Optimal asymptotic analyses on Laguerre and Hermite orthogonal approximation for functions of algebraic and logarithmic regularitiesYali

Guidong Liu; Shuhuang Xiang; Yali Zhang

arxiv: 2604.17752 · v1 · submitted 2026-04-20 · 🧮 math.NA · cs.NA

Optimal asymptotic analyses on Laguerre and Hermite orthogonal approximation for functions of algebraic and logarithmic regularitiesYali

Yali Zhang , Guidong Liu , Shuhuang Xiang This is my paper

Pith reviewed 2026-05-10 04:44 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Laguerre polynomialsHermite polynomialsorthogonal approximationasymptotic estimatesalgebraic singularitieslogarithmic singularitiesspectral projections

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The pith

Optimal asymptotic estimates are derived for the decay of Laguerre and Hermite coefficients of functions with algebraic and logarithmic singularities.

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

The paper sets out to obtain sharp rates at which the coefficients in Laguerre and Hermite series decrease when the underlying function carries an algebraic or logarithmic singularity. It reaches these rates by applying the Hilb-type formula together with van der Corput-type lemmas to the defining integrals. The resulting coefficient decay then fixes the convergence speed of the associated orthogonal projections. A reader would care because these explicit rates tell exactly how many terms are needed to reach a prescribed accuracy for singular data. Concrete examples are worked out to confirm that the predicted rates cannot be improved.

Core claim

Using the Hilb-type formula and van der Corput-type lemmas, optimal asymptotic estimates are obtained for the decay of the Laguerre and Hermite coefficients of functions possessing algebraic and logarithmic regularities; these estimates directly determine the convergence rates of the corresponding spectral orthogonal projections.

What carries the argument

Hilb-type formula and van der Corput-type lemmas applied to the integrals that define the orthogonal coefficients for singular integrands.

If this is right

The error of the Laguerre or Hermite spectral projection is bounded by an explicit rate that depends only on the type and strength of the singularity.
The derived rates are sharp, as confirmed by the numerical verification examples in the paper.
The same coefficient decay controls the accuracy of any truncated expansion or Galerkin scheme built on these orthogonal bases.

Where Pith is reading between the lines

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

The approach supplies a template that could be reused for other classical orthogonal polynomial families once comparable asymptotic formulas are available.
In applications that expand singular data in Laguerre or Hermite bases, the results give a direct way to choose the truncation degree needed for a target tolerance.

Load-bearing premise

The Hilb-type formula and van der Corput-type lemmas apply directly to the coefficient integrals and produce the sharpest decay rates possible for the algebraic and logarithmic singularities considered.

What would settle it

Numerical computation of the Laguerre or Hermite coefficients for a concrete test function such as x to a non-integer power times log x, followed by checking whether their magnitude for large degree follows the exact power-law decay predicted by the asymptotic formula.

Figures

Figures reproduced from arXiv: 2604.17752 by Guidong Liu, Shuhuang Xiang, Yali Zhang.

**Figure 2.** Figure 2: The asymptotic estimates of |an(α)| for f(x) = |x −0.3| γ lnµ |x − 0.3| with γ = 1.2, µ = 2 (first row) and γ = 3, µ = 1 (second row). In all figures, the blue dashed lines represent the asymptotic orders provided in Theorem 4.2. 4.3. The convergence rates on the Laguerre orthogonal projections For the function f(x) = x δ lnµ (x)g(x), it is straightforward to verify that f ∈ L 2 ωα [0, +∞) if α + 2δ > −1. … view at source ↗

**Figure 3.** Figure 3: The asymptotic estimates of k f − S (α) N [f]kL 2 wα [0,+∞) for f(x) = x δ lnµ (x): δ = 1.2, µ = 3 (first row); δ = 4, µ = 1 (second row). In all figures, the blue dashed lines represent the asymptotic orders provided in Theorem 4.3. Theorem 4.4. Let f(x) be defined by (31) and satisfy the assumptions in Theorem 4.2. Then, as N → ∞, the Laguerre expansion follows that k f(x) − S (α) N [f](x)kL 2 ωα [0,+∞) … view at source ↗

**Figure 4.** Figure 4: The asymptotic estimates of k f − S (α) N [f]kL 2 wα [0,+∞) for f(x) = |x − 0.3| γ lnµ |x − 0.3|: γ = 1.2, µ = 2 (first row); γ = 3, µ = 1 (second row). In all figures, the blue dashed lines represent the asymptotic orders provided in Theorem 4.4. Moreover, for the non-uniformly Laguerre-weighted Sobolev space H m,α(Ω) [12] with any integer m ≥ 0, α > −1, Ω = (0, +∞), the weighted norm of H m,α(Ω) is defin… view at source ↗

**Figure 5.** Figure 5: The asymptotic estimates of logn |hn| for f(x) = |x − 3| s lnµ |x − 3|. In all figures, the blue dashed lines represent the asymptotic orders provided in Theorem 5.1. 5.2. The convergence rates on the Hermite orthogonal projections The asymptotic behavior of the Hermite spectral expansion coefficients for functions with algebraic and logarithmic regularities at the interior, as described in Theorem 5.1, en… view at source ↗

**Figure 6.** Figure 6: The asymptotic estimates of k f(x) − S N[f](x)kL 2 ω(R) for f(x) = |x − 3| s lnµ |x − 3|. In all figures, the blue dashed lines represent the asymptotic orders provided in Theorem 5.2. Moreover, for the Hermite-weighted Sobolev space Wm (R) [5] with any integer m ≥ 0, the weighted norm of Wm (R) is defined by kvkWm(R) =    Xm p=0 Z +∞ −∞ e −x 2 [v (p) (x)]2 dx    1 2 , 27 [PITH_FULL_IMAGE:fi… view at source ↗

read the original abstract

Based on the Hilb-type formula and van der Corput-type lemmas, we present optimal asymptotic estimates for the decay of the Laguerre and Hermite coefficients for functions with algebraic and logarithmic singularities, which in turn yield the convergence rates of the corresponding spectral orthogonal projections. Numerous examples are provided to verify the optimality of these asymptotic results.

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

Zhang et al. derive optimal coefficient decay rates for Laguerre and Hermite expansions of functions with algebraic and logarithmic singularities.

read the letter

The key takeaway is that this paper gives explicit optimal decay rates for the expansion coefficients of Laguerre and Hermite series when the function has algebraic or logarithmic singularities. It then converts those into convergence rates for the orthogonal projections. What is new is the treatment of logarithmic singularities alongside the algebraic ones. Earlier work handled smoother functions or pure power singularities, but logs appear in many applications and had not received the same asymptotic treatment in these bases. The authors start from the Hilb-type asymptotic for the polynomials, substitute into the coefficient integral, and use van der Corput lemmas to extract the leading decay term. They supply several test functions and show that the observed rates match the theory. The approach is standard and the examples are a plus because they confirm sharpness rather than just giving bounds. The math relies on well-known lemmas, so the novelty is in the application and the verification that the conditions hold for these amplitudes. For algebraic singularities this is straightforward. For logs the amplitude is not a pure power, so one needs to check that the phase derivatives do not cause the lemma to fail or that any cutoff error is negligible. The paper claims the estimates are optimal and the examples support that, so I take it they handled the details. This is useful for anyone doing error analysis in spectral methods on the half-line or real line with functions that are not infinitely smooth. If you need concrete rates to decide between bases or to bound truncation error, these results are directly applicable. I would bring it to a reading group if the group has people interested in orthogonal polynomial asymptotics or numerical approximation of singular problems. It is not broad enough for a general audience. I would not cite it in my own work unless I had a specific need for these exact rates, but the estimates look solid enough to reference if relevant. It should go to peer review. The claims are precise, the method is classical, and the verification is present, so a specialist referee can assess it without much trouble.

Referee Report

2 major / 2 minor

Summary. The manuscript derives optimal asymptotic estimates for the decay rates of Laguerre and Hermite coefficients for functions possessing algebraic and logarithmic singularities. It invokes Hilb-type formulas to approximate the orthogonal polynomials and applies van der Corput-type lemmas to the resulting oscillatory integrals, thereby obtaining sharp coefficient decay and, from those, convergence rates for the associated spectral projections. Numerous examples are supplied to illustrate the optimality of the rates.

Significance. If the error analysis confirming that the Hilb approximation remainder is negligible relative to the leading term (particularly for logarithmic singularities) can be supplied, the results would furnish sharp, usable convergence rates for orthogonal polynomial approximations of functions with limited regularity at endpoints or infinity. This would be a useful addition to the literature on spectral methods for singular problems.

major comments (2)

[Abstract and sections deriving the coefficient asymptotics] The central claim of optimality rests on substituting the Hilb-type formula into the coefficient integral and controlling the resulting oscillatory integral via a van der Corput-type lemma. For logarithmic singularities the amplitude is no longer a pure power and the phase derivatives may vanish or become unbounded near the singularity; the manuscript states that the lemmas are applied directly but supplies no separate error analysis showing that the approximation error is o of the main term for the precise singularity classes considered. This analysis is load-bearing for the asserted sharpness.
[Abstract] The abstract asserts optimality via standard lemmas without exhibiting the explicit function definitions, the precise hypotheses under which the lemmas are invoked, or verification that no post-hoc restrictions on the singularity parameters were imposed to make the lemmas applicable. Without these, the support for the claim that the derived rates are optimal (rather than merely upper bounds) cannot be assessed.

minor comments (2)

Clarify the precise locations of the singularities (at 0, at infinity, or both) for each class of test functions and state the admissible ranges of the algebraic and logarithmic exponents.
Ensure that the statements of the Hilb-type formula and the van der Corput-type lemmas used are quoted with their exact hypotheses so that the reader can verify applicability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of our asymptotic results. We address the major comments point by point below.

read point-by-point responses

Referee: [Abstract and sections deriving the coefficient asymptotics] The central claim of optimality rests on substituting the Hilb-type formula into the coefficient integral and controlling the resulting oscillatory integral via a van der Corput-type lemma. For logarithmic singularities the amplitude is no longer a pure power and the phase derivatives may vanish or become unbounded near the singularity; the manuscript states that the lemmas are applied directly but supplies no separate error analysis showing that the approximation error is o of the main term for the precise singularity classes considered. This analysis is load-bearing for the asserted sharpness.

Authors: We agree that a dedicated error analysis is required to confirm that the Hilb-type remainder is negligible compared to the leading term, especially for logarithmic singularities where the amplitude is not a pure power. In the revised manuscript we will insert a new subsection (in the section on coefficient asymptotics) that derives explicit bounds on the approximation error, showing it is o of the main oscillatory integral under the stated conditions on the singularity parameters. This will rigorously justify direct application of the van der Corput lemmas and preserve the claimed sharpness. revision: yes
Referee: [Abstract] The abstract asserts optimality via standard lemmas without exhibiting the explicit function definitions, the precise hypotheses under which the lemmas are invoked, or verification that no post-hoc restrictions on the singularity parameters were imposed to make the lemmas applicable. Without these, the support for the claim that the derived rates are optimal (rather than merely upper bounds) cannot be assessed.

Authors: The abstract is intentionally concise. The explicit function classes (algebraic singularities of the form x^alpha (log x)^beta near the relevant endpoints or infinity, with alpha > -1/2 and beta real), the precise statements of the Hilb-type formulas, and the hypotheses of the van der Corput lemmas (non-vanishing higher derivatives of the phase away from the singularity and controlled amplitude growth) are fully specified in Sections 2 and 3, with no additional restrictions imposed beyond those needed for the integrals to be well-defined. To improve accessibility we will revise the abstract to include a short clause referencing these sections and the range of admissible singularity parameters. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation applies external classical lemmas (Hilb-type, van der Corput) to coefficient integrals

full rationale

The paper derives asymptotic decay rates for Laguerre/Hermite coefficients of functions with algebraic/logarithmic singularities by substituting the Hilb-type formula into the coefficient integral and controlling the resulting oscillatory integral via van der Corput-type lemmas. These are standard external results from approximation theory and oscillatory integral analysis, not defined, fitted, or justified inside the paper. No self-definitional steps, renamed known results, or load-bearing self-citations appear in the chain; the optimality claims rest on direct application of the cited lemmas to the target integrals, with examples provided for verification. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of two classical asymptotic tools to the target function classes; no free parameters or invented entities are mentioned.

axioms (2)

domain assumption Hilb-type formula applies to the integrals arising from the Laguerre and Hermite coefficient expressions for the given singular functions
Invoked as the basis for the decay estimates.
domain assumption van der Corput-type lemmas provide sharp bounds on the oscillatory integrals for the algebraic and logarithmic phases considered
Used to obtain the optimal decay rates.

pith-pipeline@v0.9.0 · 5349 in / 1249 out tokens · 31443 ms · 2026-05-10T04:44:24.768855+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Abramowitz, I.A

M. Abramowitz, I.A. Stegun, Handbook of mathematical fu nctions with formulas, graphs, and mathematical tables, Dover Publications, Inc. , New Y ork, 1966

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Agarwal, F

P . Agarwal, F. Qi, M. Chand, S. Jain, Certain integrals involving the generalized hyper- geometric function and the Laguerre polynomials, J. Comput . Appl. Math. 313 (2017) 307–317 . https: //doi.org/10.1016/j.cam.2016.09.034

work page doi:10.1016/j.cam.2016.09.034 2017

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Avazzadeh, H

Z. Avazzadeh, H. Hassani, P . Agarwal, S. Mehrabi, M.J. Eb adi, M.S. Dahaghin, An optimization method for studying fractional-order tube rculosis disease model via generalized Laguerre polynomials, Soft comput. 27(14) (2023) 9519–9531 . https://doi.org/10.1007/s00500-023-08086-z

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Avazzadeh, H

Z. Avazzadeh, H. Hassani, M.J. Ebadi, P . Agarwal, M. Pour sadeghfard, E. Naraghirad, Optimal Approximation of Fractional Order Bra in Tumor Model Us- ing Generalized Laguerre Polynomials, Iran. J. Sci. 47(2) ( 2023) 501–513 . https://doi.org/10.1007/s40995-022-01388-1

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Colton, J

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Fatyanov, A.V

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Hassani, Z

H. Hassani, Z. Avazzadeh, P . Agarwal, S. Mehrabi, M. Eba di, M.S. Dahaghin, E. Naraghirad, A study on fractional tumor-immune interactio n model related to lung can- cer via generalized Laguerre polynomials, BMC Med. Res. Met hodol. 23(1) (2023)

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https://doi.org/10.1186/s12874-023-02006-3

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J.A. Jo, Q. Fang, T. Papaioannou, J.D. Baker, A.H. Doraf shar, T. Reil, J.H. Qiao, M.C. Fishbein, J.A. Freischlag, L. Marcu, Laguerre-b ased method for anal- ysis of time-resolved ﬂuorescence data: application to in- vivo characterization and diagnosis of atherosclerotic lesions, J. Biomed. Opt. 1 1(2) (2006) 021004. https://doi.org/10.1117/1.2186045

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