[Paper Review] On the convergence of Chebyshev--Padé approximations to real-valued algebraic functions

A A Gonchar, Евгений Андреевич Рахманов+1 more|arXiv (Cornell University)|Sep 24, 2010

Numerical Methods and Algorithms9 references3 citations

TL;DR

This paper establishes the convergence of Chebyshev-Padé approximations for real algebraic functions on $[-1,1]$ via a vector equilibrium problem in potential theory. It proves diagonal rational approximants converge in capacity, with convergence speed characterized by a mixed Green-logarithmic potential-theoretic equilibrium problem, distinguishing nonlinear and linear schemes through different compacta and matrix interactions at varying parameters $\theta$. The result generalizes Stahl's theorem to non-Markovian algebraic functions.

ABSTRACT

We announce some new results on the convergence of Chebyshev--Padé approximations to real-valued algebraic function given on the segment $[-1,1]$. The rate of convergence on the segment and in the corresponding maximal domain of meromorphity of a given function is charactirized in terms of a theoretical potential equilibrium problem.

Motivation & Objective

• To extend convergence results for Padé approximations from Markov functions to general real algebraic functions on $[-1,1]$.
• To characterize the convergence speed of diagonal Chebyshev-Padé approximants on $[-1,1]$ and in the maximal domain of meromorphy.
• To establish that nonlinear and linear Chebyshev-Padé approximations correspond to different potential-theoretic equilibrium problems and distinct stationary compacts.
• To unify the convergence analysis using a mixed Green-logarithmic vector equilibrium problem parameterized by $\theta$.

Proposed method

• The paper employs a vector equilibrium problem in potential theory involving a matrix of interaction coefficients $A_\theta$ that depends on a parameter $\theta \geq 0$, governing the behavior of the approximants.
• It defines two distinct schemes: a nonlinear scheme based on Baker's method, solving a system of nonlinear equations for coefficients of $p/q$ to match first $L+M+1$ Chebyshev-Fourier coefficients.
• It defines a linear scheme via Frobenius’ method, solving a linear homogeneous system $c_k(Qf - P) = 0$ for $k = 0, \dots, L+M$, ensuring existence but not uniqueness.
• The convergence is analyzed in capacity on $[-1,1]$ and in the corresponding maximal domain of meromorphy, using logarithmic and Green potentials relative to a compact $K$ disjoint from $[-1,1]$.
• The key technical tool is the construction of a unique equilibrium measure $\lambda_K(\theta)$ solving a vector equilibrium problem with matrix $A_\theta$, which governs the convergence rate.
• The analysis distinguishes cases via the matrix $A_\theta$, with $A_0$, $A_1$, and $A_3$ representing qualitatively different interaction structures.

Experimental results

Research questions

• RQ1 Does the diagonal Chebyshev-Padé approximation converge for arbitrary real algebraic functions on $[-1,1]$?
• RQ2 How does the convergence speed of these approximants depend on the function's analytic structure and the underlying potential-theoretic framework?
• RQ3 What is the relationship between the nonlinear and linear Chebyshev-Padé schemes in terms of their associated stationary compacts and equilibrium problems?
• RQ4 How do different values of the parameter $\theta$ affect the vector equilibrium problem and the resulting convergence behavior?
• RQ5 Can the convergence of diagonal approximants be characterized uniformly using a mixed Green-logarithmic potential-theoretic equilibrium model?

Key findings

• The paper proves that diagonal Chebyshev-Padé approximants for real algebraic functions on $[-1,1]$ converge in capacity, generalizing Stahl's classical result to non-Markovian functions.
• The convergence speed on $[-1,1]$ and in the maximal domain of meromorphy is governed by a vector equilibrium problem involving a mixed Green-logarithmic potential, with the matrix $A_\theta$ encoding the interaction structure.
• Nonlinear and linear Chebyshev-Padé approximants correspond to different equilibrium problems and distinct stationary compacts, which only coincide in special cases like Markov functions.
• For $\theta = 0$, the interaction matrix $A_0 = \begin{pmatrix}1&-1\\ -1&1\end{pmatrix}$ leads to a degenerate equilibrium, while $\theta = 1$ and $\theta = 3$ yield distinct, non-degenerate configurations with $A_1$ and $A_3$ respectively.
• The existence and uniqueness of the equilibrium measure $\lambda_K(\theta)$ for each $\theta$ ensures the theoretical foundation for the convergence analysis.
• The results establish a universal framework for convergence, showing that the potential-theoretic equilibrium model captures the intrinsic convergence behavior across different function classes and approximation schemes.