The distribution of zeros of asymptotically extremal polynomials (Q1177060)

Let \(W\) be an admissible weight on a closed set \(E\subset C\). Let \(\|\cdot\|_ E\) denote the supremum norm on \(E\), and let \(\pi_ n\) denote the class of all algebraic polynomials (with complex coefficients) having degree not exceeding \(n\). For such a weight, it is known that the constant \[ t_ n(W,E):=\int\{\| W^ n P\|_ E:\;P(z)=z^ n+\dots\in\pi_ n\} \] satisfy \[ t(W,E):=\lim_{n\to\infty}[t_ n(W,E)]^{1/n}=\exp(-F) \] for a suitable contant \(F\). By an asymptotically minimal norm sequence of monic polynomials \(p_ n(z)=z^ n+\dots\in\pi_ n\), we mean a sequence that satisfies \(\lim_{n\to\infty}\| W^ n p_ n\|_ E^{1/n}=t(W,E)\). To each \(p_ n=\prod_{n=1}^ n(z-z_ k)\), we associate the normalized distribution measure \(v(p_ n)\) defined by \(v(p_ n):={1\over n}\sum_{k=1}^ n\delta_{z_ k}\), where \(\delta_{z_ k}\) is the point distribution with total mass 1 at \(z_ k\). When \(\sigma\) has balayage to \(\partial_ \infty S\) with finite energy, this balayage will be denoted by \([\sigma]_ b\). Theorem 1: Let \(S\subset C\) be compact and \(\sigma\in M(Pc(S))\). Then (i) \(\sigma\) has a balayage to \(\partial_ \infty S\). (ii) If \(\hat\sigma\) and \(\tilde\sigma\) are balayages of \(\sigma\) to \(\partial_ \infty S\), both having finite logarithmic energy, then \(\hat\sigma=\tilde\sigma\). (iii) If \(\hat\sigma\) is a balayage of \(\sigma\) to \(\partial_ \infty S\) and \(f\) is harmonic in the interior of \(P_ c(S)\) and continuous on \(P_ c(S)\), then \(\int f d\sigma=\int f d\hat\sigma\). (iv) If \(\sigma\) has finite logarithmic energy, then it has a balayage to \(\partial_ \infty S\) with finite energy. Theorem 2: Let \(p_ n(z)=z^ n+\dots\in\pi_ n\), \(n\geq 1\) be a sequence of monic polynomials. (a) If, for an increasing sequence \(A\) of integers, (i) \(\lim_{n\to\infty, n\in A}\| W^ n p_ n\|^{1/n}_{\partial_ \infty \varphi}\leq\exp(-F)\) then, for every closed \(A\subset D_ \infty(\varphi)\), \(\lim_{n\to\infty, n\in A} v(p_ n)(A)=0\). Moreover, if \(v^*\) is any weak star limit measure of \(\{v(p_ n)\}_{n\in A}\), then \(v^*\in M(P_ c(\varphi))\), \(v^*\) (as well as \(m\)) has a balayage to \(\partial_ \infty\varphi\) with finite energy, and \([v^*]_ b=[m]_ b\). In particular, if \(f\) is harmonic in the interior of \(P_ c(\varphi)\) and continuous on \(P_ c(\varphi)\), then \(\lim_{n\to\infty, n\in A} \int f dv(p_ n)=\int f dm\). (b) Suppose that \[ \lim_{{n\to\infty} \atop {n\in A}}\| W^ n p_ n\|^{1/n}_ \varphi =\exp(-F)\tag{*} \] and also that the following interior condition holds: For any closed subset \(A\) of the interior of \(P_ c(\varphi)\), \((**)\;\lim_{n\to\infty, n\in A} v(p_ n)(A)=0\). Then, in the weak-star sense \(\lim_{n\to\infty, n\in A} v(p_ n)=m\). In particular \((*)\) and \((**)\) imply that \(\varphi=\partial_ \infty\varphi\). (c) Conversely, suppose that \(\partial_ \infty\varphi\) is regular with respect to the Dirichlet problem for \(D_ \infty(\varphi)\), \(W\) is continuous on \(\varphi\), and the zeros of \(\{p_ n\}_{n\in A}\) are uniformly bounded. If every weak-star limit measure \(v^*\) of \(\{v(p_ n)\}_{n\in A}\) has a balayage \(\hat v^*\) to \(\partial_ \infty\varphi\) satisfying \(\hat v^*=[m]_ b\), then (i) holds with equality. Theorem 3: Let \(W: E\to[0,\infty)\) be admissible and \(p_ n\in\pi_ n\), \(n\geq 1\) be a sequence of polynomials, not necessarily monic. Let \(A\) b a subsequence of integers and assume that the following conditions hold: (a) \(sup_{n\to\infty, n\in A} \| W^ n p_ n\|^{1/n}_{\partial_ \infty\varphi}\leq 1\). (b) There is a point \(z_ 0\in D_ \infty(\varphi)\) such that \[ \liminf_{{n\to\infty} \atop {n\in A}}\{{1\over n}\log| p_ n(z_ 0)|_ V(m,z_ 0)- F\}\geq 0. \] Let \(v^*\) be any weak-star limit measure of \(\{v(p_ n)\}_{n\in A}\). Then \(v^*\in M(P_ c(\varphi))\) and, for balayage to \(\partial_ \infty\varphi\), \([v^*]_ b=[m]_ b\). In Theorem 1, the authors summarize some known properties of balayage of measures established by \textit{N. S. Landkof} [Foundations of modern potential theory (1972; Zbl 0253.31001)]. In Theorem 2, they compare \([m(W,E)]_ b\) with \([{v^*} ]_ b\), where \(v^*\) is any weak-star limit measure of \(\{v(p_ n)\}\). The existence of \([v^*]_ b\) is a part of the theorem. In fact the authors extend a result of \textit{H. P. Blatt}, \textit{E. B. Saff}, and \textit{M. Simkani} [J. Lond. Math. Soc., II. Ser. 38, No. 2, 307-316 (1988; Zbl 0684.41006)]. Theorem 3 is an extension of results proved by \textit{R. Grothman} [On zeros of sequences of polynomials, ibid. (to appear)]. In this paper the authors give useful lemmas and applications. These results give new directions on this line.