On the Totik–Widom Property for a Quasidisk

Abstract

Let K be a quasidisk on the complex plane. We construct a sequence of monic polynomials p n = p n (·, K) with zeros on K such that ||p n || K ≤ O(1)cap(K) n as n → ∞.

Asymptotics of Chebyshev polynomials, I: subsets of $${\mathbb {R}}$$ R

“…With the case a j ↓ 0 rapidly in mind, we call this the Koch antenna (although, so far as we know, Koch never considered it!). If lim inf a j = 0, K ∞ is not a quasidisk and [3,4] do not apply. We believe that the case a j = 3 −j is a good candidate for a situation where TW might fail.…”

“…We suggested in several talks that if TW fails, it likely fails for the Koch snowflake but this set is more regular that one might think -it is a quasidisk. Andrievski [3] and Andrievski-Nazarov [4] proved that every quasidisk has the TW property, so the Koch snowflake does not provide a counterexample. , where z 0 ∈ Ω, is a compact set whose boundary has tangled spikes and the boundary is not continuous nor everywhere accessible from the outside.…”

Widom Factors and Szegő-Widom Asymptotics, a Review

“…Next, we turn to upper bounds. A collection of very general bounds for unweighted Chebyshev polynomials were obtained by Andrievskii [5,6] and Andrievskii-Nazarov [7]. See also Totik-Varga [44].…”

Asymptotics of Chebyshev Polynomials. IV. Comments on the Complex Case

“…We suggested in several talks that if TW fails, it likely fails for the Koch snowflake but this set is more regular that one might think -it is a quasidisk. Andrievski [3] and Andrievski-Nazarov [4] proved that every quasidisk has the TW property, so the Koch snowflake does not provide a counterexample. Our point here is not that this example should be analyzed but that while searching for possible counterexamples to "every CSC set is TW", one needs to consider sets whose boundary is not a continuous curve.…”