Expected Number of Real Roots for Random Linear Combinations of Orthogonal Polynomials Associated with Radial Weights

Expected Number of Real Zeros for Random Linear Combinations of Orthogonal Polynomials

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)) logn 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...

On the Number of Real Roots of Random Polynomials

Roots of random polynomials have been studied exclusively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdős-Offord, showed that the expectation of the number of real roots is 2 π logn + o(logn). In this paper, we determine the true nature of the error term by showing that the expectation equals 2 π...

On Linear Combinations of Orthogonal Polynomials

In this expository paper, linear combinations of orthogonal polynomials are considered. Properties like orthogonality and interlacing of zeros are presented.

Exact Bounds for Orthogonal Polynomials Associated with Exponential Weights*

Orthogonal polynomials with discontinuous weights

Abstract. In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, An(z) and Bn(z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w0 ...