The expected value of the number of real zeros of a random sum of Legendre polynomials

Abstract:

It is known that the expected number of zeros in the interval $(-1,1)$ of the sum $a_0\psi _0(t)+a_1\psi _1(t)+\dotsb +a_n\psi _n(t)$, in which $\psi _k(t)$ is the normalized Legendre polynomial of degree $k$ and the coefficients $a_k$ are independent normally distributed random variables with mean 0 and variance 1, is asymptotic to $3^{-1/2}n$ for large $n$. We improve this result and show that this expected number is $3^{-1/2}n+o(n^\delta )$ for any positive $\delta$.