Widom faktörleri

In this thesis we recall classical results on Chebyshev polynomials and logarithmic capacity. Given non-polar compact set K, we define the n-th Widom factor W_n(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(y), we prove new results on behavior of the sequence (W_n(K)). By K. Schiefermayr[1], W_n(K) greater than or equal to 2 for any non-polar compact subset K of the real line. We prove that the theoretical lower bound 2 for compact sets on the real line can be achieved by W_2^s(K(y)) as fast as we wish. By G. Szegö[2], rate of the sequence (W_n(K)) is slower than exponential growth. We show that there are sets with unbounded (W_n(K)) and moreover for each sequence (M_n) of subexponential growth there is a Cantor-type set which Widom factors exceed M_n for infinitely many n. By N.I. Achieser[3][4], limit of the sequence (W_n(K)) does not exist in the case K consists of two disjoint intervals. In general the sequence (W_n(K)) may behave highly irregular. We illustrate this behavior by constructing a Cantor-type set K such that one subsequence of (W_n(K)) converges as fast as we wish to the theoretical lower bound 2, whereas another subsequence exceeds any sequence (M_n) of subexponential growth given beforehand.