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arxiv: 1906.09078 · v1 · pith:JL7UW3G6new · submitted 2019-06-21 · 🧮 math.CV

On overconvergent subsequencs of closed to rows classical Pade' approximants

Ralitza K. Kovacheva This is my paper

Pith reviewed 2026-05-25 18:36 UTC · model grok-4.3

classification 🧮 math.CV
keywords overconvergencePadé approximantspower seriesradius of convergenceTaylor polynomialscomplex analysis
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The pith

Padé approximant sequences with slowly growing degrees m(n) exhibit overconvergence.

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

The paper extends classical results by Hadamard and Ostrowski on overconvergent Taylor polynomials, along with work by López Lagomasino and Fernández Infante on fixed rows of the Padé table. It establishes overconvergence for sequences of classical Padé approximants π_{n,m_n} tied to a power series with positive radius of convergence when the degree satisfies m(n) ≤ m(n+1) ≤ m(n) and grows as o(n/log n) or O(n). A sympathetic reader would care because the result unifies diagonal-type and row-type approximations under a single slow-growth condition on the degrees. The extension applies directly to the phenomenon of convergence outside the disk of convergence for these varying-row sequences.

Core claim

For a power series f with positive radius of convergence, the classical Padé approximants π_{n,m_n} with nondecreasing m(n) satisfying either m(n) = o(n/log n) or m(n) = O(n) as n → ∞ form overconvergent sequences, extending the Hadamard-Ostrowski theorems on Taylor polynomials and the fixed-row results of López Lagomasino and Fernández Infante.

What carries the argument

The sequences of classical Padé approximants π_{n,m_n} indexed by n with slowly growing degrees m(n) that permit analytic continuation beyond the natural radius.

If this is right

  • Overconvergence holds for Padé sequences whose rows increase slowly, just as it does for the Taylor polynomials studied by Hadamard and Ostrowski.
  • Fixed-row overconvergence results extend immediately to the case of varying rows under the stated growth bounds on m(n).
  • The same overconvergence conclusion applies whether m(n) grows very slowly or reaches linear growth in n.
  • The radius of convergence of the original series remains the only obstruction to overconvergence for these degree sequences.

Where Pith is reading between the lines

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

  • The growth threshold o(n/log n) may mark a natural boundary between overconvergent and non-overconvergent regimes for general Padé sequences.
  • Explicit constructions for functions with known Padé tables, such as rational or exponential functions, could test the sharpness of the O(n) bound.
  • The result suggests that choosing m(n) adaptively within the allowed growth range might enlarge the domain of guaranteed convergence for approximation algorithms.

Load-bearing premise

The power series has positive radius of convergence and the degree sequence m(n) is nondecreasing with growth no faster than O(n).

What would settle it

A concrete power series with positive but finite radius of convergence together with an explicit sequence m(n) = o(n/log n) for which the corresponding Padé approximants fail to converge at any point outside the disk of convergence.

read the original abstract

Let $f$ be a power series with positive radius of convergence. In the present paper, we study the phenomenon of overconvergence of sequences of classical Pade' approximants pi{n,m_n} associated with f, where m(n)<=m(n+1)<=m(n) and m(n) = o(n/\log n), resp. m(n) = 0(n) as n is going to infiity. We extend classical results by J. Hadamard and A. A. Ostrowski related to overconvergent Taylor polynomials, as well as results by G. Lo'pez Lagomasino and A. Ferna'ndes Infante concerning overconvergent subsequences of a fixed row of the Pade' table.

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 / 2 minor

Summary. The paper extends classical overconvergence results of Hadamard–Ostrowski for Taylor polynomials and of López Lagomasino–Fernández Infante for fixed rows of the Padé table to sequences of classical Padé approximants π_{n,m_n} associated to a power series f with positive radius of convergence, under the assumptions that m(n) is nondecreasing and satisfies either m(n)=o(n/log n) or m(n)=O(n) as n→∞.

Significance. If the claimed extensions hold, the work would modestly enlarge the known range of overconvergence phenomena for Padé approximants beyond the classical fixed-row setting, which may be of interest to specialists in complex approximation theory and analytic continuation.

minor comments (2)
  1. [Abstract] Abstract, first paragraph: the stated growth condition is written as “m(n)≤m(n+1)≤m(n)”, which is contradictory unless m is constant; this is presumably a typographical error for “m(n)≤m(n+1)” together with the two growth regimes.
  2. [Title] The title contains several typographical errors (“subsequencs”, “closed to rows”, “Pade'”) that should be corrected before publication.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, assessment of significance, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a theoretical extension of classical overconvergence results (Hadamard–Ostrowski for Taylor polynomials; López Lagomasino–Fernández Infante for fixed-row Padé subsequences) to the stated growth regimes on the degree sequence m(n). The abstract and reader's summary indicate that the argument proceeds from the given assumptions on the power series radius and the monotonicity/slow-growth bounds on m(n), without introducing fitted parameters, self-definitional quantities, or load-bearing self-citations. All cited results are external classical theorems whose statements do not depend on the present paper's conclusions, so the derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Standard domain assumptions from complex analysis; no free parameters, new entities, or ad-hoc axioms visible in abstract.

axioms (1)
  • domain assumption f is a power series with positive radius of convergence
    Explicitly stated as the setting in the abstract.

pith-pipeline@v0.9.0 · 5658 in / 987 out tokens · 25402 ms · 2026-05-25T18:36:36.901718+00:00 · methodology

discussion (0)

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Reference graph

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