Spectral Barron spaces of vector-valued functions on compact groups

Isiaka Aremua; Yaogan Mensah

arxiv: 2512.12382 · v3 · pith:6DWBNQK6new · submitted 2025-12-13 · 🧮 math.FA

Spectral Barron spaces of vector-valued functions on compact groups

Yaogan Mensah , Isiaka Aremua This is my paper

Pith reviewed 2026-05-22 11:55 UTC · model grok-4.3

classification 🧮 math.FA

keywords spectral Barron spacesvector-valued functionscompact groupsFourier transformsSobolev spacescontinuous embeddingsbounded functions

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The pith

Spectral Barron spaces of vector-valued functions on compact groups admit continuous embeddings into Sobolev spaces and spaces of bounded functions.

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

This paper examines spectral Barron spaces consisting of vector-valued functions on compact groups, where the functions' Fourier transforms satisfy a specific summability condition. It establishes various functional properties of these spaces and proves continuous embeddings into Sobolev spaces for vector-valued functions as well as into the space of bounded vector-valued functions. A reader might care because these results extend ideas from approximation theory and harmonic analysis to settings involving vector-valued data on group structures, which appear in physics and signal processing.

Core claim

Spectral Barron spaces whose elements are made up of some vector-valued functions on a compact group whose Fourier transforms admit a certain summability property possess functional properties and admit continuous embeddings with respect to Sobolev spaces of vector-valued functions and the space of bounded vector-valued functions on compact groups.

What carries the argument

The spectral Barron space, defined via the summability property of the Fourier transform of the vector-valued function on the compact group.

If this is right

These spaces inherit regularity properties from the embedded Sobolev spaces.
Functions in these Barron spaces are bounded on the compact group.
The embeddings allow for the application of Sobolev embedding theorems in this vector-valued group setting.

Where Pith is reading between the lines

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

The construction may extend scalar-valued Barron space results to the vector-valued case on groups.
One could test the claimed embeddings by explicit computation on concrete compact groups such as the circle group.

Load-bearing premise

The summability condition on the Fourier transforms is sufficient to guarantee the claimed functional properties and continuous embeddings, as stated in the abstract without explicit verification details or counterexamples.

What would settle it

A counterexample vector-valued function on a compact group whose Fourier transform meets the summability condition but fails to lie in the target Sobolev space or bounded function space would disprove the continuous embedding.

read the original abstract

In this article, we study spectral Barron spaces whose elements are made up of some vector-valued functions on a compact group whose Fourier transforms admit a certain summability property. We investigate their functional properties and some continuous embeddings of these spaces with respect to other function spaces among which are Sobolev spaces of vector-valued functions and the space of bounded vector-valued functions on compact groups.

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.

Desk Editor's Note private letter to a colleague

The paper defines spectral Barron spaces for vector-valued functions on compact groups via Fourier summability and claims embeddings, but supplies no estimates or proofs for the key step.

read the letter

The main thing here is a definition of spectral Barron spaces for vector-valued functions on compact groups, built around a summability condition on the Fourier transforms, plus claims of continuous embeddings into Sobolev spaces and bounded continuous functions. That specific combination looks new compared to the scalar case in the literature I know. The authors get credit for framing the objects cleanly and identifying the target spaces they want to connect to. It is a direct extension attempt in the harmonic analysis setting. The soft spot is exactly where the stress-test note points: the summability condition is asserted to control the target norms, but there are no explicit comparison constants, no handling of how operator norms scale with representation dimension, and no check on whether the vector-valued reconstruction converges uniformly or in the Sobolev sense. For a Banach-space valued map the geometry of the target space V could matter, and without that worked out the central claim stays unverified. The abstract states the results but does not derive them, which makes the soundness gap real rather than minor. This is aimed at specialists already working on function spaces and Fourier analysis on groups. A reader who wants a complete theory with verifiable bounds will find it preliminary. It deserves a serious referee if the authors can add the missing norm estimates and sharpness discussion; otherwise the contribution stays too thin to carry the claims.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces spectral Barron spaces consisting of vector-valued functions on compact groups whose Fourier transforms satisfy a summability condition. It investigates functional properties of these spaces and claims continuous embeddings into Sobolev spaces of vector-valued functions and into the space of bounded vector-valued functions on the group.

Significance. If the claimed embeddings are established with explicit constants and convergence arguments, the work would extend Barron-type spaces to the setting of compact groups and vector-valued functions, providing a spectral characterization of regularity that could be useful in approximation theory and harmonic analysis on non-abelian groups. The vector-valued generalization and the use of Peter-Weyl theory are natural directions, but the current manuscript does not yet deliver the required technical verification.

major comments (2)

[Abstract] Abstract: The central claims of continuous embeddings into Sobolev spaces of vector-valued functions and into bounded vector-valued functions rest on an unshown argument that the chosen summability condition on the Fourier transforms (presumably a weighted ℓ¹ norm over irreps) controls the target norms. No explicit comparison constants or convergence estimates for the Peter-Weyl reconstruction in the V-norm are supplied.
[Main embedding result] Main embedding result (presumably §3 or §4): The sufficiency of the Fourier summability for boundedness in C_b(G,V) or Sobolev norms is asserted without norm estimates. For vector-valued f:G→V the reconstruction requires bounds on the operator norms of the projections that may depend on dim(π) or the geometry of V; these bounds are not derived or referenced.

minor comments (2)

[Definition of the space] The precise definition of the summability condition (e.g., whether it is ∑_π d_π ||ˆf(π)||_op or a weighted variant) should be stated explicitly with the associated norm on the space.
[Introduction] A brief comparison with existing results on scalar Barron spaces or Sobolev embeddings on compact groups would help situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for more explicit technical details in the embedding results. We address each major comment below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

Referee: [Abstract] Abstract: The central claims of continuous embeddings into Sobolev spaces of vector-valued functions and into bounded vector-valued functions rest on an unshown argument that the chosen summability condition on the Fourier transforms (presumably a weighted ℓ¹ norm over irreps) controls the target norms. No explicit comparison constants or convergence estimates for the Peter-Weyl reconstruction in the V-norm are supplied.

Authors: We agree that the abstract is concise and does not include the technical estimates. The proof of the embeddings relies on the Peter-Weyl theorem for V-valued functions, where the reconstruction series converges absolutely in the sup-norm under the given summability condition because unitary representations satisfy ||π(g)|| = 1. To make this fully explicit, we will revise the abstract to reference the norm comparison and add a dedicated paragraph in Section 3 deriving the constant C such that ||f||_{C_b(G,V)} ≤ C ⋅ spectral Barron norm, with C depending only on the group G. Convergence estimates for the partial sums will also be included. revision: yes
Referee: [Main embedding result] Main embedding result (presumably §3 or §4): The sufficiency of the Fourier summability for boundedness in C_b(G,V) or Sobolev norms is asserted without norm estimates. For vector-valued f:G→V the reconstruction requires bounds on the operator norms of the projections that may depend on dim(π) or the geometry of V; these bounds are not derived or referenced.

Authors: We acknowledge that the current presentation assumes familiarity with standard bounds from Peter-Weyl theory without spelling them out for the vector-valued case. The operator-norm bound ||π(g) v|| ≤ ||v|| follows directly from unitarity, and the trace is controlled by dim(π) ⋅ ||hat f(π)||. For the Sobolev embedding we use the Fourier multiplier characterization. In the revision we will insert an explicit derivation of these bounds (including the dependence on dim(π)) together with a reference to the relevant statements in Folland's Abstract Harmonic Analysis or Bump's Lie Groups. This material will appear as a new subsection or short appendix so that the estimates are self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: definitions and embeddings derived from standard Fourier analysis on compact groups

full rationale

The paper defines spectral Barron spaces directly via a summability condition on the Fourier transforms of vector-valued functions on compact groups, then proves functional properties and continuous embeddings into Sobolev and bounded function spaces. No equations reduce a claimed result to its own input by construction, no fitted parameters are relabeled as predictions, and no load-bearing steps rely on self-citations or uniqueness theorems imported from the authors' prior work. The derivation chain uses Peter-Weyl theory and standard operator-norm estimates on irreps, which are external to the present manuscript and not self-referential.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central objects rest on the standard Fourier analysis on compact groups plus the new summability condition that defines membership in the space. No numerical parameters are fitted; the summability rule itself functions as an ad-hoc domain assumption that selects the functions under study.

axioms (2)

standard math Fourier transform on compact groups is well-defined and invertible for suitable vector-valued functions
Invoked implicitly when the paper refers to Fourier transforms admitting summability properties.
ad hoc to paper The chosen summability condition on Fourier coefficients produces a Banach space with the listed embeddings
This is the defining property introduced in the abstract; its consequences are asserted without proof details here.

invented entities (1)

Spectral Barron space of vector-valued functions on compact groups no independent evidence
purpose: To collect functions whose Fourier coefficients satisfy a summability condition and to study their embeddings
New space defined by the summability property; no independent existence proof or external verification is supplied in the abstract.

pith-pipeline@v0.9.0 · 5574 in / 1515 out tokens · 47256 ms · 2026-05-22T11:55:22.574302+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 3.1: B^s_γ(G,A) = {f : (1+γ(σ)^2)^{s/2} ˆf ∈ S^1(Ĝ,A)} with norm ||(1+γ(σ)^2)^{s/2} ˆf||_{S^1}
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.12: continuous embedding B^s_γ(G,A) ↪ L^∞(G,A) via Peter-Weyl reconstruction

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

15 extracted references · 15 canonical work pages · 1 internal anchor

[1]

Math.,53, (1989) 301-307

Assiamoua VSK, Olubummo A., Fourier-Stieltjes transform of vector-valued measures on compact groups, Acta Sci. Math.,53, (1989) 301-307

work page 1989
[2]

Chair Math

Assiamoua VSK, Mensah Y., The Fourier algebraA 1(G, A) of vector valued functions on com- pact groups, In : Contemporary problems in Mathematical Physics, Int. Chair Math. Phys. Appl. (ICMPA-UNESCO Chair), Cotonou, (2008) 223-230

work page 2008
[3]

R, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans

Barron, A. R, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans. Inform. Theory, 39(3), (1993) 930-945. DOI: 10.1109/18.256500

work page doi:10.1109/18.256500 1993
[4]

R, Approximation and estimation bounds for artificial neural networks, Machine Learning, 14(2), (1994) 115-133

Barron, A. R, Approximation and estimation bounds for artificial neural networks, Machine Learning, 14(2), (1994) 115-133. DOI: 10.1007/BF00993164

work page doi:10.1007/bf00993164 1994
[5]

Functional analysis and partial differential equations in spectral Barron spaces

Choulli, M., Lu, S., Takase, H., Functional analysis and partial differential equations in spectral Bar- ron spaces, arXiv:2507.06778v1 [math.FA] 9 Jul 2025. https://doi.org/10.48550/arXiv.2507.06778

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2507.06778 2025
[6]

Math., 175 (2), (1996) 491-505

Dooley, A.H., Gupta, S.K., On norms of trigonometric polynomials onSU(2), Pacific J. Math., 175 (2), (1996) 491-505

work page 1996
[7]

Feng, Y., Lu, J., Solution theory of Hamilton-Jacobi-Bellman equations in spectral Barron spaces, arXiv:2503.18656 [math.AP], 2025 https://doi.org/10.48550/arXiv.2503.18656

work page doi:10.48550/arxiv.2503.18656 2025
[8]

Folland, G.B., A course in abstract harmonic analysis, CRC Press, Inc.,1995

work page 1995
[9]

Hewitt, E., Ross, K.A., Abstract Harmonic Analysis,II, Springer-Verlag 1970

work page 1970
[10]

Liao, Y., Ming, P., Spectral Barron space for deep neural network approximation, SIAM J. Math. Data Sci., 7 (3), (2025) 1053-1076

work page 2025
[11]

Sci., 4(1), (2010) 1-8

Mensah, Y., Assiamoua, VSK, On spaces of Fourier-Stieltjes transform of vector measures on com- pact groups, Math. Sci., 4(1), (2010) 1-8

work page 2010
[12]

Commun., 30, (2025), 153-160

Mensah, Y., Sobolev spaces of vector-valued functions on compact groups, Math. Commun., 30, (2025), 153-160

work page 2025
[13]

Mensah, Y., Vector Fourier analysis on compact groups and Assiamoua spaces, Open J. Math. Anal., 9(1), (2025), 85-91

work page 2025
[14]

10 YAOGAN MENSAH AND ISIAKA AREMUA

Ruzhansky, M., Turunen, V., Pseudo-differential operators and symmetries, Birkh¨ auser Verlag AG, 2010. 10 YAOGAN MENSAH AND ISIAKA AREMUA

work page 2010
[15]

Approx., 57, 1109-1132 (2023)

Siegel, J.W., Xu, J., Characterization of the Variation Spaces Corresponding to Shallow Neural Networks, Constr. Approx., 57, 1109-1132 (2023). https://doi.org/10.1007/s00365-023-09626-4 Department of Mathematics, University of Lom´e, Togo Email address:mensahyaogan2@gmail.com Department of Physics, University of Lom ´e, Togo Email address:claudisak@gmail.com

work page doi:10.1007/s00365-023-09626-4 2023

[1] [1]

Math.,53, (1989) 301-307

Assiamoua VSK, Olubummo A., Fourier-Stieltjes transform of vector-valued measures on compact groups, Acta Sci. Math.,53, (1989) 301-307

work page 1989

[2] [2]

Chair Math

Assiamoua VSK, Mensah Y., The Fourier algebraA 1(G, A) of vector valued functions on com- pact groups, In : Contemporary problems in Mathematical Physics, Int. Chair Math. Phys. Appl. (ICMPA-UNESCO Chair), Cotonou, (2008) 223-230

work page 2008

[3] [3]

R, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans

Barron, A. R, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans. Inform. Theory, 39(3), (1993) 930-945. DOI: 10.1109/18.256500

work page doi:10.1109/18.256500 1993

[4] [4]

R, Approximation and estimation bounds for artificial neural networks, Machine Learning, 14(2), (1994) 115-133

Barron, A. R, Approximation and estimation bounds for artificial neural networks, Machine Learning, 14(2), (1994) 115-133. DOI: 10.1007/BF00993164

work page doi:10.1007/bf00993164 1994

[5] [5]

Functional analysis and partial differential equations in spectral Barron spaces

Choulli, M., Lu, S., Takase, H., Functional analysis and partial differential equations in spectral Bar- ron spaces, arXiv:2507.06778v1 [math.FA] 9 Jul 2025. https://doi.org/10.48550/arXiv.2507.06778

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2507.06778 2025

[6] [6]

Math., 175 (2), (1996) 491-505

Dooley, A.H., Gupta, S.K., On norms of trigonometric polynomials onSU(2), Pacific J. Math., 175 (2), (1996) 491-505

work page 1996

[7] [7]

Feng, Y., Lu, J., Solution theory of Hamilton-Jacobi-Bellman equations in spectral Barron spaces, arXiv:2503.18656 [math.AP], 2025 https://doi.org/10.48550/arXiv.2503.18656

work page doi:10.48550/arxiv.2503.18656 2025

[8] [8]

Folland, G.B., A course in abstract harmonic analysis, CRC Press, Inc.,1995

work page 1995

[9] [9]

Hewitt, E., Ross, K.A., Abstract Harmonic Analysis,II, Springer-Verlag 1970

work page 1970

[10] [10]

Liao, Y., Ming, P., Spectral Barron space for deep neural network approximation, SIAM J. Math. Data Sci., 7 (3), (2025) 1053-1076

work page 2025

[11] [11]

Sci., 4(1), (2010) 1-8

Mensah, Y., Assiamoua, VSK, On spaces of Fourier-Stieltjes transform of vector measures on com- pact groups, Math. Sci., 4(1), (2010) 1-8

work page 2010

[12] [12]

Commun., 30, (2025), 153-160

Mensah, Y., Sobolev spaces of vector-valued functions on compact groups, Math. Commun., 30, (2025), 153-160

work page 2025

[13] [13]

Mensah, Y., Vector Fourier analysis on compact groups and Assiamoua spaces, Open J. Math. Anal., 9(1), (2025), 85-91

work page 2025

[14] [14]

10 YAOGAN MENSAH AND ISIAKA AREMUA

Ruzhansky, M., Turunen, V., Pseudo-differential operators and symmetries, Birkh¨ auser Verlag AG, 2010. 10 YAOGAN MENSAH AND ISIAKA AREMUA

work page 2010

[15] [15]

Approx., 57, 1109-1132 (2023)

Siegel, J.W., Xu, J., Characterization of the Variation Spaces Corresponding to Shallow Neural Networks, Constr. Approx., 57, 1109-1132 (2023). https://doi.org/10.1007/s00365-023-09626-4 Department of Mathematics, University of Lom´e, Togo Email address:mensahyaogan2@gmail.com Department of Physics, University of Lom ´e, Togo Email address:claudisak@gmail.com

work page doi:10.1007/s00365-023-09626-4 2023