Polynomials orthogonal with respect to oscillatory weights

The classical theory of Gaussian quadrature assumes a positive weight function. This implies many desirable properties of the rule: Guaranteed existence and uniqueness of the orthogonal polynomials whose zeros are the nodes of the rule, nodes that are contained in the interval of integration, as well as positive quadrature weights, which implies that the rule is stable. There has been little research on polynomials that are orthogonal with respect to non-positive weight functions, although these could be interesting for, for example, oscillatory quadrature problems. In this talk I will present some of the few results we have on this, as well as some weird and wonderful conjectures.