Real zeros of a random polynomial with Legendre elements

[摘要] LetT0∗(x),T1∗(x),…,Tn∗(x)be a sequence of normalized Legendre polynomials orthogonal with respect to the interval(−1,1). The asymptotic estimate of the expected number of real zeros of the random polynomialg0T0∗(x)+g1T1∗(x)+…+gnTn∗(x)wheregj,j=1,2,…,nare independentidentically and normally distributed random variables with mean zero and variance one is known. The present paper considers the case when the means and variances of the coefficients are not all necessarily equal. It is shown that in general this expected number of real zeros is only dependent on variances and is independent of the means.

[学科分类] 应用数学