The quadratic moment problem for the unit circle and unit disk

For the quadratic complex moment problem, we obtain necessary and sufficient conditions for the existence of representing measures supported in the unit circleT or in the closed unit disk. We explicitly construct all finitely atomic representing measures supported inT or which have the fewest atoms possible. For the quadratic-moment problem in which the moment matrixM(1) is positive and invertible, there exists an ellipse D such that the minimal (3-atomic) representing measures are supported in the complement of the interior region of . Finally, we apply these results to obtain information on the location of the zeros of certain cubic polynomials.Dedicated to the memory of Velaho D. Bowman-FialkowBoth authors were partially supported by research grants from the National Science Foundation. The second-named author was also partially supported by an award from the State of New York/UUP Professional Development and Quality of Working Life Committee.