Computing Chebyshev polynomials using the complex Remez algorithm
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.
Cite
@article{arxiv.2405.05067,
title = {Computing Chebyshev polynomials using the complex Remez algorithm},
author = {Lennart Aljoscha Hübner and Olof Rubin},
journal= {arXiv preprint arXiv:2405.05067},
year = {2025}
}
Comments
All computations have been redone with the aid of the new co-author using the arbitrary precision library mpmath. This has led to a new development in the plots of zeros of Chebyshev polynomials on the equilateral triangle. All conjectures have been reformulated as hypotheses in the text