Expected number of real zeros for random linear combinations of orthogonal polynomials
Expected number of real zeros for random linear combinations of orthogonal polynomials
复制标题
正交多项式的随机线性组合的预期实零数
DOI：
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发表时间：
2015
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通讯作者：
Xiaoju Xie
中科院分区：
文献类型：
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作者：
D. Lubinsky;I. Pritsker;Xiaoju Xie
研究了正交多项式随机线性组合的真实的零点期望个数。众所周知，Kac多项式，由具有i.i.d.高斯系数，只有$（2/pi + o（1））log{n}$预期的真实的零的程度$n$。另一方面，如果基由勒让德（或更一般地由雅可比）多项式给出，则随机线性组合具有$n/sqrt{3} + o（n）$期望的真实的零点。我们证明了后一个渐近关系对于真实的直线上的一大类随机正交多项式普遍成立，并给出了关于真实的零点期望个数的更一般的局部结果。
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 $(2/pi + o(1))log{n}$ expected real zeros in terms of the degree $n$. On the other hand, if the basis is given by Legendre (or more generally by Jacobi) polynomials, then random linear combinations have $n/sqrt{3} + o(n)$ expected real zeros. We prove that the latter asymptotic relation holds universally for a large class of random orthogonal polynomials on the real line, and also give more general local results on the expected number of real zeros.