A quantitative maximum entropy theorem for the real line

A bound B is constructed for a large and natural class of entropy integrals on the real line. The construction involves Burg's maximum entropy theorem for the circle group and techniques from harmonic analysis. As the solution of an extension problem, the construction of B leads to a further construction problem which can be viewed as a means of refining Krein's theorem on positive definite extensions.Supported in part by the NSF under Grant DMS 86-01311.