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

2015
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Proceedings of the American Mathematical Society
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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 )) 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/ √ 3 + o(n) expected real zeros. We prove that the latter asymptotic relation holds

doi:10.1090/proc/12836
fatcat:b6vs3gorknfj5exputj3vb7qem