Orthogonal Laurent polynomials and the strong Hamburger moment problem

Abstract This paper is concerned with the strong Hamburger moment problem (SHMP): For a given double sequence of real numbers C = { c n } ∞ −∞ , does there exist a real-valued, bounded, non-decreasing function ψ on (−∞, ∞) with infinitely many points of increase such that for every integer n , c n = ∝ ∞ −∞ (− t ) n dψ ( t )? Necessary and sufficient conditions for the existence of such a function ψ are given in terms of the positivity of certain Hankel determinants associated with C. Our approach is made through the study of orthogonal (and quasi-orthogonal) Laurent polynomials (referred to here as L-polynomials) and closely related Gaussian-type quadrature formulas. In the proof of sufficiency an inner product for L-polynomials is defined in terms of the given double sequence C . Since orthogonal L-polynomials are believed to be of interest in themselves, some examples of specific systems are considered.