9705851 Saccone The investigator will study the linear topological structure of a class of Banach spaces of continuous analytic functions and bounded analytic functions. The primary tool is a relatively easily understood operator which resembles a Hankel operator on a Hilbert space. It has been recently demonstrated by the investigator and by other authors that these Hankel-type operators often possess a wealth of information on the Banach space structure of the function spaces they operate on. It seems likely that these methods could yield solutions of some open problems concerning bounded analytic functions on the unit ball in several complex variables and extend the solutions to strictly pseudoconvex domains. Spaces of uniformly convergent Fourier series on the unit circle and polydisk will also be studied. The goal of this project is to further understand the relatively new techniques being used to solve difficult problems in a variety of areas of the branch of pure mathematics known as analysis. These new techniques involve the replacement of complex computational tools with methods that are more abstract and theoretical in nature and are currently becoming more and more popular due to the powerful results that can be achieved with relatively little effort. The type of results we seek to obtain are less of an exact, numerical nature, but deal more those problems that in all likelihood could not be solved by computation. At the center of the research is a tool, called a Hankel-type operator, developed by the investigator in his thesis which has found an increasing number of applications since his doctoral work. One target of these ideas would be problems in harmonic analysis, or Fourier series. Fourier series are the building blocks for the theory of wavelets, a fast growing area having applications to numerous types problems such as data compression, voice recognition, and so on. Another important component of the project will be problems of analytic func tions in one and several complex variables. Otherwise known as complex analysis, this area of mathematics is intimately connected to harmonic analysis, mentioned above. The key feature of the results being sought after in this project is the generality in which they work and also, once the tedious theoretical foundations have been laid, the ease in which they can be applied.
