Expected number of real zeros for random linear combinations of orthogonal polynomials

Expected number of real zeros for random orthogonal polynomials

“…They also stated estimates for the expected number of real zeros of random Hermite and Laguerre polynomials, but those arguments contain significant gaps. The authors recently showed [29] that if the basis is given by the orthonormal polynomials associated to a finite Borel measure with compact support on the real line, then random linear combinations have n/ √ 3 + o(n) expected real zeros under mild conditions on the weight. The second and the third authors [33] extended this asymptotic to random orthogonal polynomials associated with the Freud weights W (x) = e −c|x| λ , c > 0, λ > 1.…”

“…. , n. For a sketch of the proof of Proposition 1.1, see [29]. We note that multiple zeros are counted only once by the standard convention in all of the above results on real zeros.…”

Expected number of real zeros for random Freud orthogonal polynomials

“…In particular, they cover the case of random Hermite polynomials. Lubinsky and the authors also recently showed [27] that if the basis is given by orthonormal polynomials associated with a finite Borel measure compactly supported on the real line, then random linear combinations have n/ √ 3 + o(n) expected real zeros under some mild conditions on the weight. Interesting computations and pictures of zero distributions of random orthogonal polynomials may be found on the chebfun web page of Trefethen [33].…”

“…. , n. For a sketch of the proof of Proposition 1.1, see [27]. We note that multiple zeros are counted only once by the standard convention in all of the above results on real zeros.…”

Expected Number of Real Roots for Random Linear Combinations of Orthogonal Polynomials Associated with Radial Weights

“…where P n j are entire functions that take real values on the real line (see also [Van15,LPX15] for recent treatment of this problem). In particular, Edelman and Kostlan [EK95, §3] proved that…”