Hadamard-Type Asymptotics for Products of Best Rational Approximation Errors

Abstract: Let $ρ{n,m}(f;E)$ 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 $$ \prod{k=0}^{m}ρ_{n-m+k,k}(f;E). $$ 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.

• The paper proves a sharp Hadamard-type product formula for uniform best rational approximation errors that directly relates to the radii of meromorphy.
• It employs techniques such as Green function analysis, Hankel operator theory, and conformal mappings to connect rational approximation errors with classical Hadamard and determinant asymptotics.
• The results provide practical insights into convergence rates of rational approximants and inform applications in numerical quadrature, analytic continuation, and operator theory.

Hadamard-Type Asymptotics for Products of Best Rational Approximation Errors

Introduction and Background

The paper "Hadamard-Type Asymptotics for Products of Best Rational Approximation Errors" (2604.03854) addresses the asymptotic behavior of products of uniform best rational approximation errors for analytic functions on a compact set $E \subset \mathbb{C}$, focusing on a Hadamard-type perspective. The foundational objective is to connect the classical Hadamard theory for Hankel determinants encoding radii of meromorphic continuation to a rational approximation setting, generalizing Gonchar’s sharp asymptotic results for individual rows of the Walsh table to products involving multiple rows.

For given $f$ analytic on $E$ and non-negative integers $n, m$, set $\rho_{n,m}(f;E)$ as the error of the best uniform approximation of $f$ by rational functions with numerator and denominator degrees at most $n$ and $m$, respectively. The classical theorem of Gonchar asserts that, for each $m \geq 0$,

$$\limsup_{n\to\infty}\rho_{n,m}(f;E)^{1/n} = \frac{1}{R_m} < 1,$$

where $f$0 is the maximum radius on which $f$1 admits a meromorphic continuation with at most $f$2 poles. In the case $f$3, this recovers the Bernstein–Saff–Walsh–Varga theory for polynomial approximation.

Hadamard’s theorem for Hankel determinants links the exponential asymptotic of determinants formed from Taylor coefficients of $f$4 to the product $f$5, where $f$6 are radii of meromorphy. The paper draws an analogy between this determinant-based structure and a product of best rational approximation errors along a superdiagonal segment of the Walsh table,

$f$7

Main Results

1. Sharp Hadamard-Type Product Formula:

For $f$8 a continuum with connected complement and Jordan boundary, and $f$9 analytic on $E$0, it is shown that

$E$1

For more general Green sublevel sets $E$2, with $E$3, the asymptotics refine to

$E$4

Both estimates are sharp under the stated geometric assumptions. The alternating inclusion of approximation errors from consecutive rows of the Walsh table encodes the entire meromorphic hierarchy captured in the determinant formula of Hadamard.

2. Comparison Inequalities:

For any continuum $E$5 with connected complement and $E$6 analytic on $E$7, the product of best rational approximation errors on $E$8 and a sublevel set $E$9 satisfies: $n, m$0 with equality for $n, m$1 possessing Jordan boundary.

3. Common Extremal Subsequences:

There exists a common subsequence $n, m$2 along which, simultaneously for all $n, m$3, the error exponents as well as the partial products up to index $n, m$4 attain their maximal rates: $n, m$5 as $n, m$6, $n, m$7.

4. Connection to Hankel Determinants (Disc Case):

In the disc, the quotient of the Hankel determinant and the rational error product satisfies

$n, m$8

linking the classical moment problem to rational approximation asymptotics explicitly.

Proof Techniques

The approach unifies techniques from complex approximation theory, operator theory, and geometric function theory:

• Green Function Analysis: The level sets of the Green function serve as canonical enlargements $n, m$9 of the compact $\rho_{n,m}(f;E)$0 for regularity and comparison purposes.
• Hankel Operator and AAK Theory: Singular values of Hankel operators with symbol $\rho_{n,m}(f;E)$1 (and their weighted variants) are thoroughly analyzed, using Adamyan–Arov–Kre\u{\i}n (AAK) theory to connect singular numbers with best meromorphic approximation errors, especially on domains with analytic boundary.
• Conformal Mappings and Faber Expansions: Exterior conformal maps and Faber polynomial expansions allow reduction of the general problem on Jordan domains to the disc case.
• Comparison Lemmas and Transfer Principles: The analysis deploys delicate comparison arguments between the errors on curves and compact sets, exploiting kernel estimates and boundary regularity.
• Hadamard-Type Factorizations: The translation from Taylor/Hankel determinant asymptotics to products of rational approximation errors along specific superdiagonal rays in the Walsh table is essential.

Implications and Connections

From a theoretical viewpoint, these results solidify the connection between the meromorphic continuation structure (encapsulated by the radii $\rho_{n,m}(f;E)$2) and uniform rational approximation rates in complex domains. The main product asymptotic formulas provide a structural mirror to classical determinant identities but in the context of uniform rational rather than coefficient-based (moment) approximation.

On the practical side, these asymptotics inform the rate at which sequences of rational approximants (with varying numerator and denominator degrees) converge to a given analytic function, including in numerical quadrature, analytic continuation, and model reduction problems where control of error decay rates is critical.

In operator theory, the bridge via Hankel operators and AAK theory highlights rational approximation errors as nonlinear singular values, with consequences for the study of spectral properties and their connection to function-theoretic singularities.

Future Directions

Possible extensions of this work include:

• Expansion to classes of functions with essential singularities or branch points, where the present framework might require additional consideration of natural boundaries.
• Exploration of the effect of additional geometric structures on $\rho_{n,m}(f;E)$3 (e.g. symmetry, smoothness of the boundary) on the sharpness and universality of the asymptotic products.
• Connections to structured Padé and multipoint rational approximation and the asymptotic conditioning of these problems in numerical algorithms.

Moreover, the analytic framework deployed could be adapted to study multi-dimensional analogues or non-uniform approximation metrics (e.g., weighted norms, $\rho_{n,m}(f;E)$4 settings).

Conclusion

The paper rigorously establishes exact product asymptotic formulas for best rational approximation errors in the uniform norm, paralleling classical Hadamard determinant asymptotics, and elucidates the central role played by meromorphy radii in the exponential decay rate of these products. The results synthesize insights from classical approximation theory, potential theory, and operator methods, yielding a definitive understanding of rational error decay in relation to analytic continuation phenomena.