Szegő–Widom asymptotics of Chebyshev polynomials on circular arcs

Abstract

Thiran and Detaille give an explicit formula for the asymptotics of the sup-norm of the Chebyshev polynomials on a circular arc. We give the so-called Szegő-Widom asymptotics for this domain, i.e., explicit expressions for the asymptotics of the corresponding extremal polynomials. Moreover, we solve a similar problem with respect to the upper envelope of a family of polynomials uniformly bounded on this arc. That is, we give explicit formulas for the asymptotics of the error of approximation as well as of the extremal functions. Our computations show that in the proper normalization the limit of the upper envelope represents the diagonal of a reproducing kernel of a certain Hilbert space of analytic functions. Due to Garabedian the analytic capacity in an arbitrary domain is the diagonal of the corresponding Szegő kernel. We don't know any result of this kind with respect to upper envelopes of polynomials. If this is a general fact or a specific property of the given domain, we rise as an open question.

Ahlfors problem for polynomials

“…Recently we added to this family GMP matrices [33], which are related to approximation by rational functions with a prescribed system of poles. There is no doubt that after a suitable choice of the exponential factor the limit of the minimal deviation in such a rational approximation would lead to the same function Υ(λ), see [8], where such universality was demonstrated for the Chebyshev extremal problem. (ii) Hilbert space structure.…”

“…(ii) Hilbert space structure. log-subharmonicity is a general property of upper envelopes of families of analytic functions [16,Lecture 7], see also [8]. This explains why the following matrix formed by partial derivatives should be nonnegative…”

Pointwise Remez inequality

“…Lemma 2. 13 Let E(α, δ) with α ≤ 0 be such that the corresponding extremal polynomial A n,α,δ (z) corresponds to the case (ii) (case (iii)). Then, the infinitesimal variation ẇ(x) generated by increasing θ(1) (increasing θ(−1)) under the constraint of a constant gap length (δ = const) leads to an increase (decrease) of h 0 .…”

“…4. Nowadays, the language of potential theory is so common and widely accepted, see, e.g., [6,11,13,19,27], that we will formulate our asymptotic result using this terminology.…”

Widom Factors and Szegő-Widom Asymptotics, a Review

“…Yet, the case of sets consisting of finitely many smooth components some or all of which are arcs in general position in the complex plane has proven to be much more difficult. Widom made conjectures regarding that case, but subsequent works [41,45,19] have shown that these conjectures are incorrect. In particular, Widom expected that generically the asymptotic upper bound (3.5) is attained for sets e with finitely many smooth components when at least one of them is an arc.…”