. Yarotsky, “Error Bounds for Approximations with Deep ReLU Networks,”Neural Networks, vol. 94, pp. 103–114, Oct. 2017.doi:10.1016/j.neunet.2017.07.002 44

    work page doi:10.1016/j.neunet.2017.07.002 2017