On Chebyshev polynomials in the complex plane

Abstract

Abstract. The estimates of the uniform norm of the Chebyshev polynomials associated with a compact set K in the complex plane are established. These estimates are exact (up to a constant factor) in the case where K consists of a finite number of quasiconformal curves or arcs. The case where K is a uniformly perfect subset of the real line is also studied.

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.…”

Asymptotics of Chebyshev polynomials, II: DCT subsets of R

“…Our methods there say nothing about the complex case. In this regard, we mention the recent interesting paper of Andrievskii [2] who has proven Totik-Widom bounds for a class of sets that, for example, includes the Koch snowflake.…”

On the Totik–Widom Property for a Quasidisk

“…In particular, any quasidisk obeys the Totik-Widom bound (1.1), see also [3,Theorem 2] where quite different method for the estimate of w n (K) is used.…”