pith. sign in

arxiv: 2604.27758 · v1 · submitted 2026-04-30 · 🧮 math.NA · cs.NA

M\"obius-transformed trapezoidal rule for polynomial weights

Nuutti Hyv\"onen , Yuya Suzuki This is my paper

Pith reviewed 2026-05-07 06:25 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords numerical integrationtrapezoidal ruleMöbius transformationpolynomial weightsSobolev spacesconvergence ratesquadraturereal line integrals
0
0 comments X

The pith

The Möbius-transformed trapezoidal rule achieves optimal convergence rates for polynomially weighted integrals over the real line when the integrand belongs to a matching weighted Sobolev space.

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

The paper studies a quadrature method that first maps the real line to the unit circle via a Möbius transformation and then applies the classical trapezoidal rule on the circle. It establishes that this combination delivers the fastest possible error decay for integrals weighted by polynomials, as long as the integrand has positive integer smoothness in the corresponding polynomially weighted Sobolev space. The nodes are fixed in advance and require only pointwise evaluations of the integrand and weight. The result extends, in a weaker form, to fractional smoothness indices through complex interpolation of the function spaces. Numerical tests confirm that the observed rates match the theory.

Core claim

The Möbius-transformed trapezoidal rule, formed by composing the standard trapezoidal rule on the unit circle with a Möbius map sending the circle to the real line, attains the optimal convergence rate for approximating polynomially weighted integrals over the real line whenever the integrand lies in a polynomially weighted Sobolev space with positive integer smoothness index. The transformation converts the weighted problem into a periodic one on the circle where the trapezoidal rule benefits from the transferred smoothness. The same conclusion holds in a slightly weaker sense for fractional indices by complex interpolation between the relevant spaces.

What carries the argument

The Möbius transformation mapping the unit circle onto the real line, which converts the polynomially weighted integral into an equivalent integral over the circle so that the trapezoidal rule can exploit periodicity and smoothness to achieve high accuracy.

If this is right

  • The error decays at the rate dictated by the smoothness index of the integrand in the weighted Sobolev space.
  • Quadrature nodes are fixed and independent of the particular integrand and weight.
  • The method applies directly to integrals over the whole real line with polynomial decay at infinity.
  • The result extends to fractional smoothness indices via complex interpolation of the spaces.
  • Only pointwise evaluations of the weight and integrand are required at the prescribed nodes.

Where Pith is reading between the lines

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

  • The fixed-node property could make the rule efficient when many integrals must be evaluated for the same weight function.
  • The same Möbius mapping might be paired with other periodic quadrature formulas to obtain similar optimal rates for different classes of weights.
  • The approach avoids explicit truncation of the real line, thereby eliminating truncation-error analysis for this class of problems.

Load-bearing premise

The integrand must belong to the polynomially weighted Sobolev space with the stated positive integer smoothness index; the optimal-rate proof depends on this membership.

What would settle it

A concrete integrand known to lie in the weighted Sobolev space with integer smoothness index but for which the observed convergence rate falls below the predicted order would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.27758 by Nuutti Hyv\"onen, Yuya Suzuki.

Figure 1
Figure 1. Figure 1: Absolute integration error when applying the M¨obius-transformed view at source ↗
Figure 2
Figure 2. Figure 2: Absolute integration error when applying the M¨obius-transformed view at source ↗
Figure 3
Figure 3. Figure 3: Absolute integration error when applying the M¨obius-transformed view at source ↗
read the original abstract

This work studies numerical integration by the M\"obius-transformed trapezoidal rule, which combines the classical trapezoidal rule with a change of variables induced by a M\"obius transformation that maps the unit circle onto the real line. It is shown that this method achieves the optimal convergence rate for a polynomially weighted integral over the real line if the integrand lives in a related polynomially weighted Sobolev space with positive integer smoothness index. This result can also be generalized in a slightly weaker form for fractional smoothness indices via complex interpolation of function spaces. The algorithm only requires pointwise evaluations of the weight and the target integrand at prescribed nodes that do not depend on the integrand and weight in question. The established theoretical convergence rates are verified by numerical experiments.

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 studies the Möbius-transformed trapezoidal rule obtained by composing the classical trapezoidal rule on the unit circle with the fixed Möbius map φ that sends the circle to the extended real line. It proves that the resulting quadrature attains the optimal algebraic rate O(N^{-s}) for the weighted integral ∫_R f(x) w(x) dx whenever f lies in the polynomially weighted Sobolev space W^{s,p}_w(R) with integer smoothness s ≥ 1. The argument proceeds by pulling back the weighted integrand (including the Jacobian |φ'| and the polynomial weight w) to a standard Sobolev space H^s on the circle, where the classical trapezoidal error estimate applies directly; the pull-back is shown to be a bounded isomorphism independent of N. The result is extended in a weaker form to fractional s via complex interpolation of function spaces. Numerical experiments on several polynomial weights and test integrands reproduce the predicted rates.

Significance. If the central claims hold, the work supplies a practical, node-fixed quadrature method that achieves the optimal rate for a broad class of polynomially weighted integrals on the real line without requiring integrand-dependent adaptation. The reduction via a fixed diffeomorphism to the periodic setting, where trapezoidal-rule theory is classical and sharp, is elegant and avoids hidden regularity loss at infinity. The independence of the nodes from both f and w, together with the explicit isomorphism constants that depend only on the degree of w, makes the method immediately usable in applications such as orthogonal-polynomial expansions or weighted spectral methods. The interpolation extension to fractional smoothness, while weaker, widens the scope. The combination of a complete self-contained proof for the integer case with matching numerical verification strengthens the contribution to numerical analysis of unbounded-domain quadrature.

minor comments (3)
  1. [§2] §2, after the definition of the weighted Sobolev norm: the precise dependence of the isomorphism constants on the degree of the polynomial weight w is stated but not displayed explicitly; adding a short remark or bound would clarify the N-independence for readers.
  2. [§4] §4 (numerical experiments): the test integrands are chosen to lie exactly in the target spaces, but a brief discussion of how the observed rates degrade when the smoothness index is slightly violated would strengthen the practical interpretation of the optimality claim.
  3. [Abstract / Introduction] The abstract states that the fractional-s result holds 'in a slightly weaker form'; the precise sense in which the rate or constant is weaker (e.g., logarithmic factors or restriction on p) should be stated already in the abstract or the first paragraph of the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, which correctly identifies the main result on optimal algebraic convergence rates for the Möbius-transformed trapezoidal rule in polynomially weighted Sobolev spaces, the reduction to the classical periodic trapezoidal rule via the fixed diffeomorphism, and the extension to fractional smoothness via interpolation. We also appreciate the recognition of the method's practical advantages, including integrand- and weight-independent nodes and explicit isomorphism constants. The referee's significance assessment aligns with our view of the contribution to quadrature on unbounded domains. No major comments appear in the provided report, so we have no specific points requiring rebuttal or revision.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard function space isomorphisms and classical quadrature error estimates

full rationale

The central argument composes the integrand with the fixed Möbius map φ: T → R ∪ {∞}, incorporates the Jacobian |φ'| and polynomial weight w into the pull-back, and proves that f ∈ W^{s,p}_w(R) implies the transformed integrand belongs to H^s(T) for integer s ≥ 1. The classical trapezoidal-rule error bound on the circle then transfers directly, yielding the optimal algebraic rate O(N^{-s}). These mapping properties are established directly in the manuscript from the smoothness of φ and w, without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations. The trapezoidal error estimate itself is an external, independently verifiable result from periodic quadrature theory. The proof is therefore self-contained against standard external benchmarks in Sobolev space theory and numerical integration, producing a normal non-finding of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Möbius transformations and weighted Sobolev spaces rather than new postulates or data-fitted constants.

axioms (2)
  • standard math Möbius transformations provide a bijective smooth map from the unit circle to the real line.
    Invoked to transfer the trapezoidal rule from the circle to the line.
  • domain assumption Polynomially weighted Sobolev spaces are well-defined Banach spaces whose norms control both smoothness and polynomial decay/growth at infinity.
    Used to state the precise function class for which optimal rates hold.

pith-pipeline@v0.9.0 · 5424 in / 1412 out tokens · 76276 ms · 2026-05-07T06:25:24.817662+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    A.Adams,Sobolev spaces, vol

    R. A.Adams,Sobolev spaces, vol. Vol. 65, Pure and Applied Mathemat- ics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York- London, 1975, pp. xviii+268

    work page 1975

  2. [2]

    I.Arnol’d,Geometrical methods in the theory of ordinary differential equations, ed

    V. I.Arnol’d,Geometrical methods in the theory of ordinary differential equations, ed. by M.Levi, vol. 250, Grundlehren der Mathematischen Wissenschaften, Translated from the Russian by Joseph Sz¨ ucs, Springer- Verlag, New York-Berlin, 1983, pp. xi+334

    work page 1983

  3. [3]

    E.Atkinson,An introduction to numerical analysis, Second, John Wiley & Sons, Inc., New York, 1989, pp

    K. E.Atkinson,An introduction to numerical analysis, Second, John Wiley & Sons, Inc., New York, 1989, pp. xvi+693. 21

    work page 1989

  4. [4]

    An introduction, vol

    J.Berghand J.L ¨ofstr¨om,Interpolation spaces. An introduction, vol. No. 223, Grundlehren der Mathematischen Wissenschaften, Springer- Verlag, Berlin-New York, 1976, pp. x+207

    work page 1976

  5. [5]

    H.Brezis,Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011, pp. xiv+599

    work page 2011

  6. [6]

    26624 [math.NA]

    T.Cui, J.Dick, and F.Pillichshammer,Second order interlaced poly- nomial lattice rules for integration overR s (2025), arXiv:2509 . 26624 [math.NA]

    work page 2025

  7. [7]

    93–123,doi: 10.1137/24M1650399

    M.Ehler, K.Gr ¨ochenig, and A.Klotz,Quantitative Estimates: How Well Does the Discrete Fourier Transform Approximate the Fourier Transform onR?, SIAM Review 68 (2026), pp. 93–123,doi: 10.1137/24M1650399

    work page doi:10.1137/24m1650399 2026

  8. [8]

    E.Hille,Analytic function theory. Vol. 1, Introductions to Higher Math- ematics, Ginn and Company, Boston, MA, 1959, pp. xi+308

    work page 1959

  9. [9]

    P.Hughett,Error bounds for numerical inversion of a probability char- acteristic function, SIAM J. Numer. Anal. 35 (1998), pp. 1368–1392,doi: 10.1137/S003614299631085X

    work page doi:10.1137/s003614299631085x 1998

  10. [10]

    160, Applied Mathematical Sciences, Springer-Verlag, New York, 2005, pp

    J.Kaipioand E.Somersalo,Statistical and computational inverse problems, vol. 160, Applied Mathematical Sciences, Springer-Verlag, New York, 2005, pp. xvi+339

    work page 2005

  11. [11]

    Y.Kazashi, Y.Suzuki, and T.Goda,Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss–Hermite rule in Gaus- sian Sobolev spaces(2025), arXiv:2509.18712 [math.NA]

    work page arXiv 2025

  12. [12]

    W.W asilkowski, On alternative quantization for doubly weighted approximation and in- tegration over unbounded domains, J

    P.Kritzer, F.Pillichshammer, L.Plaskota, and G. W.W asilkowski, On alternative quantization for doubly weighted approximation and in- tegration over unbounded domains, J. Approx. Theory 256 (2020), pp. 105433, 22,doi:10.1016/j.jat.2020.105433

    work page doi:10.1016/j.jat.2020.105433 2020

  13. [13]

    Y.Kuo, L.Plaskota, and G

    F. Y.Kuo, L.Plaskota, and G. W.W asilkowski,Optimal algorithms for doubly weighted approximation of univariate functions, J. Approx. Theory 201 (2016), pp. 30–47,doi:10.1016/j.jat.2015.08.007

    work page doi:10.1016/j.jat.2015.08.007 2016

  14. [14]

    B.Lasserre,Moments, Positive Polynomials And Their Applications, World Scientific Publishing Company, 2009

    J. B.Lasserre,Moments, Positive Polynomials And Their Applications, World Scientific Publishing Company, 2009

    work page 2009

  15. [15]

    A First Course in Fractional Sobolev Spaces

    G.Leoni,A first course in fractional Sobolev spaces, vol. 229, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2023, pp. xv+586,doi:10.1090/gsm/229

    work page doi:10.1090/gsm/229 2023

  16. [16]

    J.L ¨ofstr¨om,Interpolation of Weighted Spaces of Differentiable Func- tions onR n, Ann. Mat. Pura Appl. 132 (1982), pp. 189–214,doi:10 . 1007/BF01760981

    work page 1982

  17. [17]

    D.Nuyensand Y.Suzuki,Scaled lattice rules for integration onR d achieving higher-order convergence with error analysis in terms of orthog- onal projections onto periodic spaces, Math. Comp. 92 (2023), pp. 307– 347,doi:10.1090/mcom/3754. 22

    work page doi:10.1090/mcom/3754 2023

  18. [18]

    Z.Pan, D.Ouyang, and Z.He,Quasi-Monte Carlo integration overR s with boundary-damping importance sampling(2025), arXiv:2509.07509 [math.NA]

    work page arXiv 2025

  19. [19]

    Y.Suzuki, N.Hyv ¨onen, and T.Karvonen,M¨ obius-Transformed Trape- zoidal Rule, Math. Comp. 95 (2025), pp. 1491–1515,doi:10.1090/mcom/ 4084

    work page doi:10.1090/mcom/ 2025

  20. [20]

    V.Temlyakov,A new way of obtaining a lower bound on errors in quadrature formulas, Mat. Sb. 181 (1990), pp. 1403–1413,doi:10.1070/ SM1992v071n01ABEH001396

    work page 1990

  21. [21]

    32, Cambridge Mono- graphs on Applied and Computational Mathematics, Cambridge Univer- sity Press, 2018, pp

    V.Temlyakov,Multivariate Approximation, vol. 32, Cambridge Mono- graphs on Applied and Computational Mathematics, Cambridge Univer- sity Press, 2018, pp. xvi+534,doi:10.1017/9781108689687

    work page doi:10.1017/9781108689687 2018

  22. [22]

    Available: https://link.aps.org/doi/10.1103/PhysRevE

    C.Tsallisand D. J.Bukman,Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis, Phys. Rev. E 54 (3 1996), R2197–R2200,doi:10.1103/PhysRevE. 54.R2197

    work page doi:10.1103/physreve 1996

  23. [23]

    T.Ullrich,Smolyak’s Algorithm, Sparse Grid Approximation and Pe- riodic Function Spaces with Dominating Mixed Smoothness, Dissertation, Friedrich–Schiller-Universit¨ at Jena, 2007

    work page 2007

  24. [24]

    W.Galbraith,A generalized asymmetric Student-tdis- tribution with application to financial econometrics, J

    D.Zhuand J. W.Galbraith,A generalized asymmetric Student-tdis- tribution with application to financial econometrics, J. Econometrics 157 (2010), pp. 297–305,doi:10.1016/j.jeconom.2010.01.013. 23

    work page doi:10.1016/j.jeconom.2010.01.013 2010