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Expected number of real zeros for random linear combinations of orthogonal polynomials. (English) Zbl 1337.30008
The main object is the expected number of real zeros of a large class of random polynomials. Let \(\mu\geq 0\) be a Borel measure compactly supported in \(\mathbb R\), with non-finite support and with finite moments of all orders. Let \(\{p_j\}_{j=0}^n\) denote an orthonormal system for \(\mu\), where the \(p_j\) are polynomial of degree \(j\) with positive leading coefficient. Consider then the random linear combination \[ P_n(x)=\sum_{j=0}^n c_j\, p_j(x), \] where the \(c_j\) are i.i.d. Gaussians with mean \(0\) and variance \(\sigma^2\).
Given \(E\subset\mathbb R\), let \(N_n(E)\) denote the number of real zeros of \(P_n\) in \(E\). A classical result of M. Kac [Bull. Am. Math. Soc. 49, 314–320 (1943; Zbl 0060.28602)] shows that, in the monomial case \(p_j(x)=x^j\), \[ \mathbb E[N_n(\mathbb R)]=\Big(\frac 2\pi +o(1)\Big)\log n, \quad n\to\infty. \] On the other hand, M. Das [Proc. Am. Math. Soc. 27, 147–153 (1971; Zbl 0212.49401)] showed that in the case when \(p_j\) are Legendre polynomials \[ \mathbb E[ N_n(-1,1)]=\frac n{\sqrt{3}}+o(n),\quad n\to\infty. \] The same asympotics were shown to hold in the case when \(p_j\) are Jacobi polynomials [M. Das and S. S. Bhatt, Indian J. Pure Appl. Math. 13, 411–420 (1982; Zbl 0481.60067)]. Several similar results have been obtained for other families of orthogonal functions and for other probability distributions of the coefficients.
The main result of this paper extends the latter results to a much more general setting: for certain regular measures \(\mu\) whose support is a regular (in the sense of potential theory) compact set \(K\), and for intervals \([a,b]\) where \(\mu\) is well behaved, one has \[ \lim_{n\to\infty}\frac 1n \mathbb E[N_n([a,b])]=\frac 1{\sqrt{3}} \nu_K([a,b])\;, \] where \(\nu_K\) denotes the equilibrium measure of the compact \(K\). A canonical situation where this holds is when \(K\) is a finite union of intervals and \(d\mu(x)=w(x)\; dx\), with \(w>0\) a.e. on \(K\).
The starting point in the proof of this result is a generalisation of the exact formula obtained by Kac in the monomial case (Proposition 1.1), in which \(\mathbb E[N_n([a,b])]\) is expressed as an integral over \([a,b]\) of an explicit function depending on the reproducing kernel \[ K_n(x,y)=\sum_{j=0}^n p_j(x) p_j(y) \] and some of its derivatives. With this the authors notice (Lemma 3.2) that \[ \frac 1n \mathbb E[N_n([a,b])]=\frac {1+o(1)}{\sqrt 3} \int_a^b \frac 1n K_{n+1}(x,x)\; d\mu(x)\;. \] Universality properties for the reproducing kernel proved previously by two of the authors and by V. Totik [J. Anal. Math. 81, 283–303 (2000; Zbl 0966.42017)] yield \[ \lim_{n\to\infty} \frac 1n K_{n+1}(x,x)=\frac{d\nu_K}{d\mu}(x), \] hence the result.

MSC:
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30B20 Random power series in one complex variable
60B10 Convergence of probability measures
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