Expected number of real zeros for random linear combinations of orthogonal polynomials (Q2790270)

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 NEWLINE\[NEWLINE P_n(x)=\sum_{j=0}^n c_j\, p_j(x), NEWLINE\]NEWLINE where the \(c_j\) are i.i.d. Gaussians with mean \(0\) and variance \(\sigma^2\).NEWLINENEWLINEGiven \(E\subset\mathbb R\), let \(N_n(E)\) denote the number of real zeros of \(P_n\) in \(E\). A classical result of \textit{M. Kac} [Bull. Am. Math. Soc. 49, 314--320 (1943; Zbl 0060.28602)] shows that, in the monomial case \(p_j(x)=x^j\), NEWLINE\[NEWLINE \mathbb E[N_n(\mathbb R)]=\Big(\frac 2\pi +o(1)\Big)\log n, \quad n\to\infty. NEWLINE\]NEWLINE On the other hand, \textit{M. Das} [Proc. Am. Math. Soc. 27, 147--153 (1971; Zbl 0212.49401)] showed that in the case when \(p_j\) are Legendre polynomials NEWLINE\[NEWLINE \mathbb E[ N_n(-1,1)]=\frac n{\sqrt{3}}+o(n),\quad n\to\infty. NEWLINE\]NEWLINE The same asympotics were shown to hold in the case when \(p_j\) are Jacobi polynomials [\textit{M. Das} and \textit{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.NEWLINENEWLINEThe 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 NEWLINE\[NEWLINE \lim_{n\to\infty}\frac 1n \mathbb E[N_n([a,b])]=\frac 1{\sqrt{3}} \nu_K([a,b])\;, NEWLINE\]NEWLINE 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 \(K\).NEWLINENEWLINEThe 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 NEWLINE\[NEWLINE K_n(x,y)=\sum_{j=0}^n p_j(x) p_j(y) NEWLINE\]NEWLINE and some of its derivatives. With this the authors notice (Lemma 3.2) that NEWLINE\[NEWLINE \frac 1n \mathbb E[N_n([a,b])]=\frac {1+o(1)}{\sqrt 3} \int_a^b \frac 1n K_{n+1}(x,x)\; d\mu(x)\;. NEWLINE\]NEWLINE Universality properties for the reproducing kernel proved previously by two of the authors and by \textit{V. Totik} [J. Anal. Math. 81, 283--303 (2000; Zbl 0966.42017)] yield NEWLINE\[NEWLINE \lim_{n\to\infty} \frac 1n K_{n+1}(x,x)=\frac{d\nu_K}{d\mu}(x), NEWLINE\]NEWLINE hence the result.