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


Authors: D. S. Lubinsky, I. E. Pritsker and X. Xie
Journal: Proc. Amer. Math. Soc. 144 (2016), 1631-1642
MSC (2010): Primary 30C15; Secondary 30B20, 60B10
DOI: https://doi.org/10.1090/proc/12836
Published electronically: September 9, 2015
MathSciNet review: 3451239
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Abstract: 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/\pi + o(1))\log {n}$ 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 have $ n/\sqrt {3} + o(n)$ expected real zeros. We prove that the latter asymptotic relation holds universally for a large class of random orthogonal polynomials on the real line, and also give more general local results on the expected number of real zeros.


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Additional Information

D. S. Lubinsky
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: lubinsky@math.gatech.edu

I. E. Pritsker
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: igor@math.okstate.edu

X. Xie
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: sophia.xie@okstate.edu

DOI: https://doi.org/10.1090/proc/12836
Received by editor(s): December 9, 2014
Received by editor(s) in revised form: April 27, 2015
Published electronically: September 9, 2015
Additional Notes: The research of the first author was partially supported by NSF grant DMS136208 and US-Israel BSF grant 2008399.
The research of the second author was partially supported by the National Security Agency (grant H98230-12-1-0227) and by the AT&T Foundation.
Dedicated: In memory of Al Goodman, a great complex analyst and wonderful mentor, on the centenary of his birth.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2015 American Mathematical Society

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