Real zeros of a random polynomial with Legendre elements. (1997)

• Record Type: Journal Article
• Authors: Farahmand, K.
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 10:Number 3(1997)
• Page Start: 257
• Page End: 264
• Publication Date: 1997
• Subjects: orthogonal polynomial -- weight functions -- Kac-Rice formula -- probability space -- Jepson's theorem -- Darboux-Christoffel formula -- recurrence formula; Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953397000324
• Languages: English
• ISSNs: 1048-9533

Abstract

Abstract : Let T 0 ∗ ( x ), T 1 ∗ ( x ), …, T n ∗ ( 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 polynomial g 0 T 0 ∗ ( x ) + g 1 T 1 ∗ ( x ) + … + g n T n ∗ ( x ) where g j, j = 1, 2, …, n are independent identically 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.