In the theory of orthogonal polynomials the so called “universality law” has been studied from different viewpoints and with measures satisfying less and less stringent conditions. If \(\{p_k(x)\}\) are the orthonormal polynomials with respect to a finite positive Borel measure \(\mu\) with compact support on the real line, the simplest case of this law is given in terms of the reproducing kernel \[ K_n(x,y)=\sum_{k=0}^n\,p_k(x)p_k(y),\;\tilde{K}_n(x,y)=w(x)^{1/2}w(y)^{1/2}K_n(x,y), \] and reads \[ \lim_{n\rightarrow\infty}\,{\tilde{K}_n\left(\xi+{a\over\tilde{K}_n(\xi,\xi)}, \xi+{b\over\tilde{K}_n(\xi,\xi)}\right)\over\tilde{K}_n(\xi,\xi)}={\sin{\pi (a-b)}\over\pi (a-b)}. \eqno{(\ast)} \] The measure \(\mu\) has infinitely many points in its support and \[ w={\text{d}\mu\over\text{d}x} \] is the Radon-Nikodym derivative of \(\mu\) and \((\ast)\) holds uniformly in \(\xi\) on a compact subinterval of supp\((\mu)\) and in \(a,b\) in compact subsets of the real line (for \(a=b\): the right hand side is \(1\)).
The paper starts (section 1) with a short discussion of previous results on universality and the main results given show that universality is equivalent to “universality along diagonals”:
Theorem 1.1. Let \(\mu\) be a finite positive measure on the real line with compact support. Let \(J\subset\text{supp}(\mu)\) be compact and such that \(\mu\) is absolutely continuous at each point of \(J\). The following are equivalent: A. uniformly for \(\xi\in J\) and \(a\) in compact subsets of the real line \[ \lim_{n\rightarrow\infty}\,{K_n(\xi+{a\over n},\xi+{a\over n})\over K_n(\xi,\xi)}=1 \] B. uniformly for \(\xi\in J\) and \(a,b\) in compact subsets of the complex plane \[ \lim_{n\rightarrow\infty}\,{\tilde{K}_n\left(\xi+{a\over\tilde{K}_n(\xi,\xi)}, \xi+{b\over\tilde{K}_n(\xi,\xi)}\right)\over\tilde{K}_n(\xi,\xi)}={\sin{\pi (a-b)}\over\pi (a-b)}. \]
Theorem 1.2. Let \(\mu\) be a finite positive measure on the real line with compact support. Let \(J\subset\text{supp}(\mu)\) be compact and such that \(\mu\) is absolutely continuous in an open set containing \(J\). Assume that \(w\) is bounded above and below by positive constants in that open set. Assume, moreover, that uniformly for \(\xi\in J\) \[ \lim_{s\downarrow 0}\,\int_{\xi-s}^{\xi+s}\,|w(t)-w(\xi)|dt=0. \] Then the equivalence of A and B in Theorem 1.1 remains valid.
In section 2 the main ideas behind the proof are given and after that some notation and background (section 3), some facts about entire functions of exponential type (section 4) and some lemmas on growth of polynomials (section 5). Finally, in section 6, the proofs follow.