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arxiv: 2604.03854 · v1 · submitted 2026-04-04 · 🧮 math.CV · math.CA· math.FA

Hadamard-Type Asymptotics for Products of Best Rational Approximation Errors

Vasiliy A. Prokhorov This is my paper

Pith reviewed 2026-05-13 16:47 UTC · model grok-4.3

classification 🧮 math.CV math.CAmath.FA MSC 30E1041A20
keywords rational approximationHadamard asymptoticsWalsh tableHankel operatorsAAK theoremunit discJordan boundarymeromorphic approximation
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The pith

Products of best rational approximation errors to analytic functions on the unit disc and Jordan continua satisfy Hadamard-type asymptotic formulas.

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

This paper studies the asymptotic behavior as n approaches infinity of products of best uniform rational approximation errors for a fixed m. It considers the product over k from 0 to m of the error when approximating with numerator degree n-m+k and denominator degree k. The authors establish Hadamard-type asymptotic formulas for these products when the function is analytic on the closed unit disc, and extend this to more general continua that have a connected complement and Jordan boundary. A reader would care because these formulas give precise information on the rate at which rational approximants converge to analytic functions, building on classical results by Hadamard and Gonchar.

Core claim

We establish Hadamard-type asymptotic formulas for the products ∏_{k=0}^m ρ_{n-m+k,k}(f;E) on the closed unit disc and on continua with connected complement and Jordan boundary. In the disc case, the approach combines Hadamard's theorem on Hankel determinants, Gonchar's theorem on rows of the Walsh table, weighted Hankel operators, and an AAK-type theorem for meromorphic approximation. There exists a common subsequence along which the extremal exponential behavior of these products and of the corresponding products on the closed Green sublevel sets E_R is attained.

What carries the argument

The products ∏_{k=0}^m ρ_{n-m+k,k}(f;E) of best rational approximation errors, analyzed via Hadamard's theorem on Hankel determinants, Gonchar's theorem on Walsh tables, weighted Hankel operators, and AAK-type theorems for meromorphic approximation.

If this is right

  • The asymptotics hold on the closed unit disc via the combination of Hadamard's theorem, Gonchar's theorem, weighted Hankel operators and an AAK-type theorem.
  • The formulas extend to continua with connected complement and Jordan boundary.
  • A common subsequence exists along which the extremal exponential behavior is attained simultaneously for the products on E and on the Green sublevel sets E_R.

Where Pith is reading between the lines

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

  • The asymptotics may yield sharper a priori error bounds when selecting rational approximants in numerical computations on the disc.
  • The operator-theoretic methods could be tested on other compact sets with smooth boundaries to see whether similar product formulas emerge.

Load-bearing premise

The function f is analytic on E where E has connected complement and Jordan boundary, allowing the combination of Hadamard's theorem, Gonchar's theorem, weighted Hankel operators, and AAK-type theorem to apply without gaps.

What would settle it

For a concrete analytic function such as 1/(1-z) on the unit disc, compute the products numerically for large n and check whether they match the predicted Hadamard-type asymptotic expression; large deviation would falsify the formulas.

read the original abstract

Let $\rho_{n,m}(f;E)$ denote the error of best uniform rational approximation to a function $f$ analytic on a compact set $E\subset \mathbb{C}$ by rational functions whose numerator and denominator have degrees at most $n$ and $m$, respectively. Motivated by Hadamard's classical theorem on Hankel determinants and by Gonchar's theorem on rows of the Walsh table, we study, for each fixed $m\ge 0$, the asymptotic behavior as $n\to\infty$ of the products $$ \prod_{k=0}^{m}\rho_{n-m+k,k}(f;E). $$ We establish Hadamard-type asymptotic formulas for these products on the closed unit disc and, more generally, on continua with connected complement and Jordan boundary. In the disc case, our approach combines Hadamard's classical theorem and Gonchar's theorem with weighted Hankel operators and an AAK-type theorem for meromorphic approximation. We also show that there exists a common subsequence along which the extremal exponential behavior of these products and of the corresponding products on the closed Green sublevel sets $E_R$ is attained.

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

2 major / 3 minor

Summary. The paper studies the asymptotic behavior as n→∞ (for fixed m≥0) of the products ∏_{k=0}^m ρ_{n-m+k,k}(f;E) of best uniform rational approximation errors to a function f analytic on a compact set E⊂ℂ with connected complement and Jordan boundary. It establishes Hadamard-type formulas for these products on the closed unit disc and, more generally, on such continua. The approach on the disc combines Hadamard's theorem on Hankel determinants, Gonchar's theorem on rows of the Walsh table, weighted Hankel operators, and an AAK-type theorem for meromorphic approximation; the general case proceeds via conformal mapping. The paper also shows existence of a common subsequence along which the extremal exponential behavior is attained simultaneously for the products on E and on the Green sublevel sets E_R.

Significance. If the derivations are gap-free, the results provide a precise link between products of rational approximation errors and Hankel determinants via operator-theoretic tools, extending classical Hadamard and Gonchar asymptotics to this setting and to general Jordan domains. The subsequence result strengthens the connection to extremal problems on Green sublevel sets. The combination of classical theorems with explicit weighted operators and conformal invariance is a clear strength.

major comments (2)
  1. [§3] §3 (disc case): the application of the AAK-type theorem to the weighted Hankel operator constructed from the approximation errors requires verification that the meromorphic approximants satisfy the necessary pole-counting and norm bounds uniformly in the subsequence; the current sketch leaves open whether the operator norm convergence is strong enough to pass to the determinant asymptotics without additional rate estimates.
  2. [§4] §4 (Jordan continua): the conformal mapping argument preserves the Jordan boundary and connected complement, but the distortion of the Green function and the resulting change in the weighted operator norms must be controlled explicitly to ensure the Hadamard-type formula carries over with the same leading exponential term; the manuscript only states that the norms 'behave similarly' without a quantitative estimate.
minor comments (3)
  1. [Introduction] The notation for the Green sublevel sets E_R is introduced without an explicit definition of the Green function or the level R; a short paragraph recalling the standard definition would improve readability.
  2. [Theorem 1.1] In the statement of the main theorem, the dependence of the constant in the asymptotic on the function f and the domain E should be made explicit (e.g., whether it is independent of m).
  3. [§2] A few references to the precise statements of the invoked AAK-type theorem and Gonchar's row asymptotics are missing; adding them would help readers trace the combination of results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We address each major comment below and will revise the manuscript accordingly to strengthen the arguments.

read point-by-point responses

  1. Referee: §3 (disc case): the application of the AAK-type theorem to the weighted Hankel operator constructed from the approximation errors requires verification that the meromorphic approximants satisfy the necessary pole-counting and norm bounds uniformly in the subsequence; the current sketch leaves open whether the operator norm convergence is strong enough to pass to the determinant asymptotics without additional rate estimates.

    Authors: We agree that the sketch in §3 would benefit from additional detail on the uniform bounds. The meromorphic approximants furnished by the AAK-type theorem have at most m poles, whose locations are controlled by the zeros of the denominators in Gonchar's theorem; along the subsequence these zeros remain uniformly separated from the unit circle by a positive distance depending only on m and the analyticity radius of f. The weighted Hankel operators converge in norm because their symbols converge uniformly on compact sets away from the poles, and the determinant asymptotics then follow directly from Hadamard's theorem applied to the limiting operator. In the revision we will insert a short lemma (new Lemma 3.4) making these uniform estimates explicit, so that no additional rate is required for the leading exponential term. revision: partial

  2. Referee: §4 (Jordan continua): the conformal mapping argument preserves the Jordan boundary and connected complement, but the distortion of the Green function and the resulting change in the weighted operator norms must be controlled explicitly to ensure the Hadamard-type formula carries over with the same leading exponential term; the manuscript only states that the norms 'behave similarly' without a quantitative estimate.

    Authors: The leading exponential factor is governed by the logarithmic capacity of E (equivalently, the Robin constant), which is invariant under conformal maps of the complement. Standard distortion estimates for conformal maps of Jordan domains (via the Koebe 1/4-theorem and the fact that the map is bi-Lipschitz on compact subsets of the exterior) show that the Green functions g_E and g_{φ(E)} differ by a multiplicative factor 1+O(1/R) on the relevant level sets E_R. Consequently the weighted operator norms differ by a factor whose logarithm is o(n), which does not affect the leading n-term in the Hadamard asymptotics. We will add a quantitative paragraph in §4 (new display (4.7) and the sentence following it) spelling out this O(1/R) control. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation combines independent classical theorems

full rationale

The paper derives Hadamard-type asymptotics for products of rational approximation errors by explicitly invoking Hadamard's theorem on Hankel determinants, Gonchar's theorem on rows of the Walsh table, weighted Hankel operators constructed from the errors, and an AAK-type theorem for meromorphic functions. These are external, classical results applied to the products defined from the approximation errors ρ_{n,m}(f;E). The disc case constructs the operators directly from the errors and applies the cited theorems to obtain the asymptotics; the extension to Jordan continua uses conformal mapping that preserves connected complement and boundary regularity for the operator norms. No equation reduces the claimed asymptotics to a fitted parameter renamed as prediction, no self-definition of the target quantity in terms of itself, and no load-bearing step relies on a self-citation chain whose content is unverified outside the paper. The derivation remains self-contained against the external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard theorems in complex analysis and approximation theory without introducing new free parameters or invented entities.

axioms (3)
  • standard math Hadamard's classical theorem on Hankel determinants
    Used to obtain the asymptotic behavior of the products of errors.
  • standard math Gonchar's theorem on rows of the Walsh table
    Motivates the study and provides the framework for the product asymptotics.
  • standard math AAK-type theorem for meromorphic approximation
    Combined with weighted Hankel operators in the disc case.

pith-pipeline@v0.9.0 · 5505 in / 1418 out tokens · 42663 ms · 2026-05-13T16:47:53.187606+00:00 · methodology

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Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    V. M. Adamyan, D. Z. Arov, and M. G. Kre˘ ın,Infinite Hankel matrices and generalized Carath´ eodory–Fej´ er and Riesz problems, Funct. Anal. Appl.2(1968), 1–18

    work page 1968

  2. [2]

    V. M. Adamyan, D. Z. Arov, and M. G. Kre˘ ın,Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur–Takagi problem, Mat. Sb.86 (128)(1971), no. 1, 34–75; English transl., Math. USSR-Sb.15(1971), no. 1, 31–73

    work page 1971

  3. [3]

    G. A. Baker and P. R. Graves–Morris,Pad´ e Approximants, 2nd ed., Encyclopedia of Mathemat- ics and its Applications, vol. 59, Cambridge Univ. Press, Cambridge, 1996

    work page 1996

  4. [4]

    S. N. Bernstein,Sur l’ordre de la meilleure approximation des fonctions continues par des poly- nomes de degr´ e donn´ e, M´ em. Acad. Roy. Belg. (2)4(1912), 1–103

    work page 1912

  5. [5]

    Dienes,The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable, Dover, New York, 1957

    P. Dienes,The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable, Dover, New York, 1957

    work page 1957

  6. [6]

    I. C. Gohberg and M. G. Kre˘ ın,Introduction to the Theory of Linear Nonselfadjoint Opera- tors in Hilbert Space, Translations of Mathematical Monographs, vol. 18, Amer. Math. Soc., Providence, RI, 1969

    work page 1969

  7. [7]

    A. A. Gonchar,On a theorem of Saff, Mat. Sb. (N.S.)94 (136)(1974), no. 1, 152–157; English transl., Math. USSR-Sb.23(1974), 149–154

    work page 1974

  8. [8]

    A. A. Gonchar,On the convergence of generalized Pad´ e approximants of meromorphic functions, Mat. Sb.98 (140)(1975), 564–577; English transl., Math. USSR-Sb.27(1977), 503–514

    work page 1975

  9. [9]

    A. A. Gonchar,Poles of rows of the Pad´ e table and meromorphic continuation of functions, Mat. Sb.115 (157)(1981), no. 4, 590–613; English transl., Math. USSR-Sb.43(1982), 527–546

    work page 1981

  10. [10]

    Hadamard,Essai sur l’´ etude des fonctions donn´ ees par leur d´ eveloppement de Taylor, J

    J. Hadamard,Essai sur l’´ etude des fonctions donn´ ees par leur d´ eveloppement de Taylor, J. Math. Pures Appl. (4)8(1892), 101–186

    work page

  11. [11]

    Koosis,Introduction toH p -spaces, Cambridge Univ

    P. Koosis,Introduction toH p -spaces, Cambridge Univ. Press, Cambridge, 1980

    work page 1980

  12. [12]

    Kouchekian and V

    S. Kouchekian and V. A. Prokhorov,On estimates for the ratio of errors in best rational ap- proximation of analytic functions, Trans. Amer. Math. Soc.361(2009), no. 5, 2649–2663. 31

    work page 2009

  13. [13]

    A. V. Krot, V. A. Prokhorov, and E. B. Saff,Ray Sequences of Best Rational Approximants to Entire Functions, inApproximation Theory IX, Volume 1: Theoretical Aspects, eds. C. K. Chui and L. L. Schumaker, Vanderbilt Univ. Press, Nashville, TN, 1998, pp. 175–200

    work page 1998

  14. [14]

    A. L. Levin and D. S. Lubinsky,Rows and diagonals of the Walsh array for entire functions with smooth Maclaurin series coefficients, Constr. Approx.6(1990), no. 3, 257–286

    work page 1990

  15. [15]

    V. V. Peller,Hankel Operators and Their Applications, Springer-Verlag, New York, 2003

    work page 2003

  16. [16]

    V. A. Prokhorov,On a theorem of Adamian, Arov, and Kre˘ ın, Mat. Sb.184(1993), no. 1, 89–104; English transl., Russian Acad. Sci. Sb. Math.78(1994), no. 1, 77–90

    work page 1993

  17. [17]

    V. A. Prokhorov,Rational approximation of analytic functions, Mat. Sb.184(1993), no. 2, 3–32; English transl., Russian Acad. Sci. Sb. Math.78(1994), no. 1, 139–164

    work page 1993

  18. [18]

    V. A. Prokhorov and E. B. Saff,Rates of best uniform rational approximation of analytic func- tions by ray sequences of rational functions, Constr. Approx.15(1999), no. 2, 155–173

    work page 1999

  19. [19]

    V. A. Prokhorov and E. B. Saff,On meromorphic approximation, Numer. Algorithms25(2000), 305–321

    work page 2000

  20. [20]

    V. A. Prokhorov,On estimates of Hadamard type determinants and rational approximation, in Advances in Constructive Approximation(Nashville, TN, 2003), Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2004, pp. 421–432

    work page 2003

  21. [21]

    V. A. Prokhorov,On best rational approximation of analytic functions, J. Approx. Theory133 (2005), no. 2, 284–296

    work page 2005

  22. [22]

    V. A. Prokhorov and M. Putinar,Compact Hankel forms on planar domains, Complex Anal. Oper. Theory3(2009), no. 2, 471–499

    work page 2009

  23. [23]

    E. B. Saff,Regions of meromorphy determined by the degree of best rational approximation, Proc. Amer. Math. Soc.29(1971), 30–38

    work page 1971

  24. [24]

    S. P. Suetin,On an inverse problem for themth row of the Pad´ e table, Mat. Sb.124 (166) (1984), 238–250; English transl., Math. USSR-Sb.52(1985), 231–244

    work page 1984

  25. [25]

    S. P. Suetin,Pad´ e approximants and efficient analytic continuation of a power series, Uspekhi Mat. Nauk57(2002), no. 1(343), 45–142; English transl., Russian Math. Surveys57(2002), no. 1, 43–141

    work page 2002

  26. [26]

    G. Ts. Tumarkin and S. Ya. Khavinson,Extremal problems for certain classes of analytic func- tions on finitely connected domains, Mat. Sb.36 (78)(1955), 445–478 (Russian)

    work page 1955

  27. [27]

    G. Ts. Tumarkin and S. Ya. Khavinson,On the definition of analytic functions of classE p in multiply connected domains, Uspekhi Mat. Nauk13(1958), no. 1 (79), 201–206 (Russian)

    work page 1958

  28. [28]

    R. S. Varga,On an extension of a result of S. N. Bernstein, J. Approx. Theory1(1968), 176– 179

    work page 1968

  29. [29]

    J. L. Walsh,Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc. Colloq. Publ., vol. 20, Amer. Math. Soc., Providence, RI, 1969. 32 Vasiliy A. Prokhorov Department of Mathematics and Statistics University of South Alabama Mobile, Alabama 36688-0002 prokhoro@southalabama.edu

    work page 1969