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arXiv · 2508.08449

Weighted Chebyshev Polynomials on Compact Subsets of the Complex Plane

Galen Novello·Klaus Schiefermayr·Maxim Zinchenko
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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.

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Galen Novello, Klaus Schiefermayr, Maxim Zinchenko. 2025-08-11. Weighted Chebyshev Polynomials on Compact Subsets of the Complex Plane. https://doi.org/10.1007/978-3-030-75425-9_18

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