Widom Factors

Abstract

Given a non-polar compact set K,we define the n-th Widom factor Wn(K) as the ratio of the sup-norm of the n-th Chebyshev polynomial on K to the n-th degree of its logarithmic capacity. By G. Szegő, the sequence \((W_{n}(K))_{n=1}^{\infty }\) has subexponential growth. Our aim is to consider compact sets with maximal growth of the Widom factors. We show that for each sequence \((M_{n})_{n=1}^{\infty }\) of subexponential growth there is a Cantor-type set whose Widom’s factors exceed Mn. We also present a set K with highly irregular behavior of the Widom factors.