On the zeros of asymptotically extremal polynomial sequences in the plane

Abstract

Abstract. Let E be a compact set of positive logarithmic capacity in the complex plane and let {P n (z)} ∞ 1 be a sequence of asymptotically extremal monic polynomials for E in the sense that lim supThe purpose of this note is to provide sufficient geometric conditions on E under which the (full) sequence of normalized counting measures of the zeros of {P n } converges in the weak-star topology to the equilibrium measure on E, as n → ∞. Utilizing an argument of Gardiner and Pommerenke dealing with the balayage of measures, we show that this is true, for example, if the interior of the polynomial convex hull of E has a single component and the boundary of this component has an "inward corner" (more generally, a "non-convex singularity"). This simple fact has thus far not been sufficiently emphasized in the literature. As applications we mention improvements of some known results on the distribution of zeros of some special polynomial sequences.Dedication: To Herbert Stahl, an exceptional mathematician, a delightful personality, and a dear friend.

Zeros of Faber Polynomials for Joukowski Airfoils

“…For the rest of the zeros, reasoning as in the proof of Theorem 3.2, we see that they accumulate on L + b and distribute asymptotically like (φ −1 b ) * (η| (c + ,c − ) ). ] that any weak-* subsequential limit µ of {µ n } has a balayage to ∂K which is the equilibrium measure µ K of K, Theorems 3.1 and 4.1 show that any Joukowski airfoil K with 1 < R cos θ ≤ 3/2 admits an electrostatic skeleton; i.e., a positive measure µ with closed support S in K where S has empty interior and connected complement such that the logarithmic potentials of µ and µ K agree (in our case) on C \ K. See [6] and [8] for more on this subject.…”

“…Indeed, the whole sequence {µ n } converges to µ K if L is a Jordan curve. By different methods, this last assertion was also proven to be true if L has an inner cusp, see [8,Corollary 3.2].…”

Widom Factors and Szegő-Widom Asymptotics, a Review

“…Another interesting result on convergence to the equilibrium measure is given by Saff-Stylianopoulos [34]. They prove that if ∂e has an inward pointing corner (more generally, a non-convex type singularity), then the zerocounting measures dµ n always converge weakly to dρ e .…”

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

“…It is an intriguing question to understand when the density of zeros measure converges to the equilibrium measure. An interesting result on this question is in Saff-Stylianopoulos [28] who prove that if ∂e has an inward pointing corner in a suitable sense, then the density of zeros converges to the equilibrium measure. For example, if e is a polygon that is not convex, then their hypothesis holds.…”

“…The Cauliflower is the Julia set of the map z 2 + z; see, for example, Milnor [21,Figure 2.4]. This has inward pointing cusps so, by [28], the density of zeros approaches the equilibrium measure. Since there has been previous work [6,7,9,16,1,5] on extremal polynomials on Julia sets (albeit certain disconnected Julia sets where PW fails), this might be an approachable example.…”