Orthogonal Dirichlet Polynomials

Abstract

Let (formula presented) be a sequence of distinct positive numbers. Let w be a non-negative function, integrable on the real line. One can form orthogonal Dirichlet polynomials {ϕn} from linear combinations of(formula presented), satisfying the orthogonality relation (formula presented)∫−∞∞ϕn(t)ϕm(t)¯w(t)dt=δmn. (formula presented) Weights that have been considered include the arctan density (formula presented); rational function choices of w; w(t) = e−t ; and w(t) constant on an interval symmetric about 0. We survey these results and discuss possible future directions.