On the convergence on nonlinear Pad\'e--Chebyshev approximations to the multivalued analytic functions, variation of equilibrium energy and $S$-property of stationary compacts

TL;DR

This paper investigates the convergence behavior of nonlinear Padé--Chebyshev approximations for multivalued analytic functions, revealing that they converge within the maximal domain bounded by an S-curve, with implications for approximation theory.

Contribution

The paper provides new proofs of convergence for nonlinear Padé--Chebyshev approximations and characterizes the boundary of their maximal convergence domain as an S-curve.

Findings

Approximations converge in the maximal domain of meromorphity.

The boundary of the convergence domain is an S-curve.

Results extend understanding of nonlinear approximation methods.

Abstract

Some new results on the convergence of nonlinear diagonal Pad\'e--Chebyshev approximations to multivalued analytic function given on the segment $[-1,1]$, are proved. We show that these approximations converge to the given function in the "maxmimal" domain of its meromorphity and that the boundary of this maxmimal domain is an $S$-curve. The results of the work have been announced in abs/1009.4813 Bibliography: 57 items.