Variance of real zeros of random orthogonal polynomials
نویسندگان
چکیده
We determine the asymptotics for variance of number zeros random linear combinations orthogonal polynomials degree ≤n in subintervals [a,b] support underlying orthogonality measure μ. show that, as n→∞, this is asymptotic to cn, some explicit constant c>0.
منابع مشابه
Zeros of orthogonal polynomials on the real line
Let pnðxÞ be the orthonormal polynomials associated to a measure dm of compact support in R: If EesuppðdmÞ; we show there is a d40 so that for all n; either pn or pnþ1 has no zeros in ðE d;E þ dÞ: If E is an isolated point of suppðmÞ; we show there is a d so that for all n; either pn or pnþ1 has at most one zero in ðE d;E þ dÞ:We provide an example where the zeros of pn are dense in a gap of su...
متن کاملExpected Number of Real Zeros for Random Linear Combinations of Orthogonal Polynomials
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/π + o(1)) logn 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...
متن کاملNumber Variance of Random Zeros
The main results of this article are asymptotic formulas for the variance of the number of zeros of a Gaussian random polynomial of degree N in an open set U ⊂ C as the degree N → ∞, and more generally for the zeros of random holomorphic sections of high powers of any positive line bundle over any Riemann surface. The formulas were conjectured in special cases by Forrester and Honner. In higher...
متن کاملMonotonicity of Zeros of Orthogonal Laurent Polynomials
Monotonicity of zeros of orthogonal Laurent polynomials associated with a strong distribution with respect to a parameter is discussed. A natural analog of a classical result of A. Markov is proved. Recent results of Ismail and Muldoon based on the Hellman-Feynman theorem are also extended to a monotonicity criterion for zeros of Laurent polynomials. Results concerning the behaviour of extreme ...
متن کاملZeros of Real Random Polynomials Spanned by Opuc
Let {φi}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 form
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 2021
ISSN: ['0022-247X', '1096-0813']
DOI: https://doi.org/10.1016/j.jmaa.2021.124954