Weighted Chebyshev Polynomials on Compact Subsets of the Complex Plane

TL;DR

This paper investigates weighted Chebyshev polynomials on compact subsets of the complex plane, establishing foundational properties, invariance results, and bounds related to their norms and weights.

Contribution

It provides existence, uniqueness, and characterization results for weighted Chebyshev polynomials, along with invariance and comparison theorems, extending classical polynomial approximation theory.

Findings

Established existence and uniqueness of weighted Chebyshev polynomials.

Derived invariance of Widom factors under polynomial pre-images.

Provided a lower bound for Widom factors based on the Szegő integral.

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.