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arxiv: 2604.12515 · v2 · submitted 2026-04-14 · 🧮 math.FA

Widths of embeddings of Gaussian Sobolev spaces

Van Kien Nguyen This is my paper

Pith reviewed 2026-05-10 14:28 UTC · model grok-4.3

classification 🧮 math.FA
keywords Gaussian Sobolev spacesKolmogorov widthslinear widthssampling widthsGaussian measuresembeddingsapproximation theory
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The pith

The Kolmogorov, linear, and sampling widths of embeddings from Gaussian Sobolev spaces to Gaussian Lebesgue spaces admit exact asymptotic orders in the regimes 1 ≤ q < p < ∞ and p = q = 2.

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

The paper determines the precise rates at which approximation error tends to zero for functions belonging to Gaussian Sobolev spaces when the error is measured in Gaussian Lebesgue spaces. It does so by computing the asymptotic behavior of Kolmogorov widths, linear widths, and sampling widths, which describe the best possible performance of subspace approximation, linear operators, and pointwise sampling respectively. These rates are obtained exactly for the cases where the integrability index q is strictly smaller than p and where both indices equal 2, revealing distinct scaling behaviors in each regime. The results matter because Gaussian Sobolev spaces appear naturally in high-dimensional stochastic problems, and the widths give the fundamental limits on how many degrees of freedom are needed for a prescribed accuracy.

Core claim

The central claim is that the Kolmogorov, linear, and sampling widths of the natural embedding from the Gaussian Sobolev space W^s_p(R^d, γ) into the Gaussian Lebesgue space L_q(R^d, γ) possess exact asymptotic orders as the dimension of the approximating subspace or the number of sampling points tends to infinity, and that these orders can be stated explicitly in the parameter regimes 1 ≤ q < p < ∞ and p = q = 2.

What carries the argument

The Kolmogorov, linear, and sampling widths of the embedding operator from W^s_p(R^d, γ) to L_q(R^d, γ), which quantify the minimal worst-case approximation error achievable by n-dimensional subspaces, linear maps, and n-point sampling functionals respectively.

If this is right

  • The minimal subspace dimension required to achieve a given approximation accuracy is determined asymptotically by the width.
  • Linear approximation methods attain the same rate as the best nonlinear methods in these regimes.
  • Sampling-based approximation is limited by the sampling width, which matches or differs from the other widths according to the regime.
  • The results supply sharp constants in error bounds for numerical methods applied to functions with Gaussian weights.

Where Pith is reading between the lines

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

  • The regime distinction between q < p and p = q = 2 suggests that approximation complexity changes qualitatively when the target integrability equals the source integrability.
  • The width asymptotics could be used to calibrate the number of basis functions or samples needed in uncertainty-quantification algorithms that rely on Gaussian Sobolev regularity.
  • Similar width calculations might extend to other weighted spaces or to quasi-normed versions of the same spaces by adapting the same embedding arguments.

Load-bearing premise

The analysis relies on the standard continuous embeddings and norm properties of the Gaussian Sobolev and Lebesgue spaces holding without extra restrictions on the smoothness s or dimension d.

What would settle it

A direct computation of any of the three widths for concrete values of s, d, p, q that yields a decay rate different from the stated asymptotic order would falsify the claim.

read the original abstract

In this paper, we investigate the approximation problem for functions in Gaussian Sobolev spaces $W^s_p(\mathbb{R}^d, \gamma)$ of smoothness $s > 0$, where the approximation error is measured in the Gaussian Lebesgue space $L_q(\mathbb{R}^d, \gamma)$. Such function spaces naturally arise in the analysis of high-dimensional problems with Gaussian measures and play an important role in various applications, including uncertainty quantification and stochastic modeling. Our main objective is to analyze the asymptotic behavior of fundamental quantities that characterize the complexity of the approximation problem. In particular, we determine the exact asymptotic order of several classes of widths, including Kolmogorov, linear, and sampling widths, which quantify the optimal performance of different approximation methods. The obtained results cover the parameter regimes $1 \leq q < p < \infty$ and $p = q = 2$, where distinct phenomena in terms of approximation rates can be observed.

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 / 1 minor

Summary. The manuscript determines the exact asymptotic orders of Kolmogorov, linear, and sampling widths for the embedding W^s_p(R^d, γ) → L_q(R^d, γ) in the regimes 1 ≤ q < p < ∞ and p = q = 2, for Gaussian Sobolev spaces of smoothness s > 0.

Significance. If the exact orders are rigorously established, the results would clarify the complexity of approximation under Gaussian measures, distinguishing behavior across the stated parameter regimes and supporting applications in high-dimensional uncertainty quantification and stochastic modeling.

minor comments (1)
  1. [Abstract] The abstract states that exact asymptotic orders are determined but supplies no derivation steps, error estimates, or references to specific theorems or sections containing the proofs.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our manuscript, which accurately captures our determination of exact asymptotic orders for Kolmogorov, linear, and sampling widths of the Gaussian Sobolev embedding in the regimes 1 ≤ q < p < ∞ and p = q = 2. We appreciate the recognition of the results' potential value for high-dimensional uncertainty quantification and stochastic modeling. No specific major comments were provided in the report, so we have no point-by-point responses to address at this stage. We remain available to provide additional details or clarifications that might resolve any uncertainty in the recommendation.

Circularity Check

0 steps flagged

No significant circularity detected; derivation self-contained

full rationale

The paper derives exact asymptotic orders for Kolmogorov, linear, and sampling widths of the embedding W^s_p(R^d, γ) → L_q(R^d, γ) from standard embedding theorems and norm equivalences in Gaussian Sobolev and Lebesgue spaces. The abstract and described claims contain no self-definitional steps, no fitted parameters renamed as predictions, and no load-bearing self-citations that reduce the central result to its own inputs. The parameter regimes are stated explicitly, and the widths are characterized via independent approximation-theoretic arguments rather than tautological redefinitions. This constitutes a normal, non-circular derivation chain based on external functional-analytic facts.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard definitions and embedding theorems for Gaussian Sobolev spaces and widths; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard properties of Gaussian measures, Sobolev norms, and Kolmogorov/linear/sampling widths in Banach spaces
    Invoked to define the spaces W^s_p(R^d, γ) and L_q(R^d, γ) and to analyze their embeddings.

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discussion (0)

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Reference graph

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