Approximation by random complex polynomials and random rational functions

Random Polynomials, Random Matrices and L-functions

We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function.

On Classifications of Random Polynomials

 Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...

Random Polynomials, Random Matrices, and L-functions, Ii

We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function.

Nonlinear Approximation of Random Functions

Given an orthonormal basis and a certain class X of vectors in a Hilbert space H, consider the following nonlinear approximation process: approach a vector x ∈ X by keeping only its N largest coordinates, and let N go to infinity. In this paper, we study the accuracy of this process in the case where H = L(I), and we use either the trigonometric system or a wavelet basis to expand this space. T...

Gaussian random polynomials and analytic functions