Analytically Consistent Reconstruction of Finite Data Using Pad\'e Sequences

Balma Duch; Emerson D\'iaz; Pere Masjuan

REVIEW 5 major objections 5 minor 38 references

Analytically Consistent Reconstruction of Finite Data Using Pad\'e Sequences

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Recurring spurious poles flag bad data points

desk verdict Genuinely new iterative doublet-dynamics algorithm, but the validation does not yet isolate the doublet mechanism from the Stieltjes refit—send to peer review with major revision required. read the letter →

arxiv 2608.11262 v1 pith:Z7HHQQ42 submitted 2026-08-09 physics.data-an hep-exhep-lathep-ph

Emerson Díaz , Balma Duch , Pere Masjuan This is my paper · ORCID

classification physics.data-anhep-exhep-lathep-ph

keywords AnalyticityreconstructionexperimentaldatalatticePadésequencesFroissartdoubletsStieltjesfunctionsholomorphic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the spurious pole–zero pairs known as Froissart doublets, usually discarded as artifacts of Padé approximation, can instead be read as diagnostics of where a finite dataset is analytically inconsistent. The authors build an iterative algorithm that tracks these doublets along a sequence of Padé approximants: doublets that reappear at nearly the same location mark data points to correct, while doublets that wander are treated as harmless finite-information noise. In controlled tests on Stieltjes functions, the correction reduces mean absolute error by 56% to 99.9%, depending on noise level and sampling density. In pseudo-experimental examples, a smooth Gaussian distortion is suppressed while a Breit-Wigner resonance is preserved, and the method extends to general holomorphic functions. If the recurrence heuristic holds, this gives a model-independent way to clean experimental or lattice data before extracting its analytic structure.

What carries the argument

The central object is the three-family classification of poles and zeros observed across successive diagonal Padé approximants $P_N^N$: unstable Froissart doublets (spurious pole–zero pairs) as finite-information effects, recurrent Froissart doublets as localized data inconsistencies, and stable poles compatible with the underlying analytic function. The algorithm carries the argument by splitting each approximant as $P_N^N(x)=P_M^M(x)+P_{N-M-1}^{N-M}(x)$, assigning recurring poles to the nearest sampling node, voting on nodes across the sequence, and iteratively correcting the highest-vote nodes toward the selected Stieltjes component subject to positivity and convexity constraints.

What would settle it

Take a known Stieltjes function and add a single controlled systematic bump with amplitude below the statistical noise floor. If the resulting Froissart doublet wanders across orders and the algorithm leaves the bump in place, the recurrence criterion fails in the weak-signal regime. Conversely, place a genuine complex-conjugate pole pair with very small residue so its doublet appears nearly stationary; if the algorithm corrects it away, the criterion mistakes a true singularity for an inconsistency.

Watch

Extended reading notes

Core claim

The central claim is that the dynamics of Froissart doublets along a Padé sequence separate three families: unstable doublets caused by finite information, recurrent doublets caused by localized inconsistencies in the input data, and stable poles compatible with the underlying analytic structure. From this classification, the paper derives a reconstruction algorithm that splits each Padé approximant into a Stieltjes (or holomorphic) component and a noise component, uses a voting system to rank grid nodes by how often they host a recurring pole, and iteratively updates the top-ranked nodes toward the analytic component while preserving Stieltjes positivity and convexity. The paper reports that this procedure removes localized systematic distortions such as a Gaussian bump, leaves genuine complex-conjugate poles such as a Breit-Wigner resonance in place, and keeps statistical fluctuations consistent with the underlying analytic behavior.

Load-bearing premise

The load-bearing premise is that a spurious pole–zero pair (a Froissart doublet) which reappears at nearly the same spot across successive Padé orders signals a genuine localized inconsistency in the data, while a doublet that moves is only finite-information noise; this distinction is asserted rather than proved and is validated on a handful of constructed examples.

Editorial extensions

If this is right

A dataset with localized systematic distortions can be corrected without modeling the source of the distortion, because the criterion is analytic rather than statistical.
Genuine physical structures with complex-conjugate poles, such as a Breit-Wigner resonance, survive the procedure, while smooth non-analytic deformations such as Gaussian bumps are suppressed.
The method applies beyond Stieltjes functions: for holomorphic functions, real poles inside the sampling region are treated as noise, while complex-conjugate or Szegő-curve structures are assigned to the analytic component.
The reconstructed dataset remains on the original sampling grid and keeps its statistical uncertainties, so it can be used as input to later fits or analytic continuations.
Finite sampling density and numerical precision shift the critical Padé order at which Froissart doublets appear, so the algorithm's behavior depends on the data's precision and density.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: the recurrence criterion could be used before any correction as a data-quality audit, flagging grid nodes that deserve re-measurement rather than numerical adjustment.
Inference: a natural stress test is correlated or clustered noise, since the synthetic tests use isolated random nodes; the voting tolerance would need to distinguish a genuine localized systematic from a patch of correlated fluctuations.
Inference: the stable Stieltjes order reached after convergence is an empirical estimate of how much analytic information the dataset really supports, which could inform where to stop a Padé sequence in future fits.
Inference: the same dynamics could check whether two datasets of the same observable are mutually compatible, flagging regions where they cannot share a common analytic sequence.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the iterative use of Froissart-doublet recurrence to locate bad data points is a genuinely new idea, and the paper is worth engaging seriously. But the validation as written does not yet show that the doublet dynamics, rather than the generic Stieltjes refit, is what does the work.

What is new: prior work treated Froissart doublets as artifacts to filter or as statistical indicators; this paper instead tracks their positions along Padé sequences, uses recurrence across orders to flag inconsistent nodes, and iteratively corrects those nodes while preserving Stieltjes (or holomorphic) structure. That is a real departure, and the algorithm is concrete enough to implement. The tests on log(1+x)/x with multiplicative noise show large MAE reductions (62–99.9%), and the contrast between suppressing a Gaussian bump and preserving a Breit-Wigner resonance is a sensible sanity check. The pseudo-experimental setup with Poisson statistics is a good effort to mimic realistic data.

Where the soft spots are. The central premise—recurrent doublets signal localized inconsistencies, unstable ones are finite-information effects—is asserted in Sec. 2 as a dynamical classification ('not causal but dynamical') and is never proven or tied to a theorem. The quantitative definition of 'recurrent' is just voting with adjustable thresholds (currentTolerance 0.45 Δx, vote threshold >1, diagOrders 5–12), so the algorithm has several free knobs. That alone would be okay if the validation isolated the mechanism; it does not. Table 2 and Fig. 1 are single 'representative' realizations with no error bars. There is no comparison against a baseline where nodes are chosen randomly or by plain residual size. For a smooth, convex Stieltjes function, replacing selected data points with the best Stieltjes fit will reduce MAE regardless of how those points are selected, so the reported improvements do not establish that doublet dynamics is the causal ingredient. The Breit-Wigner preservation is also anecdotal: no statistic separates stable complex poles of a resonance from pole stability that a localized systematic bump might produce. Finally, the abstract promises an open GitLab repository, but the text never gives the link or repository identifier; the appendix references 'the associated GitLab repository' without a URL. For a methods paper, that is a serious reproducibility gap.

Who benefits: people in hadron spectroscopy, lattice QCD, and scattering analyses who already use Padé approximants for analytic continuation and want a practical tool to separate systematic distortions from physics. The idea has enough potential that I would send it to a good referee with the expectation of major revision: add ensemble statistics, baselines (random-node and residual-based node selection), a sensitivity scan over thresholds, and the actual repository link. The central heuristic also needs a crisper statement of what recurrence means quantitatively, or at least empirical support from a designed experiment that varies the node-selection rule. Not a desk reject; not yet a definitive method.

Referee Report

5 major / 5 minor

Summary. The paper proposes a Padé-based iterative algorithm for reconstructing the analytic structure of a function from finite, noisy datasets. The central idea is that Froissart doublets appearing in diagonal Padé sequences should be treated as diagnostic objects: unstable doublets are associated with finite-information effects, while recurrent doublets indicate localized inconsistencies in the input data. The algorithm votes on pole recurrences to identify anomalous nodes, corrects those nodes using a selected Stieltjes component, and iterates until the pole structure stabilizes. The method is validated on Stieltjes functions, on pseudo-experimental data with Gaussian and Breit-Wigner distortions, and on a few holomorphic functions, with the full implementation released as a Mathematica notebook in a GitLab repository.

Significance. If the central premise is correct, the paper offers a practical, largely model-independent tool for separating genuine analytic features from finite-information and systematic effects, which would be useful across experimental and lattice-data analyses. The strengths of the manuscript are its concrete algorithmic implementation, open code, and a broad set of test cases including realistic pseudo-experimental ensembles. The reported improvements are large (MAE reductions up to about 99.9% in controlled examples), and the Breit-Wigner/Gaussian contrast is a sensible test of the method's diagnostic power. However, the load-bearing classification of Froissart doublets is heuristic, and the validation does not currently isolate the contribution of that classification from the effect of the Stieltjes refit.

major comments (5)

[Sec. 2 and App. A.2] The central premise of the paper is that recurrent Froissart doublets across Padé orders correspond to localized data inconsistencies, whereas unstable doublets are finite-information effects. This premise is asserted rather than demonstrated: the manuscript states that the distinction is 'not causal but dynamical,' and its operational implementation relies on adjustable thresholds (currentTolerance 0.45 Δx, vote threshold >1, diagOrders 5–12 in Table 6). Since every correction decision in the algorithm is conditional on this split, the paper needs either a proof, a rigorously quantified definition of recurrence, or a direct validation showing that the recurrence criterion correlates with injected inconsistencies rather than with other features of the data.
[Sec. 3.1, Table 2] The controlled validation consists of single 'representative' realizations. No ensemble averages, standard deviations, or seed dependence are reported, and the text itself concedes that other configurations with the same density and noise count 'may yield even better improvement.' Consequently the reported improvements, ranging from 62.31% to 99.90%, cannot be distinguished from favorable sampling. The authors should repeat the noise-generation procedure many times and report the distribution of MAE improvements, not just one example per configuration.
[Sec. 3, Eq. (2)] The reported improvements do not isolate the diagnostic role of Froissart-doublet dynamics. The correction step replaces flagged nodes with the selected Stieltjes component under positivity and convexity constraints. For a positive, decreasing, convex Stieltjes function such as log(1+x)/x, such a replacement can reduce the MAE even if the flagged nodes are chosen by an unrelated rule. A necessary control is a baseline that corrects the same number of randomly selected nodes, or nodes chosen by ordinary residual magnitude, using the same Stieltjes refit. Without such an ablation, the improvements in Table 2 could be attributed to the Stieltjes refit rather than to the recurrence classification.
[Sec. 4.2] The preservation of the Breit-Wigner resonance is attributed to the stability of complex poles across Padé orders, but no statistic is provided that quantifies this stability or separates it from the stability that a localized systematic bump could also produce in Padé approximants. The reported RMSE and pull values show that the resonance is not removed, but they do not demonstrate that the algorithm's pole-stability criterion was the operative cause. A null test with a non-resonant localized feature of comparable width and height, analyzed with the same pole-stability metric, would strengthen the claim.
[Table 3, first row] The reported post-reconstruction MAE for the holomorphic example g(z)=(z+1)^{3/2} with n=5 noisy points is 8.0×10^-6, which is identical to the value reported in Table 2 for the Stieltjes example log(1+z)/z with the same density and noise count, despite the two functions having very different scales over the interval [0,10]. Please verify this number and report the correct value; as printed, it suggests a copy error in a validation table that is central to the holomorphic-extension claim.

minor comments (5)

[Abstract and Sec. 2] There are typographical errors in the text, including 'doubltes' in the abstract and 'without lose of generality' in Section 2.
[Fig. 1 captions and Eq. (4)] The figure captions describe the perturbation as 'additive noise,' while Eq. (4) defines multiplicative noise f(z_i)(1+ε_i). Please make the terminology consistent.
[Sec. 5.2] The function tan^2(x) is listed as a holomorphic test function, but it has real poles at π/2 + kπ inside the interval [0,10]. Either choose a genuinely holomorphic example or state explicitly that the sampling domain excludes those poles.
[Sec. 3, MAE definition] The MAE is defined with respect to the exact analytic function, which is unknown in realistic applications. Please state explicitly that this metric is used only for validation in controlled settings.
[App. A.2, Tables 5-6] The algorithm is described as requiring no model for the origin of the inconsistencies, yet many parameters, including sigma, prec, currentTolerance, and vote threshold, are dataset-dependent. A brief discussion of sensitivity to these parameters would help users apply the method to new datasets.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the load-bearing Froissart-doublet classification is an input heuristic validated against known generating functions, and the reported MAE/RMSE improvements are measured against truth functions not used by the algorithm.

full rationale

The paper's central reconstruction claim is validated externally: Tables 2 and 3 compare mean absolute errors before and after reconstruction against the known analytic generating functions f(z)=log(1+z)/z and g(z)=(z+1)^{3/2}, and Section 4 compares RMSE against the known background F(x)=log(1+x)/x. These truth functions are not used by the algorithm to set its parameters; the algorithm instead fits Padé sequences to the noisy data and corrects points flagged by pole-recurrence voting. The classification of Froissart doublets into unstable, recurrent, and genuine families is explicitly presented as a dynamical observation rather than a derived theorem: 'The distinction between the first two classes is not causal but dynamical and, thus, observed from the PA sequence.' This is an input heuristic, not a conclusion derived from itself, and its usefulness is tested by the controlled experiments. The Stieltjes-component selection and positivity/convexity constraints are applications of standard Stieltjes-function properties, not results equivalent to the algorithm's output. Self-citations (e.g., Refs. [13-18]) provide background Padé convergence and application context and are not used as load-bearing justifications for the correction scheme or to forbid alternative reconstructions. No prediction in the paper reduces by construction to a fitted parameter or to a self-citation chain, so no circular step is present.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The 'diagnostic object' status of Froissart doublets is a reinterpretation of existing mathematical objects, and the Stieltjes/noise decomposition is a bookkeeping device, not an entity. The free parameters are algorithm hyperparameters and the data-dependent selection of the Stieltjes order M; they are tuned per dataset and affect the reported improvements.

free parameters (5)

vote threshold = default >1 (Table 6)
Minimum number of times a pole must recur at a node before that point is flagged for correction; set by hand and dataset-dependent.
pole acceptance tolerance (currentTolerance) = 0.45 * Delta x (Table 6)
Maximum distance between a PA pole and a grid node for the pole to be retained; hand-set, affects which nodes are corrected.
correction Gaussian width (sigma) = 1e-5 (Table 5)
Width of the stochastic update around the analytic reference during correction; fixed small value.
data precision (prec) = 1e-4 in Sec. 3.1; 1e-6 in one test (Table 5)
Truncation precision of synthetic data and acceptance threshold; controls where Froissart doublets appear.
Stieltjes component order M = adaptive; up to 3 in examples
Selected per iteration as the highest-order component satisfying the Stieltjes criterion, with ties broken by minimum MAE to the data; a data-dependent model-selection step central to the correction.

assumptions (5)

standard math Diagonal Padé approximants to Stieltjes functions converge and satisfy ordering and bounding properties.
Used to justify that PA sequences encode the analytic structure; cited to Ref. [1].
domain assumption The input data lie on the real axis and the underlying function belongs to the Stieltjes class, or in the generalization to the holomorphic class with no poles in the sampling domain.
The entire signal/noise assignment in Eq. (2) depends on this class; the paper states it as the setting in Secs. 3 and 5.
ad hoc to paper Recurrent Froissart doublets across the Padé sequence indicate localized data inconsistencies, while unstable doublets are finite-information effects.
Stated in Sec. 2 as the basis of the method, but not proven; the algorithm corrects data to remove exactly this class of poles.
domain assumption Finite sampling density and finite numerical precision produce the same observable effect (unstable Froissart doublets) and cannot be separately distinguished from the approximants.
Used to justify treating finite-information effects as a common noise contribution; Sec. 2.
ad hoc to paper Statistical fluctuations that are consistent with the assumed analytic structure should be preserved; only localized inconsistencies should be corrected.
Design principle behind iterative correction; it is a normative choice, not derived.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Analytically Consistent Reconstruction of Finite Data Using Pad\'e Sequences." pith.science (2026). https://pith.science/paper/Z7HHQQ42

@misc{pith2026260811262,
  author       = {Pith},
  title        = {Pith review of: Analytically Consistent Reconstruction of Finite Data Using Pad\'e Sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z7HHQQ42}},
  note         = {Machine review of arXiv:2608.11262}
}

read the original abstract

Reconstructing the analytic structure of a function from finite datasets is a fundamental problem across theoretical, numerical, and experimental physics. While Pad\'e approximants provide a natural framework, finite-information effects, as well as statistical and systematic uncertainties, may obscure the underlying analytic structure and limit reconstruction reliability. In this work, we reinterpret the appearance of Froissart doublets not merely as numerical artifacts but as \textit{diagnostic objects} carrying information about the analytic consistency of the input data. Accordingly, we develop a general Pad\'e-based algorithm that exploits the dynamics of Froissart doublets along Pad\'e sequences to identify localized inconsistencies and iteratively reconstruct the analytic structure most compatible with the data. The method requires no model for the origin of the inconsistencies and distinguishes genuine analytic features from spurious structures induced by finite-information effects. We validate it using Stieltjes functions, realistic pseudo-experimental datasets with statistical and systematic uncertainties, and general holomorphic functions. The complete algorithm is provided as a supplementary Mathematica notebook in an open GitLab repository.

Figures

Figures reproduced from arXiv: 2608.11262 by the authors.

**Figure 2.** Representative data subset of the Stieltjes function in Eq. 3, generated under [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Comparison between the original dataset (red triangles) and the reconstructed [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Original data (red triangles) and analytically reconstructed data (blue points) with [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: Application of the analytic reconstruction algorithm to subtract background (log [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: Extraction of the Breit-Wigner resonance after subtraction of the background (log [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: Representative data subsets of the holomorphic function in Eq. 13, generated under [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 20 canonical work pages

[1]

G. A. Baker and P. Graves-Morris,Pad´ e Approximants, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1996, Section 5.4, Theorem 5.4.2

work page 1996
[2]

Froissart,Approximation de Pad´ e

M. Froissart,Approximation de Pad´ e. Application ` a la physique des particules ´ el´ ementaires, Les rencontres physiciens-math´ ematiciens de Strasbourg - RCP25, Vol- ume 9, Talk no. 2, 13 pp., 1969

work page 1969
[3]

Bessis, J

D. Bessis, J. Comput. Appl. Math.66(1996), 85-88

work page 1996
[4]

Gonnet, S

P. Gonnet, S. G¨ uttel and L. N. Trefethen, SIAM Rev.55, no. 1, 101-117 (2013)

work page 2013
[5]

On rational functions without Froissart doublets

B. Beckermann, G. Labahn and A. C. Matos, Numer. Math.138, 615-633 (2018) [arXiv:1605.00506 [math.NA]]

work page Pith review arXiv 2018
[6]

Gilewicz and M

J. Gilewicz and M. Pindor, J. Comput. Appl. Math.87, 199-214 (1997) [doi:10.1016/S0377-0427(97)00185-4]

work page doi:10.1016/s0377-0427(97)00185-4 1997
[7]

Gilewicz and Y

J. Gilewicz and Y. Kryakin, J. Comput. Appl. Math.153, 235-242 (2003) [doi:10.1016/S0377-0427(02)00674-X]

work page doi:10.1016/s0377-0427(02)00674-x 2003
[8]

Signal induced Symmetry Breaking in Noise Statistical Properties of Data Analysis

L. Perotti, D. Vrinceanu and D. Bessis, [arXiv:1009.6021 [physics.data-an]]

work page Pith review arXiv

Show all 38 references

[9]

Boito, A

D. Boito, A. Cucchieri, C. Y. London and T. Mendes, JHEP02(2023), 144 doi:10.1007/JHEP02(2023)144 [arXiv:2210.10490 [hep-lat]]

2023 arXiv
[10]

Yamada and I

H. Yamada and I. Kensuke, International Journal of Computational Mathematics 2014 (2014)

2014
[11]

A. F. Falc˜ ao and O. Oliveira, Phys. Rev. D106(2022) no.11, 114022 doi:10.1103/PhysRevD.106.114022 [arXiv:2209.14815 [hep-lat]]

2022 arXiv
[12]

Binosi and R

D. Binosi and R. A. Tripolt, Phys. Lett. B801(2020), 135171 doi:10.1016/j.physletb.2019.135171 [arXiv:1904.08172 [hep-ph]]

2020
[13]

Masjuan, Ph.D

P. Masjuan, Ph.D. thesis, Universitat Aut` onoma de Barcelona, 2010 [arXiv:1005.5683 [hep-ph]]

2010 arXiv
[14]

Peris, Phys

S. Peris, Phys. Rev. D74(2006), 054013 doi:10.1103/PhysRevD.74.054013 [arXiv:hep- ph/0603190 [hep-ph]]

2006
[15]

Boito, P

D. Boito, P. Masjuan and F. Oliani, JHEP08(2018), 075 doi:10.1007/JHEP08(2018)075 [arXiv:1807.01567 [hep-ph]]. 23

2018 arXiv
[16]

Boito, C

D. Boito, C. Y. London and P. Masjuan, JHEP01(2022), 054 doi:10.1007/JHEP01(2022)054 [arXiv:2110.09909 [hep-ph]]

2022 arXiv
[17]

Masjuan and S

P. Masjuan and S. Peris, Phys. Lett. B686(2010), 307-312 doi:10.1016/j.physletb.2010.02.069 [arXiv:0903.0294 [hep-ph]]

2010 arXiv
[18]

Boito, C

D. Boito, C. Y. London, P. Masjuan and C. Rojas, Phys. Rev. D110(2024) no.7, 074012 doi:10.1103/PhysRevD.110.074012 [arXiv:2405.13638 [hep-ph]]

2024 arXiv
[19]

Masjuan and S

P. Masjuan and S. Peris, JHEP05(2007), 040 doi:10.1088/1126-6708/2007/05/040 [arXiv:0704.1247 [hep-ph]]

2007 arXiv
[20]

Masjuan and S

P. Masjuan and S. Peris, Phys. Lett. B663(2008), 61-65 doi:10.1016/j.physletb.2008.03.040 [arXiv:0801.3558 [hep-ph]]

2008 arXiv
[21]

Masjuan, J

P. Masjuan, J. J. Sanz-Cillero and J. Virto, Phys. Lett. B668(2008), 14-19 doi:10.1016/j.physletb.2008.07.106 [arXiv:0805.3291 [hep-ph]]

2008 arXiv
[22]

Masjuan, S

P. Masjuan, S. Peris and J. J. Sanz-Cillero, Phys. Rev. D78(2008), 074028 doi:10.1103/PhysRevD.78.074028 [arXiv:0807.4893 [hep-ph]]

2008 arXiv
[23]

Masjuan, Phys

P. Masjuan, Phys. Rev. D86(2012), 094021 doi:10.1103/PhysRevD.86.094021 [arXiv:1206.2549 [hep-ph]]

2012 arXiv
[24]

Escribano, P

R. Escribano, P. Masjuan and P. Sanchez-Puertas, Phys. Rev. D89(2014) no.3, 034014 doi:10.1103/PhysRevD.89.034014 [arXiv:1307.2061 [hep-ph]]

2014 arXiv
[25]

Masjuan and P

P. Masjuan and P. Sanchez-Puertas, [arXiv:1504.07001 [hep-ph]]

arXiv
[26]

Escribano, P

R. Escribano, P. Masjuan and P. Sanchez-Puertas, Eur. Phys. J. C75(2015) no.9, 414 doi:10.1140/epjc/s10052-015-3642-z [arXiv:1504.07742 [hep-ph]]

2015 arXiv
[27]

Escribano, S

R. Escribano, S. Gonz` alez-Sol ´ ıs, P. Masjuan and P. Sanchez-Puertas, Phys. Rev. D94 (2016) no.5, 054033 doi:10.1103/PhysRevD.94.054033 [arXiv:1512.07520 [hep-ph]]

2016 arXiv
[28]

Masjuan and P

P. Masjuan and P. Sanchez-Puertas, JHEP08(2016), 108 doi:10.1007/JHEP08(2016)108 [arXiv:1512.09292 [hep-ph]]

2016 arXiv
[29]

Masjuan and P

P. Masjuan and P. Sanchez-Puertas, Phys. Rev. D95(2017) no.5, 054026 doi:10.1103/PhysRevD.95.054026 [arXiv:1701.05829 [hep-ph]]

2017 arXiv
[30]

Gonz` alez-Sol ´ ıs and P

S. Gonz` alez-Sol ´ ıs and P. Masjuan, Phys. Rev. D98(2018) no.3, 034027 doi:10.1103/PhysRevD.98.034027 [arXiv:1805.11262 [hep-ph]]

2018 arXiv
[31]

Gonz` alez-Sol ´ ıs, P

S. Gonz` alez-Sol ´ ıs, P. Masjuan and C. Rojas, Phys. Rev. D104(2021) no.11, 114041 doi:10.1103/PhysRevD.104.114041 [arXiv:2110.06153 [hep-ph]]

2021 arXiv
[32]

Vaziri, M

H. Vaziri, M. R. Shojaei and P. Masjuan, Eur. Phys. J. ST235(2026) no.9, 2093-2109 doi:10.1140/epjs/s11734-026-02193-2 [arXiv:2512.23107 [hep-ph]]

2026 arXiv
[33]

Costin, G

O. Costin, G. V. Dunne and M. Meynig, J. Phys. A55(2022) no.46, 464007 [arXiv:2208.02410 [math-ph]]

2022 arXiv
[34]

Gilewicz and M

J. Gilewicz and M. Pindor, J. Comput. Appl. Math.87(1997), 199-214

1997
[35]

Abbott, S

R. Abbott, S. Fields, W. I. Jay, P. Oare and M. Saccardi, [arXiv:2605.20509 [hep-lat]]

arXiv
[36]

Gammel, in P.R

J.L. Gammel, in P.R. Graves-Morris, Ed., Pad´ e Approximants (Inst. of Physics, London,
[37]

Gammel and J

J.L. Gammel and J. Nuttal, J. Math. Anal. Appl. 43 (1973) 694-696

1973
[38]

Gilewicz and B

J. Gilewicz and B. Truong-Van, in B. Sendov, Ed., Constructive Theory of Functions ’87, Proc. Internat. Conf. on Constructive Theory of Functions, Varna, 1987 (Publishing House of the Bulgarian Academy of Sciences, Sofia, 1988) 145-151. 25

1987

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

G. A. Baker and P. Graves-Morris,Pad´ e Approximants, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1996, Section 5.4, Theorem 5.4.2

work page 1996

[2] [2]

Froissart,Approximation de Pad´ e

M. Froissart,Approximation de Pad´ e. Application ` a la physique des particules ´ el´ ementaires, Les rencontres physiciens-math´ ematiciens de Strasbourg - RCP25, Vol- ume 9, Talk no. 2, 13 pp., 1969

work page 1969

[3] [3]

Bessis, J

D. Bessis, J. Comput. Appl. Math.66(1996), 85-88

work page 1996

[4] [4]

Gonnet, S

P. Gonnet, S. G¨ uttel and L. N. Trefethen, SIAM Rev.55, no. 1, 101-117 (2013)

work page 2013

[5] [5]

On rational functions without Froissart doublets

B. Beckermann, G. Labahn and A. C. Matos, Numer. Math.138, 615-633 (2018) [arXiv:1605.00506 [math.NA]]

work page Pith review arXiv 2018

[6] [6]

Gilewicz and M

J. Gilewicz and M. Pindor, J. Comput. Appl. Math.87, 199-214 (1997) [doi:10.1016/S0377-0427(97)00185-4]

work page doi:10.1016/s0377-0427(97)00185-4 1997

[7] [7]

Gilewicz and Y

J. Gilewicz and Y. Kryakin, J. Comput. Appl. Math.153, 235-242 (2003) [doi:10.1016/S0377-0427(02)00674-X]

work page doi:10.1016/s0377-0427(02)00674-x 2003

[8] [8]

Signal induced Symmetry Breaking in Noise Statistical Properties of Data Analysis

L. Perotti, D. Vrinceanu and D. Bessis, [arXiv:1009.6021 [physics.data-an]]

work page Pith review arXiv