Zeros of orthogonal polynomials on harmonic algebraic curves

Abstract

Starting with an infinite, complex, positive-definite, hermitian matrix, we build a triangular +1 matrix $\hat D_n$ generalising the tridiagonal one of the Hankel case. We apply this construction to moments matrices arising from distributions on curves Im(A(z))=0, A(z) θ C[z]. We study the matrix $A\left( {\hat D_n } \right)$ and its relationship with the truncated matrix of the correspondinding symmetric operator and we give conditions for the zeros to be on the curve.