Widom Factors

Abstract

Given a 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. By G. Szegő, the sequence (W n (K)) ∞ n=1 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 of subexponential growth there is a Cantor-type set whose Widom's factors exceed M n . We also present a set K with highly irregular behavior of the Widom factors.

On Chebyshev polynomials in the complex plane

“…Let now K ⊂ R , where R is the real line, consist of an infinite number of components. According to [13,Theorem 4.4] in this case {t n (K)} can increase faster than any sequence {t n } satisfying t n ≥ 1 and lim n→∞ (log t n )/n = 0 . Therefore, in order to have particular bounds for t n (K) some additional assumptions on K are needed.…”

Turán type oscillation inequalities in $L^q$ norm on the boundary of convex domains

“…There are several known estimates for capacities and the so-called "Widom factors" (see e.g. [27,64] and the references therein) between Chebyshev constants and corresponding powers of the transfinite diameter: for us, these more precise estimates are not needed, as (7) suffices. There are some known explicit computations or comparisons and estimates of capacities from other geometric parameters of the respective sets: a few most basic ones can be found e.g.…”