ABSTRACT
Orthogonal Laurent polynomials (L-polynomials) are introduced and studied by Jones and Thron [1] in connection with the study of the strong moment problems, which is a generalization of the classical moment problems. See also [2, 3]. Since orthogonal L-polynomials and the ordinary orthogonal polynomials share a lot of similar properties, many results on orthogonal polynomials have been adapted to the orthogonal L-polynomial setting. For example, the recurrence relations, Favard’s theorem, Christoffel-Darboux identities, the separation of zeros property, and Gaussian quadrature have all been established for orthogonal L-polynomials. For survey articles on orthogonal L-polynomials, see [4, 5, 6]. One of the items missing in the above list is a separation theorem of Chebyshev-Markov-Stieltjes type that relates the Christoffel numbers in the Gaussian quadrature, a solution to the strong moment problem, and the separation property of the zeros of orthogonal L-polynomials, although the validity of such separation theorem has been conjectured by many researchers. In this paper, we will establish such a separation theorem. The key of the proof is an existence theorem for one-sided interpolation L-polynomials (see Theorem 4 below), which is of interest in its own.
