Norm estimates for Chebyshev polynomials, I

In this paper, we extend the sharp upper bound of Christiansen et al. (2017) and the sharp lower bound of Schiefermayr (2008) to the case of weighted Chebyshev polynomials on subsets of [−1,1] for the weight w(x)=1−x². We then analyse the norm of Chebyshev polynomials on a circular arc, prove monotonicity of the corresponding Widom factors, find exact values of their supremum and infimum, and obtain a new proof for their limit.