Expected Number of Level Crossings Of Legendre Polynomials

Abstract

The aim of this paper is to estimate the number of real zeros of a orthogonal random polynomial under different condition when the coefficients belong to the domain of attraction of orthogonal properties. Let be random polynomial such that [y₀ (w), y₁ (w),yn(w)] is a sequence of mutually independent, normally distributed random variables with mean zero and variance unity and [Ψ₀ (t),……. Ψ_n (t)] be a sequence of normalized orthogonal Legendre polynomials, defined by    where P_n(t) is the classical Legendre polynomial. Then, for any constant K such that (K²/n)→0 as n→∞, the mathematical expectation of number of real zeros of the equation =k is asymptotic to .