Widom Factors

Abstract

In this thesis we recall classical results on Chebyshev polynomials and logarithmic capacity. 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. We consider results on estimations of Widom factors. By means of weakly equilibrium Cantor-type sets, K(γ), we prove new results on behavior of the sequence (Wn(K))∞ n=1.By K. Schiefermayr[1], Wn(K) ≥ 2 for any non-polar compact K ⊂ R. We prove that the theoretical lower bound 2 for compact sets on the real line can be achieved by W2s (K(γ)) as fast as we wish. By G. Szeg˝o[2], rate of the sequence (Wn(K))∞ n=1 is slower than exponential growth. We show that there are sets with unbounded (Wn(K))∞ n=1 and moreoverfor each sequence (Mn)∞ n=1 of subexponential growth there is a Cantor-type set which Widom factors exceed Mn for infinitely many n. By N.I. Achieser[3][4], limit of the sequence (Wn(K))∞ n=1 does not exist in the case K consists of two disjoint intervals. In general the sequence (Wn(K))∞ n=1 may behave highly irregular. We illustrate this behavior by constructing a Cantor-type set K such that one subsequence of (Wn(K))∞ n=1 converges as fast as we wish to the theoretical lower bound 2, whereas another subsequence exceeds any sequence (Mn)∞ n=1 of subexponential growth given beforehand