ABSTRACT Proposal: DMS-9600085 PI: Duren In the area of Bergman spaces, Duren will continue his recent work by investigating properties of canonical divisors and their analogues in more general invariant subspaces. Specific questions will include structural properties of extremal functions and the determination of generators for invariant subspaces. Also to be studied are sampling sequences and interpolation sequences in Bergman spaces. For planar harmonic mappings, the project will investigate the connection between dilatation and boundary correspondence, will seek to develop improved forms of both the existence and uniqueness assertions in the harmonic version of the Riemann mapping theorem, and will continue a study of harmonic mappings of multiply connected domains. Duren also plans to investigate a curvature problem for minimal surfaces and to make further studies of the connections between minimal surfaces and their underlying harmonic mappings. Finally, a basic coefficient conjecture for harmonic mappings will be addressed. Bergman spaces often arise in connection with complex analysis, approximation theory, and operator theory, so it is important to understand their structure. Sampling sequences make contact with the modern theory of wavelets, which has broad applications to image processing, signal analysis, and data storage. Conformal mappings have long been used to change coordinates in solving problems of fluid flow, heat conduction, and the like. Harmonic mappings are significant generalizations of conformal mappings which share some of their properties but present many new phenomena. They are particularly useful in the study of minimal surfaces (area-minimizing surfaces), represented physically by soap films spanning loops of wire.
