Asymptotic properties of zeros of orthogonal rational functions

A sequence {x(m) : m = 0,1,2,…} of real numbers gives rise to an absolutely continuous measure ψN(T) on the unit circle given by dψ(T)N(θ)dθ = 12φ|∑N−1m=0x(m)Tme−>imθ|2, −φ⩽θ⩽φ where N is a natural number, T ∈ (0, 1). Each measure ψN(T) determines a sequence {φn(N,T)} of monic orthogonal rational functions with prescribed poles outside the closed unit disk. (In particular these rational functions may be polynomials.) A measure ψ0 with support consisting of the points ξ1,…,ξn0 on the unit circle is given. Let n>n0. Under a suitable condition on weak∗ convergence of the measures and with a proper ordering of the zeros zk(n,N,T) of φn(N,T), it can be shown that limT→1 [limT→∞ zk(n,N,T)] = ξk, k = 1,·,n0 and that there exists a positive number ϱn < 1 such that ¦zk(n,N,T)¦ ⩽ ϱn for k = n0 + 1,…,n, for all N and all T ∈ (0, 1). The theory is applied to the problem of determining unknown frequencies in a trigonometric signal x(m) =∑Ij= −IAjeimwj, m = 0, 1, 2