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

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JACOB S. CHRISTIANSEN∗, BARRY SIMON† AND MAXIM ZINCHENKO‡ Abstract. We make a number of comments on Chebyshev polynomials for general compact subsets of the complex plane. We focus on two aspects: asymptotics of the zeros and explicit Totik–Widom upper bounds on their norms.

1

Introduction

Let e ⊂ C be a compact, not finite, set. For any continuous, complex-valued function, f , on e, let (1.1)

f e = sup |f (z)|. z∈e

The Chebyshev polynomial, Tn , of e is the (it turns out unique) degree n monic polynomial that minimizes Pe over all degree n monic polynomials, P. We define (1.2)

tn ≡ Tn e .

We will use Tn(e) and tn(e) when we want to be explicit about the underlying set. We let C(e) denote the logarithmic capacity of e (see [36, Section 3.6] or [5, 17, 19, 20, 28] for the basics of potential theory; in particular, we will make reference below to the notion of equilibrium measure). This paper continues our study [11, 12, 13] of tn and Tn . Those papers mainly (albeit not entirely) dealt with the case e ⊂ R. In this paper, we make a number of comments on the general complex case focusing on two aspects, upper bounds ∗ Research supported in part by Project Grant DFF-4181-00502 from the Danish Council for Independent Research and by the Swedish Research Council (VR) Grant No. 2018-03500. † Research supported in part by NSF grants DMS-1265592 and DMS-1665526 and in part by Israeli BSF Grant No. 2014337. ‡ Research supported in part by Simons Foundation grant CGM-581256.

207 ´ JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 141 (2020) DOI 10.1007/s11854-020-0120-9

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J. S. CHRISTIANSEN, B. SIMON AND M. ZINCHENKO

on tn , which we called Totik–Widom bounds (henceforth, sometimes, TW bounds), and the asymptotics of zeros of Tn (z). As is often the case in complex analysis, there is magic in simple observations. Larry Zalcman has long been a master magician in this way, so we are pleased to provide this present to him recognizing his long service as editor-in-chief of Journal d’Analyse Math´ematique. We begin by sketching the uniqueness proof for Tn which extends the argument when e ⊂ R (a case that appears in many places including [11]). We call z ∈ e an extreme point for P if and only if |P(z)| = Pe . We claim that any norm minimizer, P, a monic polynomial of degree n, must have at least n + 1 extreme points. For, if there are only z1 , . . . , zk with k ≤ n distinct extreme points for P, by Lagrange interpolation, we can find a polynomial Q with degree k − 1 so that (1.3)

Q(zj ) = P(zj ),

j = 1, . . . , k.

Then for ε small and positive, it is easy to see that P − εQe < Pe violating the fact that P is a norm minimizer. (We note that for e = [−1, 1], Tn has exactly n + 1 extreme points although for many sets, e.g., e = D, each Tn has infinitely many extreme points.) Suppose now that f and g are both norm minimizers among monic polynomials of degree n. Then so is h = 12 (f + g). Pick {zj }n+1 j=1 distinct extreme points for h. Since |h(zj )| = tn and |f (zj )|, |g(zj)| ≤ tn , we must have that f (zj ) = g(zj

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