Weighted Chebyshev Polynomials on Compact Subsets of the Complex Plane
Complex Variables
2025-08-13 v1
Abstract
We study weighted Chebyshev polynomials on compact subsets of the complex plane with respect to a bounded weight function. We establish existence and uniqueness of weighted Chebyshev polynomials and derive weighted analogs of Kolmogorov's criterion, the alternation theorem, and a characterization due to Rivlin and Shapiro. We derive invariance of the Widom factors of weighted Chebyshev polynomials under polynomial pre-images and a comparison result for the norms of Chebyshev polynomials corresponding to different weights. Finally, we obtain a lower bound for the Widom factors in terms of the Szeg\H{o} integral of the weight function and discuss its sharpness.
Cite
@article{arxiv.2508.08449,
title = {Weighted Chebyshev Polynomials on Compact Subsets of the Complex Plane},
author = {Galen Novello and Klaus Schiefermayr and Maxim Zinchenko},
journal= {arXiv preprint arXiv:2508.08449},
year = {2025}
}