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arxiv: 2605.30796 · v4 · pith:FTQ4KHFMnew · submitted 2026-05-29 · 🧮 math.NA · cs.NA

Lightning Plus Polynomial Approximation: Optimal Root-Exponential Convergence for Singular Functions in Corner Domains

Shuhuang Xiang , Jun Xiang , Shunfeng Yang , Yuee Zhong This is my paper

Pith reviewed 2026-06-28 21:51 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords lightning approximationroot-exponential convergencecorner singularitiesrational approximationsingular functionspolynomial approximationconvergence ratessector domains
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The pith

Lightning plus polynomial approximations achieve optimal root-exponential convergence for singular functions at corners.

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

The paper establishes that rational approximations formed by polynomials plus lightning terms with exponentially clustered poles near the singularity attain the optimal root-exponential convergence rate for functions of the form g(z) z^α and g(z) z^α log z on sector domains. This rate is O(exp(-π sqrt(2(2-β) N α))), where β relates to the sector angle and α to the singularity strength, and the optimal clustering parameter is explicitly given. The proof relies on isolating the singular part using a decomposition for corner domains and then applying known results or analysis for the prototype functions. Sympathetic readers care because this provides a theoretical basis for efficient numerical methods on non-smooth domains, where convergence would otherwise be limited by the singularity.

Core claim

For prototype functions g(z)z^α or g(z)z^α log z with g analytic near the sector, the lightning plus polynomial scheme with σ_opt = sqrt(2(2-β)π)/sqrt(α) achieves the convergence O(e^{-sqrt(2(2-β)Nα) π}), which is optimal and matches Stahl's rate when β=0. This extends to corner domains via the Gopal-Trefethen decomposition framework.

What carries the argument

The lightning approximation consisting of rational functions with preassigned tapered exponentially clustered poles near the corner, combined with a polynomial term.

If this is right

  • When the sector angle corresponds to β=0, the rate reduces to Stahl's optimal rate for x^α on [0,1].
  • The same optimal rate holds for the logarithmic singularity cases.
  • Explicit formulas for the optimal pole clustering parameter are provided for use in computations on corner domains.
  • The conjectures from the 2023 SIAM J. Numer. Anal. paper are confirmed.

Where Pith is reading between the lines

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

  • The approach may generalize to other singular functions if they can be decomposed similarly into analytic plus singular prototype parts.
  • Practical codes for solving PDEs on polygonal domains could adopt this σ_opt to achieve the fastest possible convergence with this method.
  • It raises the question of whether even better rates are possible with different pole placements or more general rationals, but the paper shows this simple choice is already optimal.

Load-bearing premise

The singular function on the corner domain can be decomposed into an analytic part plus the prototype singular function g(z) z^α or with log, allowing the lightning approximation to be applied directly to the singular part.

What would settle it

Numerical computation of the error in approximating z^{0.5} on a 90-degree sector for increasing numbers of poles N, checking if the observed rate matches or exceeds the predicted sqrt(N) in the exponent.

Figures

Figures reproduced from arXiv: 2605.30796 by Jun Xiang, Shuhuang Xiang, Shunfeng Yang, Yuee Zhong.

Figure 1.1
Figure 1.1. Figure 1.1: Sector domain (left): Sβ = n z : z = xe± θπ 2 i x ∈ [0, 1], θ ∈ [0, β] o and V-shaped domain (right): Vβ = n z : z = xe± βπ 2 i , x ∈ [0, 1]o for fixed β ∈ [0, 2). The red points illustrate the distributions of exponentially clustered poles. Another two specific pole-distribution models on the interval [−C, 0] were introduced in Tre￾fethen, Nakatsukasa, and Weideman [30]. One is a uniform exponential clu… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: A curvy domain with an interior angle φkπ, determined by the tangent rays extending from the common vertex. In this paper, we provide a complete proof of Conjecture 1.2 for the sector domain Sβ by employing Poisson’s summation formula [7, 23] in conjunction with Cauchy’s integral theorem and Runge’s approximation theorem. Furthermore, we demonstrate that selecting the poles in (1.4) with σ = π √ 2(2−β) √… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Fig.1.3 [PITH_FULL_IMAGE:figures/full_fig_p005_1_3.png] view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: The distance of the poles (1.4) from the singularity z = 0 with various values of σ = η −1σopt with η −1 = 4, 2, 1, 0.5 and fixed α = π 10 and β = 1 2 : σopt = π p 2(2 − β)/ √ α. uniformly for z ∈ Sβ. In particular, for α being a positive integer, it holds |reN (z) − z α log z| = ( O(e −σα√ N ), σ ≤ σopt, O(e −πη√ 2(2−β)Nα), σ > σopt. (1.10) 0 10 20 30 10-15 10-10 10-5 100 0 10 20 30 10-15 10-10 10-5 100… view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: ∥z α log z − rN,σ(z)∥C(Sβ) with various values of σ, fixed α = π 10 , β = 1 2 (left) and α = π 2 , β = 7 4 (right), respectively: N1 = (2 : 1 : 30)2 and N2 = ceil(1.3 √ N1) [PITH_FULL_IMAGE:figures/full_fig_p006_1_5.png] view at source ↗
Figure 1.6
Figure 1.6. Figure 1.6: ∥z α log z − rN,σ(z)∥C(Sβ) with various values of σ, fixed α = π 6 , β = 2 3 and σopt = π √ 2(2−β) √ α . By employing the rational schemes as presented in Theorem 1.1 and Theorem 1.2, we can directly apply the identities |x| 2 = x 2 , |x| 2α = x 2α , 2πη√ Nα = 4π 2 σ √ N to establish the following corollary similar to [38, Theorem 1.3] for all α > 0. Corollary 1.1. Let rN be defined in (1.1), then rN (x … view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: ∥z α − rNt (z)∥C(Sβ) (left) and ∥z α log z − reNt (z)∥C(Sβ) (right) with σ = π √ 2(2−β) √ α and Nt = ⌈(κ + 1)2N1⌉ where κ is defined by (2.11). = X N1 j=1  pj |pj | αh[ √ jh − T] l 2 √ jh(z − pj ) + |pj | αh[ √ jh − T] l 2 √ jh  Ym k=1 z − sk sk − pj ! (2.20) + X Nt j=N1+1 z|pj | αh[ √ jh − T] l 2 √ jh(z − pj ) Ym k=1 z − sk sk − pj ! = X N1 j=1 a (l) j z − pj + X Nt j=N1+1 z|pj | αh( √ jh − T) l 2 √ j… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Neighborhood Ωρ ⊃ Sβ. radius 1 + d), which implies |z| ≤ 3 2 , |z − sk| ≤ |z| + sk ≤ sk + 3 2 for z ∈ Ωρ, and max z∈Ωρ Ym k=1 |z − sk| ≤ Ym k=1  sk + 3 2  = Ym k=1  δ + ω + 3 2 + ω cos (2k − 1)π 2m  ≤  δ + ω + 3 2 m = ℵ m by the arithmetic mean-geometric mean inequality. Particularly, the above upper bound for β = 0 can be replaced by 2 √ 2 + 1 2 m = ℵ m. We now establish an upper bound for |r (l)… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The integral contours Γ − h (blue) and Γ + h (red). The poles of f (l,1)(u, z) lying in the interior of contour Γ ∓ h for the two integrals in (3.30) are u(2j, k), u(2j − 1, k), j = 1, · · · , m0, k = 0, 1, · · · , m. In view of Lemma 3.1 and Lemma 3.2, it suffices to estimate the third integral in (3.32). The third integral can be bounded by the representation given below Z 2m0πα 0 h f (l,1)( √ h + it, … view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Error decay rates (left), contours with pole distributions (middle), and pointwise errors (right) for LP solution rn(z) to Laplace equation on a domain Ω that looks like a violet. 6 Conclusions Building upon the rigorous analysis of root-exponential convergence for lightning schemes in ra￾tional approximation of corner singularities, this paper establishes the optimal convergence rate for efficient light… view at source ↗
read the original abstract

This paper presents a rigorous convergence analysis for the lightning plus polynomial approximation scheme, which employs rational approximations constructed with preassigned tapered, exponentially clustered poles. This pole placement strategy was originally introduced by Trefethen and his collaborators for the resolution of corner singularities. Ample numerical results indicate that this scheme achieves root-exponential convergence, and in particular attains the same optimal convergence rate as the best rational approximation to $x^\alpha$ on $[0,1]$ established by Stahl.% which is conjectured in [SIAM J. Numer. Anal., 61:2580-2600, 2023]. In this work, we establish optimal root-exponential convergence for the class of prototype functions of the form $g(z)z^\alpha$ or $g(z)z^\alpha\log z$, where $g$ is analytic on a neighborhood of the sector domain. These results confirm the validity of Conjectures 3.1 and 5.3 stated in [SIAM J. Numer. Anal., 61:2580-2600, 2023], and demonstrate that the choice $\sigma_{\mathrm{opt}} =\frac{\sqrt{2(2 - \beta)}\pi}{\sqrt{\alpha}}$ achieves the theoretically optimal convergence rate $\mathcal{O}\left(e^{-\sqrt{2(2 - \beta)N\alpha}\pi}\right)$. Notably, for the specific case of $\beta = 0$, the scheme recovers Stahl's optimal convergence rate for $x^\alpha$. Furthermore, working within the decomposition framework for corner domains proposed by Gopal and Trefethen, this paper provides a rigorous proof of optimal root-exponential convergence for lightning plus polynomial approximation problems on corner domains, and explicitly derives the optimal pole clustering parameter.

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

1 major / 1 minor

Summary. The paper claims to provide a rigorous convergence analysis for the lightning plus polynomial approximation scheme applied to singular functions. It establishes optimal root-exponential convergence for prototype functions of the form g(z)z^α or g(z)z^α log z (with g analytic in a neighborhood of the sector), confirms Conjectures 3.1 and 5.3 from prior work, derives the optimal pole clustering parameter σ_opt, recovers Stahl's rate for β=0, and extends the result to general corner domains within the Gopal-Trefethen decomposition framework.

Significance. If the proofs are complete and the decomposition transfer holds, the work is significant for supplying the theoretical foundation for an observed optimal approximation technique for corner singularities, matching the best rational approximation rate and providing an explicit σ_opt that generalizes Stahl's result.

major comments (1)
  1. [Abstract / corner domains paragraph] Abstract (paragraph on corner domains) and the corresponding extension section: the rigorous proof of optimality for corner domains requires that the analytic remainder after the Gopal-Trefethen decomposition admits an approximation rate strictly faster than O(e^{-√[2(2-β)Nα]π}); no explicit bound, lemma, or argument is supplied showing the remainder error is o of the singular prototype rate, which is load-bearing for the central claim on general corner problems.
minor comments (1)
  1. [Abstract] Abstract: the stated formula σ_opt = √[2(2-β)] π / √α is inconsistent with the form √[2(2-β)π]/√α needed to produce the claimed rate O(e^{-√[2(2-β)Nα]π}); this should be aligned with the derivation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting this important point regarding the extension to general corner domains. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract / corner domains paragraph] Abstract (paragraph on corner domains) and the corresponding extension section: the rigorous proof of optimality for corner domains requires that the analytic remainder after the Gopal-Trefethen decomposition admits an approximation rate strictly faster than O(e^{-√[2(2-β)Nα]π}); no explicit bound, lemma, or argument is supplied showing the remainder error is o of the singular prototype rate, which is load-bearing for the central claim on general corner problems.

    Authors: We agree that a complete proof of optimality on general corner domains requires showing that the analytic remainder term (after Gopal-Trefethen decomposition) admits an approximation rate strictly faster than the root-exponential rate of the singular prototype. The current manuscript invokes the decomposition framework and states that the overall scheme achieves the optimal rate, but does not supply an explicit lemma, bound, or argument establishing that the remainder error is o(e^{-√[2(2-β)Nα]π}). In the revision we will add a short section or lemma (likely in the extension section) that bounds the remainder contribution, for instance by appealing to standard polynomial or rational approximation results on a slightly enlarged domain where the remainder is analytic and the lightning-plus-polynomial scheme can be shown to converge faster than the target rate. This will close the gap while preserving the rest of the argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external Stahl result and Gopal-Trefethen framework

full rationale

The paper derives the optimal σ_opt and the root-exponential rate O(e^{-√[2(2-β)Nα]π}) for the prototype functions g(z)z^α (and log variants) by direct analysis of the lightning-plus-polynomial scheme with tapered poles. This is presented as a proof that confirms prior conjectures rather than assuming them. The extension to corner domains is explicitly conditioned on the Gopal-Trefethen decomposition (an external reference), with the claim limited to “within the decomposition framework.” No equation is shown to reduce to a fitted parameter or self-citation by construction; the β=0 case is noted to recover Stahl’s independent result rather than presuppose it. The central claims therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on analyticity of g in a neighborhood of the sector, validity of the Gopal-Trefethen corner decomposition, and Stahl's optimal rate for the model problem on [0,1]; no new free parameters are introduced beyond the given singularity exponent α and angle parameter β.

axioms (2)
  • domain assumption g is analytic in a neighborhood of the sector domain
    Invoked to separate the singular factor z^α from the regular part (abstract).
  • domain assumption Gopal-Trefethen decomposition framework isolates the corner singularity
    Used to reduce the corner-domain problem to the prototype sector case (abstract, corner-domains paragraph).

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