This project has two components. The first is a study of the operator theory and function theory of the symmetric Fock space. The principal investigator and his collaborators have recently developed a geometric characterization of the Carleson measures for that space. They believe that this result, together with the discretization techniques developed in obtaining it, will provide effective tools for the major immediate goal, which is to describe the interpolating sequences for this Fock space. The second component is the study of the relation between the Rankin-Cohen bracket operation and the direct sum decomposition of tensor products of Hardy and Bergman spaces. Both the bracket and the decomposition are implemented using the same bilinear differential operator, the transvectant. Hence it is natural to speculate that the algebraic structure induced on the graded space of automorphic forms by the bracket (an associative, noncommutative product structure) is mirrored by a new algebraic structure in the tensor product decomposition. The principal investigator proposes to identify that structure and develop its algebraic and analytic properties.
The first component of this project, studying the symmetric Fock space, is part of a mathematical program that has evolved in function theory and harmonic analysis for thirty years -- the use of discretization techniques to study continuous phenomena. Beyond its contribution to theoretical mathematics, this program has led to major advances in signal processing techniques and other areas of data analysis. Wavelet analysis is the best known product of this program, but there are many others. The work on the Fock space is squarely in that tradition. It will be an exploration of how a particular discretization technique that is known to be very effective in theoretical mathematics and in numerical applications can be adapted to a much more sophisticated geometric setting. The second component of the project, the study of the relation between the Rankin-Cohen bracket and the decomposition of certain tensor products, is a response to the observation that the same complicated computational constructs ("transvectants") show up in these two, so far, unrelated contexts. A basic principle in mathematics is that such "accidents" are almost always the first indication of unexpected, and sometimes deep, connections between what had been seen as unrelated areas. The applicant proposes to establish that that is the situation here and to develop the new insights suggested by those connections.
