Weighted Chebyshev Polynomials on Compact Subsets of the Complex Plane

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.