随机正交多项式实零点的期望个数
Expected number of real zeros for random orthogonal polynomials
摘要：
研究了正交多项式随机线性组合的实零点期望个数。众所周知，Kac多项式，由具有i.i.d.高斯系数，在次数$n$方面仅具有$(2/π+ o(1))\log{n}$期望的实零。如果基由与实直线上的紧支撑的Borel测度相关联的正交多项式给出，或者与定义在整个实直线上的Freud权重相关联，则随机线性组合具有$n/\sqrt{3} + o(n)$期望的实零。我们证明了相同的渐近关系适用于所有随机正交多项式在实直线上与一个大类的重量，并给出局部结果的期望实零点的数目。我们还表明，这些随机多项式的适当规模的零的计数措施收敛弱的乌尔曼分布或反正弦分布。
英文摘要：
We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $(2/π+ o(1))\log{n}$ expected real zeros in terms of the degree $n$. If the basis is given by the orthonormal polynomials associated with a compactly supported Borel measure on the real line, or associated with a Freud weight defined on the whole real line, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real zeros. We prove that the same asymptotic relation holds for all random orthogonal polynomials on the real line associated with a large class of weights, and give local results on the expected number of real zeros. We also show that the counting measures of properly scaled zeros of these random polynomials converge weakly to either the Ullman distribution or the arcsine distribution.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
