Zeros of real random polynomials spanned by OPUC

Abstract

Let {ϕ i } ∞ i=0 be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure µ. We study zero distribution of random linear combinations of the formwhere η 0 , . . . , η n−1 are i.i.d. standard Gaussian variables. We use the Christoffel-Darboux formula to simplify the density functions provided by Vanderbei for the expected number real and complex of zeros of Pn. From these expressions, under the assumption that µ is in the Nevai class, we deduce the limiting value of these density functions away from the unit circle. Under the mere assumption that µ is doubling on subarcs of T centered at 1 and −1, we show that the expected number of real zeros of Pn is at most (2/π) log n + O(1), and that the asymptotic equality holds when the corresponding recurrence coefficients decay no slower than n −(3+ǫ)/2 , ǫ > 0. We conclude with providing results that estimate the expected number of complex zeros of Pn in shrinking neighborhoods of compact subsets of T.

An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

“…When α = 0, one can clearly see from (8) that Φ m (z; 0) = z m and therefore Kac-Geronimus polynomials (10) specialize to Kac polynomials (1). For random polynomials (6) with f m (z) = ϕ m (z) given by (8)-(9) it can be easily shown using the Christoffel-Darboux formula, see [27,Theorem 1.1], that (7) can be rewritten as…”

“…Since the functions f m (z) in ( 6) must be real-valued on the real line, we are only interested in real recurrence coefficients, i.e., α m ∈ (−1, 1) for all m ≥ 0. It is known [27] that when m p |α m | is a bounded sequence for some p > 3/2, estimate (3) remains valid for random polynomials (6) with f m (z) = ϕ m (z) given by ( 8)- (9). Moreover, if the recurrence coefficients decay exponentially, it was shown by the authors in [1] that the expected number of real zeros has a full asymptotic expansion of the form (4) with the constant term still given by (5).…”

“…For random polynomials (6) with f m (z) = ϕ m (z) given by ( 8)-( 9) it can be easily shown using the Christoffel-Darboux formula, see [27,Theorem 1.1], that (7) can be rewritten as (12)…”

“…as t → 0, |G p (t)| is bounded above by a polynomial of degree p + 1 independent of n, N, while | G N (t)| is bounded above on I n by a polynomial of degree N + 1 whose coefficients depend on N but not on n. We now get from (20), (37), and (39), that (34) and (35) do hold for N ≥ 3 and functions H p (t) and H N (t) that can be computed via (27)-(28) and whose moduli satisfy the described bounds. The vanishing of H p (t) as t → 0 can be verified by using (27), (38), (40), and (41). To see that H N (t) = O(t 2 ), observe that…”

Real zeros of random sums with i.i.d. coefficients

“…Asymptotics for the density function ρ n (x) in the case when the random variables {η k } are i.i.d. standard Gaussian has been well studied when the spanning functions are trigonometric functions [36], polynomials orthogonal on the real line ( [6], [7], [2], [28], [29], [31], [39]), and polynomials orthogonal on the unit circle ( [39], [1], [38]). As an application we consider the case…”