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Uniform spacing of zeros of orthogonal polynomials for locally doubling measures. (English) Zbl 1263.42005
The authors consider measures $\mu$ with compact support on the real line, that have the doubling property on an interval $[a,b]$: For some constant $L$, we have $\mu(2I)\leq L\,\mu(I)$ for all intervals $2I\subseteq [a,b]$, where $2I$ is an interval twice the length of $I$ and with midpoint at the midpoint of $I$. It is tacitly assumed that $\mu$ is not identically zero on $[a,b]$, whence the doubling property implies that $[a,b]$ must be a part of the support of $\mu$. In this paper, the authors prove the following theorem. Theorem 1.1. Let $\mu$ be a measure with compact support on the real line and with the doubling property on $[a,b]$. Then for every $\delta>0$ there exists a constant $A$, independent of $m$, such that $${1\over Am}\leq x_{m,j+1}-x_{m,j}\leq{A\over m},\ j=k,k+1,\dotsc,l-1,\eqno{(1)}$$ where $x_{m,k}<x_{m,k+1}<\dotsb<x_{m,l}$ are the zeros of the orthogonal polynomial $p_m$ (with respect to $\mu$, of degree $m$) located in the interval $[a+\delta,b-\delta]$. This generalizes a result by {\it G. Mastroianni} and {\it V. Totik} [Constructive Approximation 16, No. 1, 37--71 (2000; Zbl 0956.42001)], where the support of $\mu$ is $[-1,1]$ and $\mu$ has the doubling property on this interval, and some results by {\it Y. Last} and {\it B. Simon} [Commun. Pure Appl. Math. 61, No. 4, 486--538 (2008; Zbl 1214.42044)]. Recalling the definition of the $m$-th Christoffel function associated with $\mu$ $$\lambda_m(\xi):=\inf_{p(\xi)=1,\text{deg}\,p\leq m}\,\int\,p^2(x)d\mu(x),$$ and the Cotes numbers $$\lambda_{m,k}:=\lambda_m(x_{m,k}),$$ the authors prove the following theorem. Theorem 1.7. Let $\mu$ be a measure with compact support on the real line and with the doubling property on $[a,b]$. Then, for every $\delta>0$, there exists a constant $B$, independent of $m$, such that $${1\over B}\leq{\lambda_{m,k}\over\lambda_{m,k+1}}\leq B,\eqno{(2)}$$ whenever $x_{m,k}$ and $x_{m,k+1}$ belong to $[a+\delta,b-\delta]$. Together the Theorems 1.1 and 1.7 have the following converse. Theorem 1.8. Let $\mu$ be a measure with compact support. If (1) and (2) hold on every interval $[a+\delta,b-\delta]\subset \operatorname{supp} (\mu),\ \delta>0$ (with some $A$ and $B$ that may depend on $\delta$), then $\mu$ has the doubling property on every such interval.

42C05General theory of orthogonal functions and polynomials
40A05Convergence and divergence of series and sequences
40C05Matrix methods in summability
40D05General summability theorems
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