Computing Chebyshev polynomials using the complex Remez algorithm

TL;DR

This paper uses the generalized Remez algorithm to experimentally analyze high-degree Chebyshev polynomials in the complex plane, revealing patterns and proposing a fundamental relationship with Faber polynomials.

Contribution

It introduces a comprehensive experimental study of Chebyshev polynomials at high degrees using Tang's algorithm, uncovering new patterns and a potential fundamental relationship with Faber polynomials.

Findings

Identification of repeating patterns in high-degree Chebyshev polynomials

Demonstration of the accuracy and precision of Tang's algorithm for complex polynomials

Proposal of a fundamental relationship between Chebyshev and Faber polynomials

Abstract

We employ the generalized Remez algorithm, initially suggested by P. T. P. Tang, to perform an experimental study of Chebyshev polynomials in the complex plane. Our focus lies particularly on the examination of their norms and zeros. What sets our study apart is the breadth of examples considered, coupled with the fact that the degrees under investigation are substantially higher than those in previous studies where other methods have been applied. These computations of Chebyshev polynomials of high degrees reveal discernible, repeating patterns, which indicate a typical behavior of Chebyshev polynomials in a general setting. The use of Tang's algorithm allows for computations executed with precision, maintaining accuracy within quantifiable margins of error. Additionally, as a result of our experimental study, we propose what we believe to be a fundamental relationship between Chebyshev and Faber polynomials associated with a compact set.