Chebyshev polynomials on equipotential curves
Erwin Mi\~na-D\'iaz, Olof Rubin

TL;DR
This paper demonstrates that Chebyshev polynomials on certain equipotential curves converge to Faber polynomials associated with an analytic function, using approximation theory and Laurent series analysis.
Contribution
It establishes the convergence of Chebyshev polynomials on equipotential curves to Faber polynomials as the potential level tends to infinity.
Findings
Chebyshev polynomials converge to Faber polynomials as r→∞.
Zero is the unique best polynomial approximation to z^n on the unit circle.
The proof utilizes Laurent series and approximation theory techniques.
Abstract
For an analytic function with a Laurent expansion at of the form \begin{equation*} \phi(z)=z+c_{0}+\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+\cdots, \end{equation*} the Faber polynomial of degree associated to is the polynomial part of the Laurent series at of . We prove that the th Chebyshev polynomial for the equipotential curve converges to as . The proof makes use of the fact that zero is the strongly unique best approximation to the monomial on the unit circle by polynomials of degree less than .
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Taxonomy
TopicsMathematical functions and polynomials · Meromorphic and Entire Functions · Holomorphic and Operator Theory
