论文标题
随机正交多项式：局部普遍性和预期的实际根数
Random orthonormal polynomials: local universality and expected number of real roots
论文作者
论文摘要
我们考虑随机正交多项式$$ f_ {n}（x）= \ sum_ {i = 0}^{n}ξ_}ξ_{i} p_ {i} $（2+ \ ep）$ omments和$（p_n）_ {n = 0}^{\ infty} $是针对真实行上的固定紧凑型措施的正顺序多项式的系统。 在许多类别的经典多项式系统满足的温和技术假设下，我们为全球和本地的$ f_n $平均根源数量的主要渐近学数量建立了普遍性。 在本文之前，这些结果仅对具有高斯系数的随机正交多项式\ cite {lubinsky2016linear}使用KAC-RICE公式，这种方法并未扩展到我们纸张的一般性。
We consider random orthonormal polynomials $$ F_{n}(x)=\sum_{i=0}^{n}ξ_{i}p_{i}(x), $$ where $ξ_{0}$, \dots, $ξ_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep)$ moments, and $(p_n)_{n=0}^{\infty}$ is the system of orthonormal polynomials with respect to a fixed compactly supported measure on the real line. Under mild technical assumptions satisfied by many classes of classical polynomial systems, we establish universality for the leading asymptotics of the average number of real roots of $F_n$, both globally and locally. Prior to this paper, these results were known only for random orthonormal polynomials with Gaussian coefficients \cite{lubinsky2016linear} using the Kac-Rice formula, a method that does not extend to the generality of our paper.
