Asymptotics of Chebyshev polynomials, II: DCT subsets of R

Abstract

We prove Szegő-Widom asymptotics for the Chebyshev polynomials of a compact subset of R which is regular for potential theory and obeys the Parreau-Widom and DCT conditions.

Asymptotics of Chebyshev Polynomials, III. Sets Saturating Szegő, Schiefermayr, and Totik–Widom Bounds

“…We can also answer when equality in the upper or lower bound occurs asymptotically along a subsequence. In our paper with Yuditskii [3], we focused on subsequences {n j } ∞ j=1 where the zeros of T n j in gaps had limits. There is at most one zero in each gap, K [2, Theorem 2.3].…”

“…Let G denote the set of all gaps of e, i.e. bounded components of R \ e. In [3], we defined what we called a gap collection, a subset G 0 ⊂ G and for each K ∈ G 0 , a point x K ∈ K. We considered subsequences, T n j , so that for K ∈ G \ G 0 , as n j → ∞, either T n j has no zero in K or the zero goes to the one of the two edges of K and so that for K ∈ G 0 , there is a zero for large n j which goes to x K as n j → ∞. This describes all possible limit points of the set of zeros.…”

“…In [3], we discussed limits of T n / T n e for e ⊂ R under a stronger condition than PW called DCT. If e has what we called a canonical generator, which holds in a generic sense, then [3, Theorem 5.1] every Blaschke product occurs as a limit point of the normalized Chebyshev polynomials which means one has a limit with any set of simple zeros in any set of gaps.…”

“…The Chebyshev polynomial, T n , of e is the (it turns out unique) degree n monic polynomial that minimizes P e over all degree n monic polynomials, P . We define t n = T n e (1.2) This paper continues our study [2,3] of t n and T n , especially their asymptotics as n → ∞. We let C(e) denote the logarithmic capacity of e (see [13,Section 3.6] or [1,5,6,7,11] for the basics of potential theory).…”

Computation of Chebyshev Polynomials for Union of Intervals

“…We reserve the notation T K N for the Chebyshev polynomial normalized to have max-norm equal to one on K, i.e., (2) T K N =…”